TETRAHEDRA AND THEIR NETS
MATHEMATICAL AND PEDAGOGICAL IMPLICATIONS
Derege Haileselassie Mussa
Submitted in partial fulfillment of the
Requirements for the degree of
Doctor of Philosophy
Under the Executive Committee
Of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2013
© 2013
Derege Haileselassie Mussa
All Rights Reserved
ABSTRACT
TETRAHEDRA AND THEIR NETS
MATHEMATICAL AND PEDAGOGICAL IMPLICATIONS
Derege Haileselassie Mussa
If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any
two of the lengths sum to a value strictly larger than the third length one can make a triangle.
Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can
build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of
three lengths selected from the six, one still may not be able to build a tetrahedron. At the other
extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different
(incongruent) tetrahedra with the six lengths.
Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over
thousands of years, there are surprisingly many simple questions about them that still have not
been answered. This thesis answers some new questions about tetrahedra, as well raising many
more new questions for researchers, teachers, and students. It also shows in an appendix how
tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and
combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M).
In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics
addressed are a new classification system for tetrahedra based on partitions of an integer n,
existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of
tetrahedra, and other combinatorial aspects of tetrahedra.
TABLE OF CONTENTS
List of Figures…………………………………………………………………………...……….iv
List of Tables ………………………………………………………………………………….....xi
Acknowledgements…………………………………………………………………………...…xii
Dedication……………………………………………………………………………………….xiv
CHAPTER I …………………………………………………………………………..…………1
Introduction………………………………………………………………………………………..1
Purpose of the study………………………………………………………………………………2
Procedure of the study…………………………………………………………………………….3
CHAPTER II ……………………………………………………………………………..……...5
New mathematical insights into tetrahedra…………………….………………………………….5
Polygons …………………………………………………………………………………………..5
Polyhedra………………………………………………………………………………………….7
Unfoldings of polyhedra
……………………………………………………………………11
Tetrahedron .….…………………………………………………………………………………..13
Nets of tetrahedron ……………………………………………………..………………..……...18
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CHAPTER III …………………………………………………………………………………..32
Tetrahedron………………………………………………………………………………………32
Standard form of labeling a tetrahedron…………………………………………………………39
Partition type of tetrahedron……………………………………………………………………..40
Partition of Tetrahedron……………………………………………………………………….…62
Degenerate tetrahedron………………………………………………………………….………126
CHAPTER IV…………………………………………………………….. ………… …….....139
Nets of tetrahedra……………………………………………………………………….…….…139
CHAPTER V ……………………………………………………………………………….….164
Summary, conclusion and Recommendation………………………………………………...….164
REFERENCES……………………………………………………............................................169
APPENDICES ………………………..…………………………………………………….......172
New ideas to augment traditional topics and pedagogy
Appendix A ………………………………………………………………...……………..…….172
Geometry course
Appendix B……………………………………………………………..…………………..…..176
Unit I: Back ground information
Points, Lines, and Polygons
Appendix C……………………………………………………………..…………………..…..212
Unit II: Classification of degenerate tetrahedra in the Euclidean plane
ii
Appendix D ……………………………………………………………………………….…….217
Unit III: Nets of tetrahedron
Appendix E ……………………………………………………………………………….…….218
Unit IV: Classification of tetrahedron in 3D
iii
LIST OF FIGURES
Figure 2.1: Different Polygons…………………………………….…………………………….6
Figure 2.2: Convex 4-gon………………………………………………………………………..8
Figure 2.3: Non convex 7-gon…………………………………….. …………………………....9
Figure 2.4a: spanning cut tree and their nets ……………………………………………………12
Figure 2.4b: spanning cut tree and their nets……………………………………………………13
Figure 2.5a: Labeling of Tetrahedron …………………………………………………………..14
Figure 2.5b: Labeling of Tetrahedron …………………………………………………………..15
Figure 2.6: Star net…………………………………………………….. ……. . ……………. …17
Figure 2.7: Edges of a spanning tree……………………………………………………………..19
Figure 2.8: Star net and Path net ………………………………………………………………...20
Figure 2.9: Overlapping tetrahedron…………………………………………………………….21
Figure 2.10: Grünbaum overlapping tetrahedron………………………………………………. 22
Figure 2.11 :{ 2, 1, 1, 1, 1} Overlapping tetrahedron……………………………………………23
Figure 2.12: Incongruent tetrahedra ……………………………………. ………………………24
Figure 2.13: {3, 3}Tetrahedra ………………….……………………………………………......25
Figure 2.14 :{2, 2, 1, 1}Tetrahedra ……………………………………………………………..26
iv
Figure 2.15: Dual tetrahedra ……………………………………………………………………29
Figure 2.16: Degenerate tetrahedra …………………………………………………………….30
Figure 2.17: Degenerate tetrahedron ……………………………………………………………31
Figure 3.1: Potential tetrahedron………………………………………………………………..33
Figure 3.2: Tetrahedron …………………………………………………………………………34
Figure 3.3: Tetrahedron in terms of faces ………………………………………………….…..35
Figure 3.4: Tetrahedron in terms of vertices ……………………………………………………36
Figure 3.5: Tetrahedron with negative determinant …………………………………………….37
Figure 3.6: Tetrahedron with non facial ………………………………………………………...37
Figure 3.7a: Dual tetrahedron…………………………………………………………..……….38
Figure 3.7b: Dual tetrhedron……………………………………………………………………..39
Figure 3.8: labeling of tetrahedra………………………………………………………………..40
Figure 3.9: Construction of congruent tetrahedra …………………………………………........50
Figure 3.10: Congruent tetrahedra ……………………………………………………………....54
Figure 3.11: Dual tetrhedra ……………………………………………………………………...60
Figure 3.12: {2, 1, 1, 1, 1} tetrhedron with equal product matching edge………………………61
v
Figure 3.13a :{ 1, 1, 1, 1, 1, 1} Tetrahedron with equal product matching edge……………......61
Figure 3.13b :{ 1, 1, 1, 1, 1, 1} Tetrahedron with equal product matching edge… ……………62
Figure 3.14: 11 Partition classes of tetrhedra ………………………….. ….…………………..68
Figure 3.15: 25 Types of tetrahedra by partition type …………………………………………….82
Figure 3.16a: {3, 2, 1} Tetrahedron regard to the numbers of congruent triangles……………..83
Figure 3.16b: {3, 2, 1} Tetrahedron regard to the numbers of congruent triangles……….. …..84
Figure 3.17: Congruent face partition with edge partition behavior……………………………88
Figure 3.18a: Congruent face partition with edge partition behavior…………………………..90
Figure 3.18b: Congruent face partition with edge partition behavior………. ………………....91
Figure 3.18c: Congruent face partition with edge partition behavior……….. ………………...92
Figure 3.18d: Congruent face partition with edge partition behavior………. ………………....93
Figure 3.19: Partitions of a tetrahedron with integer lengths …………………………………106
Figure 3.20: Incongruent Scalene tetrahedron with integer length ……………………………114
Figure 3.21: {2, 1, 1, 1, 1} Incongruent tetrahedra with integer length………………………..118
Figure 3.22 a : monotone up and monotone down series………………………………………122
Figure 3.22 b: Monotone up and Monotone down series……………….. …………………….125
Figure 3.23a: Degenerate tetrahedra…………………………………………………………....136
Figure 3.23b: Degenerate tetrahedra……………………………………………………............137
Figure 3.23c: Degenerate tetrahedra …………………………………………………………...137
Figure 3.24: labeled tetrahedron………………………………………………………………..138
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Figure 4.1: Spanning cut tree and their nets………………………………………………........139
Figure 4.2: Spanning cut tree and their nets………………………………………...………….140
Figure 4.3: Spanning cut tree ……………………………………………………...…….......... 140
Figure 4.4: {5, 1} Tetrahedron ……………………………………………………………. …..141
Figure 4.5a: Star net for {5, 1}………………………………………………………………....142
Figure 4.5b: Path net for {5, 1}…………………………………………………………..…….143
Figure 4.6a :{2, 1, 1, 1, 1} Type1 tetrahedron………………………….………………..........144
Figure 4.6b :{2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………………….…144
Figure 4.6a :{ 2, 1, 1, 1, 1} Type 1 tetrahedron……………………………………………....145
Figure 4.7a: Star net for {2, 1, 1, 1, 1} Type 1 tetrahedron….……………………………..…146
Figure 4.7b: Path net for {2, 1, 1, 1, 1} Type 1 tetrahedron……………………………….....150
Figure 4.6b :{2, 1, 1, 1, 1} Type 2 tetrahedron……………………………………………….151
Figure 4.8a: Star net for {2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………..153
Figure 4.8b: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….157
Figure 4.9a: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….158
Figure 4.9b: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron………………………………….158
Figure 4.10a: Path net for {2, 1, 1, 1, 1} Type 2 tetrahedron…………………………………159
Figure 4.10b: Path net for {2, 1, 1, 1, 1} Type 1 tetrahedron………………………………...159
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Figure 4.11: Dual tetrahedron with the same path for {2, 1, 1, 1, 1} type……… …………….160
Figure 4.12: Dual tetrahedra with a matching edge for {2, 1, 1, 1, 1} and their nets…………..161
Figure 4.13a: {2, 1, 1, 1, 1} Tetrahedron……….………………………….…………………….162
Figure 4.13b :{ 2, 1, 1, 1, 1} Overlapping tetrahedron ….….…………………………………163
Figure 1.1: A point.......………………………………………………………………………....176
Figure1. 2: A line..……………………………………………………………………………...176
Figure 1.3: Line segments …………………………………………….………………………..177
Figure 1.4: A plane …………………………………………………….………….……………177
Figure 1.5: Three collinear points ………………………………….…………………………..177
Figure1. 6: Coplanar points..…………………………………………………………………...178
Figure 1.7: Rays ………………………………………………………………………………..178
Figure 1.8: Two congruent segments ……………………………………………………….....179
Figure 1.9: Midpoint of line segment………………………………………………………….179
Figure 1.10: Angle …………………………………………………………………………….180
Figure1.11: Right angle………………………………………………………………………..180
Figure 1.12: Acute angle………………………………………………………………………181
Figure 1.13: obtuse angle……………………………………………………………………...181
Figure 1.14: Reflex angle……………………………………………………………………...182
Figure 1.15: Adjacent…………………………………………………………………………..182
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Figure 1.16: Vertical angles…………………………………………………………………..183
Figure 1.17: Complementary angles………………………………………………………….183
Figure 1.18: Supplementary angles…………………………………………………………184
Figure 1.19: Point, line, and Plane……………………………………………………………184
Figure 1.20: Line segment……………………………………………………………………185
Figure 1.21a: Polygons…………………………………………………………………….….186
Figure 1.21b: Polygons………………………………………………………………………..187
Figure 1.22: Diagonal of Polygons…………………………………………………………....187
Figure 1.23: Non Convex polygon……………………………………………………………188
Figure 1.24: Equilateral Polygon…………………………………………………………….188
Figure 1.25: Equiangular polygon…………………………………………………………….189
Figure 1.26: Regular polygon…………………………………………………………………189
Figure 1.27: Triangle………………………………………………………………………….190
Figure 1.28: Kite……………………………………………………………………………….191
Figure 1.29: Isosceles trapezoid……………………………………………………………….192
Figure 1.30: Rhombus…………………………………………………………………………192
Figure 1.31: Parallelogram…………………………………………………………………… 193
Figure 1.32: Rectangle…………………………………………………………………………194
Figure 1.33: Square……………………………………………………………………………194
Figure 1.34: Triangle ………………………………………………………………………….195
Figure 1.35: Equilateral, Isosceles, and Scalene triangles……………………………………..196
Figure 1.36: Triangle…………………………………………………………………………..196
ix
Figure 1.37a: Congruent triangles………………………………………………………………197
Figure 1.37b: Congruent triangles……………………………………………………………...198
Figure 1.38a: Congruent triangles………………………………………………………………198
Figure 1.38b: Congruent triangles……………………………………………………………...199
Figure 1.39: Right angle triangle ………………………………………………………………199
Figure 1.40: Heronian triangles…………………………………………………… …………..200
Figure 1.41a: Quadrilaterals……………………………………………………………………202
Figure 1.41b: Quadrilaterals………………………………………………….………………..204
Figure 1.41c: Quadrilaterals…………………………………………………..………………...207
Figure 1.42: Hierarchies of quadrilaterals……………………………………………………...207
Figure 1.43a: Cyclic quadrilaterals……………………………………………………………. 209
Figure 1.43b: Cyclic quadrilaterals…………………………………………………………….210
Figure 1.44: Traiangle…………………………………………………………………………..212
Figure 1.45: Degenerate tetrahedra…………………………………………………………….215
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LIST OF TABLES
Table 3.1: Congruence Property by face and vertex…. . ..……………………………..…….85
Table 3.2: Congruent face partition with edge partition behavior…………………………...87
Table 3.3: Congruent vertex partition with edge partition behavior…………………………89
Table 5.1: Congruence Property by face and vertex.. . .………………………………………180
Table 5.2: Congruent vertex partition with edge partition behavior ………………………….181
Table 5.3: Congruent face partition with edge partition behavior……………………………..182
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ACKNOWLEDGEMENTS
It is a pleasure to thank the many people who made this thesis possible.
My time, three years and seven months, at Teachers college, Columbia University very enjoyable
and my work there became a part of my life. I am indebted to many Students and colleagues for
providing a stimulating and exciting environment in which to learn and grow.
At Teachers College I had the opportunity to study under and work with prestigious and highly
qualified mathematicians, mathematics educators and leaders in other educational fields.
They have made a great impact in my achievements and consolidation of career goals and
dreams. I thank them all.
I am particularly honored to have studied with Prof. Joseph Malkevitch who has facilitated and
inspired me to deepen my knowledge in the branches of geometry. I have been amazingly
fortunate to have him as my advisor. I want to extend my thanks to Prof. Bruce Vogeli for
helping me during the writing of the dissertation proposal process, for the valuable advice,
suggestions, and comments he made on my work.
I feel especially very blessed and profoundly happy with all the advice and the tremendous work
Prof. Malkevitch has dedicated to my research work as my dissertation Sponsor. I want to thank
him and all the members of the Dissertation Committee: Prof. Vogeli, Prof. J. Smith,
Prof. O. Roger Anderson and Prof Patrick Gallagher for their encouragement, insightful
comments and guidance which were helpful in refining the draft version of my dissertation into
its final form. I am deeply indebted to my friends whose encouragement made this work possible
towards its final stages. They have consistently helped me keep perspective on what is important
in life. They always gave me support throughout my studies.
Most importantly none of this would have been possible without the support of my parents.
xii
I would like to thank my parents for all their love. They guided me in all my pursuits.
They raised me, supported me, taught me, and loved me. I wish my father could have lived to see
me finish this work .I know it would have meant so much to him. This thesis is dedicated to him.
X111
DEDICATION
This dissertation is dedicated to my father Ato Haileselassie Mussa
XlV
1
CHAPTER I
INTRODUCTION
This thesis investigates new mathematical properties of tetrahedra motivated by showing novel
approaches to traditional questions addressed by mathematics educators interested in geometry
and how geometry is connected to other parts of Mathematics.
Definitions of geometric objects such as polygon and polyhedron have evolved since antiquity in
response to new mathematical problems. Ordinary usage of such terms contrasts with usage in
the mathematics research literature and the definitions found in texts at different levels of school
mathematics. This may cause confusion for students. Variations in definitions from one school
level to another are especially troublesome.
This study investigates these issues in the context of the transition between 2-dimensional
geometrical objects and 3-dimensional objects. In particular it addresses representing threedimensional polyhedra in the plane. Historically, such representations have included isometric
drawings, projective drawings (period of the Renaissance in the work of artists such as da Vinci),
nets (Durer), and the use of graph theory ideas. One focus of this study builds on the idea of
using partitions of an integer to provide organized thinking about geometrical objects such as
quadrilaterals and tetrahedra. Specifically studying the behavior of the existence of tetrahedra in
a partition environment is an excellent framework for implementation of small cooperative
learning projects for the goal of improving student skills with visualization and the role of
definitions.
2
A polyhedron is a 3-dimensional object that contains flat faces and straight edges. Examples
include the Platonic solids and tetrahedra in general, combinatorial cubes, and other convex
polyhedra. The tetrahedron is the simplest three dimensional solid, having four vertices, four
faces (all triangles) and six edges. Partitions of 6, the number of edges of a tetrahedron offers a
way to classify tetrahedra, and to address questions of research interest to mathematicians while
offering K-12 students a “laboratory” to make models, and get insight into representation of
geometrical objects, visualizing these objects, and telling when the objects are the same or
different. New theorems about tetrahedra are developed with these educational goals as
motivation.
Purpose of the study
The purpose of this study is to give a new mathematical approach to the study of tetrahedra and
their nets for use by students, teachers and researchers.
Goals include:
* showing ways to transition from geometrical problems arising in 2-dimensions (2D) to ones in
3-dimensions (3D)
* showing connections between different mathematics topics (algebra, geometry, combinatorics,
etc.) that should be treated in a more integrated way
* enhancing sense making for the role of definitions in geometry and to design activities
containing exploration problems and classroom activities that enable students to study geometry
3
by investigating, exploring, and establishing geometric conjectures, which eventually will
develop into proofs.
* showing that there are new mathematical questions about polygons and tetrahedra accessible at
the college and pre-college level.
The following research issues will be considered in the study:
1. How to classify tetrahedron into different types based on the notion of a partition.
2. Existence questions for tetrahedron with integer lengths.
3. Relating the existence of nets of tetrahedra to the type of spanning tree that is used to generate
the net, and under what circumstances attempts to generate a net for a tetrahedra fail due to
overlap of the faces when the tetrahedron is flattened into the plane. (Some tetrahedra have an
edge unfolding which overlaps, a rather unexpected recent discovery in elementary geometry.)
4. Which partition types of tetrahedra admit spanning tree with cuts that lead to overlaps?
5. What are current views about pedagogy involving three-dimensional polyhedra, in particular
tetrahedra?
Procedure of the study
In order to address these research questions the following procedure was used
1. Develop new results, examples, and theorems related to triangles, quadrilaterals and tetrahedra
which can provide to mathematicians, mathematics educators and students a model for seeing
how mathematics grows and develops.
4
2. Generate research questions for high school students and mathematicians.
Different tasks will be considered and a set of problems and motivational activities will be
designed to enhance student understanding of the geometry they learn. Activities for exploration
will be accompanied by comments intended to help teachers use this approach. This material
appears in the appendices.
3. Exploration of the mathematics of polygons and polyhedra.
Polyhedra and drawings of them in the plane were studied. The use of partitions enables
classification of tetrahedra and addresses mathematical research questions that offer ways to
obtain new insights about geometrical objects.
4. Develop student activities involving triangles, quadrilaterals, and tetrahedra which grow out of
the new mathematical work.
5
CHAPTER II
NEW MATHEMATICAL INSIGHTS INTO TETRAHEDRON
Some Definitions
Some basic definitions and concepts needed in what follows in more detail are introduced rather
than be extremely formal these terms are introduced in an intuitive way that would benefit their
use in school.
Polygons
A polygon is a collection of points A1 , A2 , A3 , …., An called vertices joined by line segments
A1 A2 , A2 A3 , …., An A1 joining the points as in Figure 1 where for convenience a notation
avoiding subscripts is used. The points are known as the vertices of the polygon and the
segments are known as the edges of the polygon. The segments are said to make up the boundary
of the polygon. Thus, when the polygon has 3 points it is called a triangle, when it has 4 points it
is called a quadrilateral, and when it has n points it is called an n-gon. Usually, the points are
such that no three of them are allowed to be on a line (no 3 collinear). So a triangle with an extra
vertex on one of its sides will not be thought of as a triangle but as a 4-gon, though in some
situations it can be considered as a "degenerate triangle." Usually, when talks about a polygon P
means only the "rods" that make up the boundary of the polygon. If wants to refer to the interior
region that the rods bound, strictly speaking should talk about the interior of the polygon, but
will be more informal here and what is meant should be clear from the context. A polygon whose
vertices lie in a single plane is called planar. Note, that the approach given restricts our attention
6
to planar polygons though more general polygons are of great interest to researchers in geometry.
(a)
(b)
( c)
Figure 2.1: Different Polygons
A simple polygon P is one whose edges only meet at vertices. A simple polygon P divides the
plane into two regions called the interior and exterior of P. In Figure 2.1a and 2.1b shows simple
polygons but Figure 2.1b is non-simple, it intersects itself. Sometimes is useful to view polygons
from a graph theory point of view. From this point of view an n-gon is a cycle with n vertices.
The interior of the polygon is "bounded" since it can be completely enclosed in a suitably large
circle and the exterior region is "unbounded." The exterior region cannot be enclosed in a circle
with a finite radius. Note that for a simple polygon the points at which two edges (line segments)
meet are vertices of the polygon. Two edges that meet at a vertex are known as adjacent edges.
Two of the edges in Figure 2.1b meet but this point is not a vertex of the polygon, which is not
simple. Polygons which are not simple cannot be assigned an area in a natural way.
A polygon that contains a pair of adjacent edges or, for that matter non-adjacent edge along a
straight line is called a degenerate polygon. Typically this is ruled out by the assumption that no
three points are collinear; however, for many interesting questions about polygons allowing
7
adjacent or non-adjacent sides of the polygon to lie along the same line is of interest - for
example, in discussions of "art gallery theorems."
Polyhedra
A polyhedron (plural, polyhedra) is the three-dimensional version of polygon and of the more
general objects that can be defined in arbitrary dimensions. Polyhedra appear early in the history
of geometry, including in Euclid’s Elements. The word “polyhedron” (in geometry) derives from
the Greek poly (many) plus the Indo-European hedron (seat).
The regular convex polyhedra or Platonic solids (polyhedra whose faces are regular convex
polygons and which have the same number of edges at each vertex) have been known for at least
2500 years, and were studied in detail in Euclid's Elements.
A set X is convex if when M and N are points of X then the whole line segment joining M and N
is completely contained in X. From this it follows that in a convex polygon each of its interior
angles has measure at most 180 degrees. (When the polygon is “degenerate “there can be straight
angles in the boundary of the polygon.) Note that when thinks of a polygon only as the rods that
makes it up, then polygons strictly speaking cannot be convex. Strictly speaking one has to talk
about a polygon and its interior in asking if convexity holds. Figure 2.1c shows a convex
polygon and Figure 2.1a is non-convex. Figure 2.1 b is not simple and, thus, of necessity is nonconvex.
8
Convex 4-gon
(a)
convex 4-gon and its diagonal
(b)
Figure 2.2: Convex 4-gon
I will be discussing non-convex simple polygons in addition to convex polygons. Figure 2.2a
shows a convex polygon (4-gon) ABCD and Figure 2.2b the same polygon with its two
diagonals. AC and BD are the diagonals. A 4-gon, together with its two diagonals is known as a
complete quadrilateral.
A non-convex polygon is a polygon that has at least one interior angle that measures more than
180 degrees (but less than 360 degrees). These angles are known as reflex angles.
9
Figure 2.3: Non convex 7-gon
Figure 2.3 shows a non-convex 7-gon with three reflex angles.
The convex regular solids are the tetrahedron (4 faces which are equilateral triangles), cube (6
faces which are squares), octahedron (8 faces which are equilateral triangles), icosahedron (20
faces which are equilateral triangles), and the dodecahedron (12 faces which are regular
pentagons).The faces of each of these solids are congruent to each other, and they have the same
number of faces meeting at every vertex which means that, considered as graphs, they have
regular graphs. No general definition of polyhedron appeared until much later than Euclid.
Euclid proves theorems without telling the reader that he is dealing only with convex polyhedra.
Even Euler discussed his famous formula (Euler's polyhedral formula) V – E+ F = 2
(V = vertices, E = edges, F = faces) without specifying the type of polyhedra he had in mind.
During the 19th century, the center of attention concerning polyhedra expanded beyond the
convex ones. Ernst Steinitz formulated the criteria that are necessary and sufficient for the
10
existence of a convex 3-dimensional polyhedron in the period from 1910 -1920. His famous
result, Steinitz's Theorem, is that a (vertex-edge) graph is the graph of some convex
3-dimensional polyhedron (also known as a 3-polytope) if and only if the graph is planar (can be
drawn in the plane with edges meeting at vertices) and 3-connected (for every pair of vertices u
and v in the graph there are at least three paths between the vertices u and v which have only u
and v in common). In the 19th century, certain non-convex polyhedra including various ones that
intersected themselves were discussed and described by different authors; however no one gave a
meaningful definition of non-convex polyhedron. Even in the discussion of non-convex regular
polyhedra, 19th century work does not meet modern standards of rigor.
Here we will adopt a version of an informal definition due to Joseph O’Rourke (Smith College)
who has written about polyhedra for audiences at many levels. A polyhedron is the surface of a
3-dimensional object composed of flat, convex polygon faces such that:
Every side of a polygon belongs to just one other polygon.
The faces that share a vertex form a "chain" of polygons in which every pair of consecutive
polygons share a side.
Summarizing the discussion above in slightly different terms, an object (polygon or polyhedron)
is convex if for every pair of points that belongs to the shape (its boundary or its interior), the
object contains the whole straight line segment connecting the two points. Convex polygons have
no reflex vertices. A reflex angle is an angle whose measure is greater than 180 and less than 360
(Figure 2.3). Every face of a convex polyhedron is simply connected (no holes). The tetrahedron
has 4 vertices, 4 faces and 6 edges. The interior angle at vertex v in a face incident to v is called
the face angle. The “curvature” at the vertex v is 2π - (the sum of the face angles at v).
11
Descartes’ Theorem: the sum of the curvature for a polyhedron of genus zero (topologically like
a sphere) is 4π.
Sometimes the curvature at a vertex is called the angle defect at the vertex. The genus of a graph
is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with
n handles (an oriented surface of genus n).A planar graph has genus zero because it can be drawn
on a sphere without self-crossings. Intuitively, the genus of a surface counts the number of holes
or handles it has. Thus, a sphere and the plane have genus zero since they have no holes or
handles. A donut, or torus, has genus 1 because it can be thought of as having one hole, or being
a sphere with one handle.
One of the most important theorems for polyhedra is the Euler formula relating the number of
vertices, edges and faces of a convex polyhedron.
For any convex polyhedron P with no holes, |V (P) - |E (P)| +|F (P)| = 2 where V (P), E (P),
and F (P) represent the sets of vertices, edges and faces of the polyhedron. Here |X| denotes the
number of elements in set X.
Unfolding of a Polyhedron
One unfolds a polyhedron P by cutting along some of its edges and flattening out its faces into
the plane without overlap. The result is called a simple unfolding or a net and it is created by
cutting edges of the polyhedron P that form a spanning tree of the vertex-edge graph of P. Note
that nets are treated as metrical objects when drawn in the plane, and that nets include fold lines
that show up as diagonals of the polygon that makes up the net.
12
A spanning tree is a sub graph of a graph G that is a tree (a connected graph which has no
circuits) and includes all the vertices of G. The set of cut edges must be connected because the
cut edges unfold to the boundary of a polygon and the boundary is connected. Thus, the cut
edges form a spanning tree. If one cuts edges containing a circuit the result would be to separate
what is obtained into two pieces. The boundary of a polygon of the unfolding (Figure 4) consists
of twice the number of cut edges. Each cut edge appears exactly twice on the boundary.
.
Figure 2.4a: spanning cut tree and their nets
For the cut edges of CABD and BCAD (in Figure 2.4a and 2.4b) the boundary of a polygon of
the unfolding consists of twice the number of cut edges, and is hexagon, which need not be
convex.
13
Figure 2.4b: spanning cut tree and their nets
Conjecture (Geoffrey Shephard’s Conjecture): Every convex polyhedron can be cut along some
of its edges (a spanning tree) and unfolded into the plane without overlap.
Many mathematicians have attempted to prove the conjecture. However, a counterexample has
been found for every algorithm proposed that some special type of spanning tree leads to a
simple unfolding for any convex polyhedron. On the other hand experimental results suggest that
a random spanning tree of a random polytope causes overlap with probability approaching one as
the number of vertices approaches infinity (C. Schevon).
Tetrahedra
A tetrahedron is a three-dimensional solid having four vertices, four triangular faces and six
edges that don’t lie in a single plane. Tetrahedron is of Greek origin (tetra- four and hedra -seat)
and refers to its four plane faces.
The tetrahedron is the only convex polyhedron that has four faces. It is one kind of pyramid,
which is a polyhedron with a flat polygons base (n- gon) and triangular faces connecting the base
14
to a common point so a tetrahedron is a triangular pyramid. There are two types of angles
connected with a tetrahedron: six dihedral angles formed by all pairs of faces and four trihedra
angles formed by all possible triples of faces. The sum of the dihedral angles varies between
2πand 3π while the sum of the trihedral angles varies between 0 and 2π.
Standard Forms of Labeling a Plane Diagram of a Tetrahedron
There are two standard forms for drawing a labeled tetrahedron in the plane. Both involve
isomorphic copies of K4 (the complete graph with 4 vertices) with typically different symmetry
groups as drawings in the plane.
Figure 2.5a: Labeling of Tetrahedron
Figure 2.5a has 8 symmetries and it enables one to see the edge of the perfect matching’s of a
tetrahedron more clearly. A perfect matching of a graph with an even number of vertices is a
collection of edges that are disjoint but include all of the vertices in the graph.
15
(Note the following edges are equal: b=c=d and a=e=f.)
Figure 2.5b: Labeling of Tetrahedron
The other drawing (Figure 2.5b) can have as many as 6 symmetries but in Figure 2.5b as shown
there are two symmetries.
According to the labeling of vertices and edges shown in the Figures 2.5a and Figure 2.5b can
determine the existence of a tetrahedron with the 6 labels representing values for the edge
lengths.
Arthur Cayley and later Karl Menger showed how could prove the existence of a tetrahedron T



with a specific collection of edge lengths (x, y, z, X , Y , Z ) or using vertex coordinates by
computing a 5 x 5 determinant to find the “volume “ of T.
16
Later William H. McCrea showed how instead a 3 x 3 determinant could be evaluated to achieve
the same goal. Details can be found in Wirth and Drieiding , 2009. The form of the McCrea
determinant is shown below with the letters representing lengths as in Figure 2.5a and Figure
2.5b.
 2d 2
d 2 + e 2 c 2 f 2 + d 2 b 2 


D(S ) =  d2 + e 2 c 2
2e 2
e 2 + f 2 a 2 
 f 2 + d 2 b 2 e 2 + f 2 a 2
2 f 2 
By using the McCrea determinant, a tetrahedron representing a facial (faces obey the strict
triangle inequality) sextuple S = (a, b, c, d, e, f) exists if and only if D(S), the determinant of the
McCrea matrix, is positive.
17
Thus, treating the McCrea determinant as a "black box" it is easy to tell when a tetrahedron
exists with 6 particular length sticks.
Wirth and Drieiding, 2009 show that a tetrahedron corresponding to a facial sextuple
S = (a, b, c, d, e, f) exists if and only if the star net (Figure 2.6 ) has an acute vertex triple
obeying the A-angle inequality (it has at least one acute vertex triple). An acute vertex triple is an
angle whose triple of angles at a vertex each has measure greater than zero and less than 90. The
A-angle inequality refers to requiring that the three angles A, B, C in a triple (A, B, C) obey the
inequalities: A + B < C, B + C < A, A + C < B. Facial means that the lengths corresponding to
the four faces of the proposed tetrahedron obey the strict triangle inequality.
Figure 2.6: Star net
If the McCrea determinant is zero then there is a degenerate tetrahedron whose 4 vertices lie in a
plane. If the McCrea determinant is negative there is no tetrahedron geometrically even when the
facial condition holds. The significance of a negative value of the McCrea determinant of
a tetrahedron with a given set of edge lengths is that one can be certain that a tetrahedron cannot
18
exist with the given edge lengths. Having a positive McCrea determinant for 6 stick lengths is a
necessary but not sufficient condition for the existence of a tetrahedron with these 6 stick lengths
as edge lengths.
The second form of labeling a tetrahedron (Figure 2.5b) enables us, using the McCrea
determinant as a "black box" to check relatively easily if a given set of six edge lengths can be
used for “making “ a tetrahedron.
Nets of Tetrahedra
Nets for a tetrahedron are obtained by cutting three edges at a vertex of the tetrahedron or along a
sequence (path) of three edges that visit each vertex exactly once. Thus, there are two types of
nets: star nets and path nets. Star nets are obtained by cutting three edges of tetrahedron at one
vertex of the tetrahedron. Path nets are obtained by cutting three edges of a tetrahedron along a
sequence of three edges of a tetrahedron - in graph theory terms the edges form a hamiltonian
path. More precisely, define an edge unfolding as a "development" of the surface of a
polyhedron to a plane such that the surface becomes a flat polygon bounded by segments that
derive from edges of the polyhedron (Demaine and O’Rourke, 2007).
Figure 2.7 shows examples of the two ways that one can cut edges of the same (physical)
tetrahedron to obtain a net. Figure 2.8a shows a star net at vertex A Figure 2.7b and Figure 2.8b
shows the path net that arises from cutting the edges in the path CABD (or DBAC).
19
(a)
(b)
Figure 2.7: Edges of a spanning tree
The three edges of a spanning tree which are to be cut to obtain a net are shown as bold lines.
The nets which arise can be either a convex or non-convex polygons.
(a)
20
(b)
Figure 2.8: Star net and Path net
Note that in a star net one of the vertices of the tetrahedron appears in the net in three positions.
'
'
'
'
'
As polygons the nets can be written: A B A C A D and A BDB AC (Figure 2.8a and Figure
2.8b). The net also includes (metrically) the three diagonals of the polygonal 6-gon that bounds
the net as shown in Figure 2.8. Does every spanning tree of a tetrahedron lead to a net?
How many nets can we find for a tetrahedron?
This thesis addresses these mathematical questions. As we have seen from Figure 2.7 and Figure
2.8 there are nine edges and six vertices in a net; this is because the boundary of the unfolding
consists of twice the number of cut edges. Each cut edge appears exactly twice on the boundary
which has six vertices. If takes the star net which arose from a physical tetrahedron, one can
always fold it up to a tetrahedron. However, for a path net there is no guarantee that one
21
can unfold to a non-overlapping polygon. This does not contradict Shephard's Conjecture. It just
means that some spanning trees may not lead to an unfolding. There can be a spanning tree (in
the form of a path) for a tetrahedron which when cut will lead to a way to open up the
tetrahedron which will overlap (Figure 2.9).
Figure 2.9: Overlapping tetrahedron
In response to a conjecture by K.Fukuda that overlap couldn’t occur for a tetrahedron, M.Namiki
found an example where overlap does occur. Figure 2.10 shows an explicit example of this
phenomenon that was published by Branko Grünbaum (Figure 2.10) inspired by Namiki’s
example.
22
Figure 2.10: Grünbaum overlapping tetrahedron
This thesis also found an overlapping unfolding for a tetrahedron which is not scalene (not all
edge lengths different). Previously known examples all produced scalene (Figure 2.10).
23
Figure 2.11 :{ 2, 1, 1, 1, 1} overlapping tetrahedron
How can tell when two tetrahedra are “different” or not congruent?
If two figures are called congruent when there exists a one - to - one distance preserving
correspondence between their point sets. Recall that two triangles are congruent if the sides of
one are congruent (equal in length) to the sides of the other. This is the SSS congruence
condition for triangles. SAS and AAS also suffice to show congruence where A refers to an
angle and S refers to a side.
24
D(S) = 10280448
D(S) = 10870398
Figure 2.12: Incongruent tetrahedra
The faces of the two tetrahedra in Figure 2.12 are not congruent even though the six edges of T1
(Tetrahedron 1) are congruent or equal to the six edges of T2 (Tetrahedron 2). Figure 2.12a has
two isosceles and two scalene triangles and Figure 2.12b has three isosceles and one scalene
triangles. They are incongruent tetrahedra. Also, note that to check if there are actually physical
tetrahedra that correspond to these two diagrams one has to check that the facial condition holds
as well as check that the McCrea determinant is positive. In general when one moves the
positions of the lengths in Figure 2.5b, even keeping the set of lengths the same, there is no
guarantee that if a physical tetrahedron exists for the initial drawing that a physical tetrahedron
will exist for the altered drawing!
One Goal of this thesis s to investigates the number of different partition types (see below) in
which tetrahedra can be classified. It is shown that for the 11 different partitions of 6 (the number
of edges of a tetrahedron) some of these partitions allow a “finer“way to tell tetrahedra apart. For
25
example given the partition {3, 3} one can tell apart tetrahedra where the two pairs of three equal
length edges form a) two paths b) three edges meeting at a vertex and three edges forming an
equilateral triangle.
Figure 2.13: {3, 3}Tetrahedra
Partition Types of Tetrahedra
A partition of a positive integer n is a way of writing n as the sum of positive integers. For
example {2, 1} is a partition of 3. Note that {1, 2} represents the same partition as {2, 1}.
The partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}.
The tetrahedra (6 edges) can be classified into eleven classes of partition type in terms of their
edges: {6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1},
{2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}. If we take {2, 2, 1, 1} one can interpret the partition as a
tetrahedron with 4 different edge lengths, two different sets of two edges having the same length
but different from each other, and different from the other two lengths of edges. Both tetrahedra
26
in Figure 2.12 represent the partition {2, 2, 1, 1}. For each partition of 6 one can find
incongruent tetrahedra corresponding to this partition. Note that one way of being certain that the
two physical tetrahedra shown in Figure 2.14 are incongruent is that their McCrea determinants
(shown) are not equal. Eewven without invoking the McCrea determinant, one can see using
simpler ideas that the tetrahedra are not the same.
D(S) = 10723552
D(S) = 11941600
Figure 2.14 :{ 2, 2, 1, 1}Tetrahedra
For the partition {2, 2, 1, 1} one can find two incongruent tetrahedra using the same edge
lengths. The refined approach, taking some geometric information (but not relative size of edges)
into account, leads to different potential types as has been determined in this thesis. This thesis
not only enumerates the refined number partition classes but also determines for each of these
classes if there is an integer collection of lengths for a tetrahedron in this class. There are actually
27
several issues here which have some pedagogical implications in addition to the new
mathematical results that have been obtained.
Integer Lengths
For pedagogical purposes and making physical models of tetrahedra, working with integer
lengths instead of using general real numbers is preferable. The discussion here is valid for real
numbers as long as the lengths of each of the four faces of the tetrahedron obeys the triangle
inequality and the McCrea determinant is positive. If the tetrahedron exists with integer lengths
then one can design (for the benefit of students) a simple way of labeling a tetrahedron to
determine the number of distinct ways of using those integer lengths for different partition
classes of tetrahedra. With six distinct lengths, there are at most 30 incongruent tetrahedra. Since
the labels in Figure 2.5b can be assigned in 720 (6!) different ways (the first choice can be made
in 6 ways, the next choice in 5 ways, etc.) this means that for each of the 30 different tetrahedra
there are 24 drawings which lead to a congruent version of a given tetrahedron. Another way of
thinking of this that the tetrahedron, whose graph is the complete graph on 4 vertices, as a graph,
can have 24 symmetries (automorphisms).
From a given tetrahedron one can find a “dual tetrahedron” taking into a account the triples that
are the edges at a vertex and the triples that form the faces of a tetrahedron. For example the
tetrahedron in the Figure 17a has faces 28, 30, 45; 30, 35, 42; 28, 36, 42; and 35, 36, 42. The
dual has faces with sides corresponding to the numbers (edge lengths) at the vertices in Figure
2.15b.
28
Since the tetrahedron exists, which is facial and the McCrea determinant is positive then we can
try to find a dual tetrahedron and that has faces with sides that correspond to the number at the
vertices of the original tetrahedron.
D(S) = 2250079808
Faces 28,30,45 ; 30,35,42 ;28,36,42 ; and 35,36,42
(a)
29
D(S) = 2341239800
Vertex 28, 30, 45; 30, 35, 42; 28, 36, 42; and 35, 36, 42
(b)
Figure 2.15: Dual Tetrahedra
Degenerate Tetrahedra
When the McCrea determinant of the tetrahedron is zero the four points of the "potential"
tetrahedron will lie in a single plane or lie on a single line so the tetrahedron will become
degenerate.
30
Figure 2.16: Degenerate tetrahedra
Note that the parallelogram shown in Figure 2.16 cannot be made metrically while the trapezoid
realization in the plane is metrically possible.
31
fig1
(a)
(b)
Figure 2.17: Degenerate tetrahedron
Figure 2.17b shows the degenerate tetrahedron that is determined by four vertices with six edges
and four faces such that three vertices are collinear in Figure 2.17a.
32
CHAPTER III
TETRAHEDRA
Tetrahedron: A tetrahedron (plural: tetrahedra) is a three-dimensional solid having four vertices,
four triangular faces and six edges which don’t lie in a single plane. Figure 1 may help with
following the development adopted here.
33
Figure 3.1: Potential Tetrahedron
Figure 3.2 below is a diagram of a "potential" tetrahedron with edges of length
S = (a, b, c, d, e, f) positioned as shown.
34
Figure 3.2: Tetrahedron
Using the McCrea determinant, a sextuple S = (a, b, c, d, e, f) will allow the construction of a
tetrahedron if and only if D(S) is positive and S is facial. D(s) refers to the determinant of the
matrix shown below. If M is a matrix then |M| will denote the determinant of M.
D(S ) =

2d 2
 2
2
2
 d + e c
 f 2 + d 2 b 2
d 2 + e 2 c 2
2e 2
e 2 + f 2 a 2
f 2 + d 2 b 2 

e 2 + f 2 a 2 

2f 2

If all four face triples (Figure 3.2) obey the strict triangle inequality then S is facial. In order to
ensure that a sextuple S exists as a tetrahedron we must test if the tetrahedron is facial and that
35
the McCrea determinant is positive. If one takes a sextuple of a fixed tetrahedron S and one with
6 distinct edge lengths, the lengths can be placed on the edges in 720 different ways and at most
30 of these can exist as non-congruent tetrahedra. Thus, for any particular legal tetrahedron,
there are in fact 24 other ways to label that the edges which gives rise to the same tetrahedron.
To study these ideas consider the sextuple S = (35, 30, 42, 28, 36, 45)
D(S) = 2250079808
Figure 3.3: Tetrhedron in terms of faces
We can check that this tetrahedron Figure 3 exists, which involves if D(S) positive and S facial.
The faces will have the side lengths: 28, 30, 45; 28, 36, 42; 30, 35, 42; and 35, 36, 45 and they
obey the strict triangle inequality.
Now, suppose we list the numbers of S in reverse order, giving the sextuple:
S = (45, 36, 28, 42, 30, 35)
36
D(S) = 2341239800
Figure 3.4: Tetrahedron in terms of vertices
In the same way using the McCrea determinant one can check also that this tetrahedron exists.
Thus D(S) is positive and S is facial. Its faces will have the edge lengths: 28, 30, 42; 28, 36, 45;
30, 35, 45; and 35, 36, 42 (see Figure 3.3 and 4.4 ). However these two tetrahedron are not
congruent. This raises to the question of what is the largest number of tetrahedron that a
particular collection of 6 distinct edge lengths might give. In fact, for the partition of the
sextuples of S = (35, 30, 42, 28, 36, 45) one can find exactly 30 incongruent tetrahedra and they
all exist.
One can generalize that for a sextuple S of a legal tetrahedron with six distinct lengths, one can
find at most 30 incongruent (scalene) tetrahedra.
For the partition of the sextuple of S = (61, 80, 90, 100, 110, 120) we find 29 of the possible 30
incongruent (scalene) tetrahedra associated with these 6 length can exist. The sextuples which
does not work is S = (110, 61, 90, 120, 100, 80).
37
If one creates other sextuples S there is no guarantee that S is a legal tetrahedron. For example, if
we consider the sextuple S= (5, 4, 3, 5, 3, 4) it is facial but D(S) = - 2450. The fact that the
McCrea determinant is negative means that no tetrahedron exists.
Figure 3.5:Tetrahedron with negetaive determinant
Furthermore the sextuple S = (1, 1, 3, 6, 1, 3) which is not facial but D(s) = 952 (positive), also
doesn’t exist as a tetrahedron.
Figure 3.6: Tetrahedron with non facial
38
Consider a tetrahedron which is facial and the McCrea determinant is positive such as the
tetrahedron (T ' ) in Figure 4 with faces 28, 30 ,42; 28, 36, 45; 30 , 35, 45; and 35, 36, 42.
These numbers correspond to the numbers at the vertices of the original tetrahedron ( T ) in
Figure 3.3.These edges are vertices are 28, 30, 42; 28, 36, 45; 30, 35, 45 ; and 35, 36, 42.We can
think of these as dual tetrahedra .
If S is a sextuple for (potential) tetrahedron T, S = (a, b, c, d, e, f) then T has faces a, b, c; a, e, f;
b, d, f and c, d, e and the edges at the vertices has the pattern a, b, f; a, c, e; b, c, d and d, e, f .
If we have a potential tetrahedron T ' and where the pattern of faces and vertices is interchanged
then T ' is called the dual of tetrahedron T.
Theorem: If the potential tetrahedron T has sextuple S = (a, b, c, d, e, f) then the sextuple
( f, e, d, c, b, a ) gives rise to the potential dual tetrahedron.
Figure 3.7a:Dual tetrahedron
Faces : a, c, b; b, d, f; a, e, f and c, d, e
Vertices: a, c, e; b, c, d; d, e, f and a, b, f
39
Figure 3.7b:Dual Tetrhedron
Vertices
: a, c, b; b, d, f; a, e, f and c, d, e
Faces
: a, c, e; b, c, d; d, e, f and a, b, f
they are dual tetrahedra.
Standard form of labeling a tetrahedron
There are two standard forms for drawing a labeled tetrahedron in the plane. Both involve
isomorphic copies of K 4 with different symmetry groups as drawings in the plane.
40
(a)
(b)
Figure 3.8: labeling of tetrahedra
Figure 3.8a has 8 symmetries and it enables one to see the edges of the perfect matching of a
tetrahedron more clearly while Figure 3.8b enables us more easily to assign the given number of
six distinct edge lengths in a visually appealing manner.
Partition types of tetrahedra
Using only the partition information tetrahedra would lie in one of the 11 classes. These 11
classes exist as 3D types but not as degenerate 2D types because one type doesn’t exist in the
plane.
A new mathematical question is what is the largest number of incongruent tetrahedron with six
edge lengths can be formed for each partition type?
41
Suppose a tetrahedron with a given 6 distinct edge lengths S = {a, b, c, d, e, f} for partition
{1, 1, 1, 1, 1, 1}. If I fix c, f, e there are 6 permutations of the remaining 3 edges. Assume choice
c as the smallest edge length. Consider the edges of length e and f which are a matching edge of
the tetrahedron. Now there are 6 permutation edges of the other 3 edges a, b, d giving rise to the
6 patterns shown in the diagram below. Note, in each of the 6 diagrams each pair of letters from
a, b,…, f appear in two triangles .Now having chosen c (top edge) as the smallest and the other
edge e as one of the diagonal edges then there are 4 other ways to pair e with another letters as
the other “diagonal “edges e and f, e and d, e and b, and e and a one can obtain an additional 24
incongruent tetrahedron. The edges opposite to it can be chosen in 3 ways and each of them
admits 2 edges (e and c as matching edges) then there are 6 in congruent tetrahedron.
1.
42
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43
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44
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46
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47
4.
.
48
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49
5.
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3ce
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50
Figure 3.9: Construction of congruent tetrahedra
In terms of symbols there are at most 30 potentially incongruent tetrahedra with 6 distinct edge
lengths, and each tetrahedron is congruent with 24 tetrahedra corresponding to the 24 ways of
labeling the four vertices for a total of 720 tetrahedra. The 24 congruent tetrahedra for a given
sextuple S = (a, b c, d, e, f) appear in Figure 3.10.
51
52
53
(
54
Figure 3.10: congruent tetrahedra
55
If we Consider a tetrahedron for the partition {2, 1, 1, 1, 1} with edge length
{16, 12, 9, 12, 8, 18 } one can find a “dual tetrahedron” taking into a account the triples that are
the edges at a vertex and the triples that form the faces of a tetrahedron for a matching edge
however there is no guarantee that the dual tetrahedron exists .
S = (16, 12, 9, 12, 8, 18)
.
T1
D(S) = 844312
56
S = (12, 9, 12, 16, 18, 8)
T2
D(S) = 1489752
57
S = (16, 18, 12, 8, 9, 12)
T3
D(S) =- 3675992
58
S = (18, 16, 9, 8, 12, 12)
T4
.
D(S) = - 5581152
59
S = (18, 16, 12, 8, 12, 9 )
T5
.
D(S) = - 1912032
60
S = (18, 16, 12, 8, 9, 12)
T6
D(S) = - 4353408
Figure 3.11: Dual tetrhedra
Tetrahedron 1 and 2 exist and are dual. Tetrahedron 4 and 5 are dual and doesn’t exist;
Tetrahedron 3 and 6 are self-dual and doesn’t exist.
Conjecture: If S is a self-dual {2, 1, 1, 1, 1} tetrahedron then S can’t exist.
If we consider the tetrahedron for the partition of {2, 1, 1, 1, 1} and the sextuple
S= ( 8, 12, 9, 18, 12, 16) where the product of the matching edge lengths equal
12 x 12 = 9 x 16 = 8 x 18 = 144 and all triples are facial then one can generate at most 15
tetrahedron. (Refer to page 115)
61
Figure 3.12: {2, 1, 1, 1, 1} tetrhedron with equal product matching edge
If we consider for the partition { 1, 1, 1, 1, 1, 1 } with edge length (36, 28, 30, 35, 45, 42 ) and
the product of the matching are equal, 35 x 36 = 42 x 30 = 28 x 45 = 1260, and all triples are
facial then
D(s) = 6441334200
Figure 3.13a :{ 1, 1, 1, 1, 1, 1} tetrahedron with equal product matching edge
62
One can generate 30 scalene tetrahedron and all exists. (Refer to page 108)
Figure 3.13b :{ 1, 1, 1, 1, 1, 1} tetrahedron with equal product matching edge
Conjecture: If ad = be = cf and D (a, b, c, d, e, f ) > 0 and S (a, b, c, d, e, f) is facial then all the
30 assoiciated tetrahedron sextuples exsit.
Partition of Tetrahedron
The partitions of n are the ways of writing n as the sum of positive integers. we classify the
tetrahedron according to the edges since the tetrahedron has 6 edges then there are 11 partitions
{6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1}, (2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1}. Since a tetrahedron has 6 edges, one can interpret the partition {2, 2, 1, 1} as
that one has a tetrahedron with 2 edges of the same length, two edges of the same length but
different from the first two edges of equal length and two different other edges whose length
differs from the other 5 edges. For example one might have the lengths a, a, b, b, c, and d for the
63
edges of the tetrahedron, which represents {2, 2, 1, 1}. In the same way the lengths a, a, a, b, b
and c for the edges of a tetrahedron, can represent a {3, 2, 1} partition. Using only the partition
information, tetrahedra would lie in one of the 11 classes. These 11 classes all exists as 3 D types
but not as a degenerate 2D type because a 3D type may not exist in the plane.
Tetrahedra for each of the 11 partition classes
1. { 6 }
64
{5, 1}
3.
{4, 2}
65
4. {4, 1, 1}
5. {3, 3}
66
6. {3, 2, 1}
7. {3, 1, 1, 1}
67
8. {2, 2, 2}
9. {2, 2, 1, 1}
68
10. {2, 1, 1, 1, 1}
11. {1, 1, 1, 1, 1, 1}
Figure 3.14: 11 partition classes of tetrhedra
69
However with edge lengths that belongs to a particular partition we will consider whether or not
there exists a tetrahedron that takes into account other geometrical information. For example, for
the partition {2, 2, 1, 1} we are looking to differentiate between the cases where the edge lengths
constitute a perfect matching or paths of length 2. So there will be 4 types of tetrahedron that will
be different from each other. Using the partition information any legal tetrahedron would lie in
one of a bigger number partition classes.
We can also consider the partition for a particular tetrahedron based on congruence of triangles.
Thus we might have two congruent triangles of one type, and two congruent triangles of another
type. This system can be further refined to take into account whether the triangles are equilateral,
isosceles or scalene. We will use the notation E for equilateral, I for isosceles and S for scalene
so when we have a 2I E S tetrahedron it means two congruent isosceles triangles and one
equilateral triangle and scalene triangle.
The refined approach taking some geometric information (but not relative size of edges) into
account leads to potentially 25 classes as was determined in this thesis.
Theorem: There are 25 different partition classes of tetrahedra taking into account graph
theoretical aspects of the position of the edges, and all 25 types exist.
70
2 5 Tvpes of tetrahedra bv partition tvpe
1. {6}
(6)
Faces 4E:
Vertices 4E:
2. {5, 1}
{!J,1}
F:acP.2E 21
Vertices 2C 21
71
3-1. {4, 2}
3-2. {4, 2}
72
4-1. {4, 1, 1}
4-2. {4, 1, 1}
73
5-1. {3, 3}
5-2. {3, 3}
74
6-1. {3, 2, 1}
6-2. {3, 2, 1}
75
6-3. {3, 2, 1}
6-4. {3, 2, 1}
76
7-1. {3, 1, 1, 1}
7-2. {3, 1, 1, 1}
77
7-3. {3, 1, 1, 1}
8-1. {2, 2, 2}
78
8-2. {2, 2, 2}
8-3. {2, 2, 2}
79
9-1. {2, 2, 1, 1}
9-2. {2, 2, 1, 1}
80
9-3. {2,2,1,1}
9-4. {2, 2, 1, 1}
81
10-1. {2, 1, 1, 1, 1}
10-2. {2, 1, 1, 1, 1}
82
11. {1, 1, 1, 1, 1, 1}
Figure 3.15: 25 types of tetrahedra by partition type
We can use a partition pair by edges together with a congruence type partition to pin down
exactly which of the 25 types of tetrahedra we are looking at. We can classify faces by partition
type with regard to congruence.
There are 4 faces, and 4 can be partitioned in 5 ways.
Noting partitions of 4
{4}, {3, 1}, {2, 2}, {2, 1, 1}, and {1, 1, 1, 1}
we can classify tetrahedra using numbers of congruent triangles.
{4} 4 congruent triangles ,
{3, 1} 3 congruent; 1 congruent
{2, 2} 2 congruent; 2 congruent
83
{2, 1, 1} 2 congruent; 1 congruent; 1 congruent
{1, 1, 1, 1} 1congruent ;1congruent ; 1congruent ; 1 congruent ; 1 congruent .
For example, if we consider {3, 1}, there are three triangles congruent to each other, and another
triangle which is different, not congruent to the other 3.
If we consider {2, 1, 1}, there are two triangles congruent to each other, and another two
triangles which is different, not congruent to each other and the other 2 triangles.
E, I, 2S
{2, 1, 1}
Figure 3.16a: {3, 2, 1} tetrahedron regard to the numbers of congruent triangles
However, one could imagine an additional refinement to this system as in Figure 3.16a.
This is because triangles come in three types: equilateral, isosceles, and scalene.
When you see 2I, 2S, we are looking at a tetrahedron with two congruent isosceles triangles and
two congruent scalene triangles as faces (Figure 3. 16b).
84
2I, 2S
{2, 2}
Figure 3.16b: {3, 2, 1} tetrahedron regard to the numbers of congruent triangles
I use the convention that larger numbers in the partition appear first-thus, 2S, 1I, 1I rather than
1I, 1I , 2S using alphabetical order and also which also goes from" most symmetric"
(equilateral) to least symmetric" (scalene).
Each vertex of a tetrahedron can also be classified as to whether it is equilateral or isosceles or
scalene and by congruence type. This information is recorded in a table below. Thus, the one
type {6} also is labeled faces 4E, vertices 4E. Note that it is possible that we can have for faces:
1I, 1I, 1I, 1S and for vertices 1I, 1I, 1I, 1S but the letters involved can be different. Thus, the
triangles from the face point of view have labels aab, acc, bbc, abc but the vertices have the
labels aac, abb, bcc, abc .
85
Congruence Property by face and vertex
vertex
{4}
{3,1}
{2,2}
{2,1,1}
{1,1,1,1}
No
No
No
Face
{4}
{6}, {4,2}, No
{2,2,2}
{3,1}
No
{3,3}
No
No
No
{2,2}
No
No
{5,1}, {4,1,1}
No
No
{4,2},**{3,2,1)
No
{3,3}, {3,2,1}
{2,2,1,1}
{2,1,1}
No
No
No
{2,2,2},
{2,2,1,1}
{1,1,1,1}
No
No
No
No
{4,1,1},{3,2,1},
***{3,1,1,1}
{2,2,2},
**{2,2,1,1},
**{2,1,1,1,1},
{1,1,1,1,1,1}
E - Equilateral
I – Isosceles
S – Scalene
Table 3.1: Congruence Property by face and vertex
86
The number of stars implies the number of different face-vertex congruence types for this
partition, for example
for** {2, I, I} we have 2S E I and 2I I I. (Refer to page 87 and 89)
87
Congruent face partition with edge partition behavior
Face Edge
{4}
{3,1}
{2,2}
{2,1,1}
{1,1,1,1}
{6}
4E
No
No
No
No
1
{5,1}
No
No
2E 2I
No
No
1
{4,2}
4I
No
No
2I E S
No
2
{4,1,1}
No
No
2I 2I
No
EISS
2
{3,3}
No
3I E
2I 2I
No
No
2
{3,2,1}
No
No
2I 2S
2S E I
I III
4
E S SS
3
partition
2I I I
{3,1,1,1}
No
No
No
No
IISS
I II S
{2,2,2}
4S
No
No
2S I I
I II S
3
{2,2,1,1}
No
No
No
2S I S
I S SS
4
IISS
IISS
{2,1,1,1,1}
No
No
No
No
S SSS
2
I S SS
{1,1,1,1,1,1}
No
No
No
No
S SSS
1
3
1
4
5
12
25
Where E – Equilateral, I – Isosceles and S – Scalene.
Table 3.2: Congruent face partition with edge partition behavior
88
One can interpret the entry in the above chart {3, 3} by {3, 1} using the Figure 3.16b.
(5th row and 2nd column)
Figure 3.17: Congruent face partition with edge partition behavior
We have 3I E as the congruent triangle pattern type here.
89
Congruent vertex partition with edge partition behavior
Vertex
{4}
{3,1}
{2,2}
{2,1,1}
{1,1,1,1}
{6}
4E
No
No
No
No
1
{5,1}
No
No
2E 2I
No
No
1
{4,2}
4I
No
No
2I E S
No
2
{4,1,1}
No
No
2I 2I
No
EISS
2
{3,3}
No
3I E
2I 2I
No
No
2
{3,2,1}
No
No
2I 2S
2I I I
I III
4
E S SS
3
Edge partition
2S E I
{3,1,1,1}
No
No
No
No
IISS
I II S
{2,2,2}
4S
No
No
2S I I
I II S
3
{2,2,1,1}
No
No
No
2S I S
I S SS
4
SSII
IISS
{2,1,1,1,1}
No
No
No
No
S SSS
2
I S SS
{1,1,1,1,1,1}
No
No
No
No
S SSS
1
3
1
4
5
12
25
Where E - Equilateral
I – Isosceles
S – Scalene.
Table 3.3: Congruent vertex partition with edge partition behavior
90
Theorem1: A tetrahedron T cannot have exactly three congruent scalene triangles and one
equilateral triangle. Having (1E, 3S) tetrahedron is impossible.
Figure 3.18a: Congruent face partition with edge partition behavior
Proof:
From the given sides of six edges, three of them must be equal.
Let the three sides AB=AC=BC = a.
For ABD and ACD to be scalene two of AD, BD, and CD must be different from a, say b
and c.
By congruence type CD = BD=b and AD = c which is impossible.
91
Theorem 2: A tetrahedron T cannot have two faces which are each equilateral but have different
side lengths.
Figure 3.18b: Congruent face partition with edge partition behavior
Proof:
If one triangle T is equilateral (side a), the three other faces have an edge in common with T. If
any edge of these triangles is of length b ∑ a, these triangles can’t be equivalent.
Let the three sides of an equilateral triangle be represented by a AC = AD = CD= a and
let BC = b. Then by congruence type these triangles can’t be equivalent, which is impossible.
92
Theorem 3: A tetrahedron T cannot have exactly three equilateral triangles as faces.
Figure 3.18c: Congruent face partition with edge partition behavior
Proof:
From the given six edges, five must be equal in length.
Let AB = AC = BC = CD = BD = a
By congruence type AD = b which is impossible
93
Theorem 4: A tetrahedron T cannot have two congruent equilateral triangles and 2 scalene
triangles as faces.
Figure 3.18d: Congruent face partition with edge partition behavior
Proof: Having two congruent equivalent triangles means we have the situation in Figure 3.18.
Let the five sides AB = AC = AD = BD = CD = a
And BC = b ∑ a, then by congruence type this is impossible.
Remember if the triangular inequality holds for all triples of six lengths there may be no
tetrahedron with the given six edge lengths.
Another question is which integer sextuples can be edge lengths of tetrahedron?
This thesis not only enumerates the refined number of partition classes but also determines for
94
each of these classes of partition types if there is an integer collection of lengths for a tetrahedron
in this class exists.
Partitions of a tetrahedron with integer lengths
{6}
D(s) = 400000
95
{5, 1}
D(S) = 47200
{4, 2}
D(S) = 321408
96
D(S) = 329182
{4, 1, 1}
D(S) = 243000
97
D(S) = 270444
{3, 2, 1}
D(S) = 226750
98
D(S) = 218638
D(S) = 103390
99
D(S) = 202752
{3, 3}
D(S) = 657886
100
D(S) = 679936
{3, 1, 1, 1}
D(S) =1726200
101
D(S) =1598292
D(S) = 1795968
102
{2, 2, 2}
D(S) = 1968300
D(S) = 1904992
103
D(S) = 1747980
{2, 2, 1, 1}
D(S) = 285750
104
D(S) =202158
D(S) = 151092
105
D(S) = 564408
.
{2, 1, 1, 1, 1}
D(S) = 124558
106
D(S) =227200
{1, 1, 1, 1, 1, 1}
D(S) = 301150
Figure 3.19: Partitions of a tetrahedron with integer lengths
107
We see that all 25 partition types exist with integer lengths. Now we show that for the
{1, 1, 1, 1, 1, 1} partition type and the sextuple ( 28, 30, 35, 36, 42, 45), there are a total of 30
incongruent tetrahedra.
(The 30 Incongruent Scalene tetrahedron with integer length).For each case the value of the
McCrea determinant is given.
1.
2.
22500809
22278809
108
3.
4.
2785555008
5.
2314736352
6.
27898509
23412409
109
3.
4.
51468809
9.
55755809
51538909
10.
56182609
110
3.
4.
56667409
13.
57024209
14.
5335041996
5305835052
111
15.
16.
5804836812
17.
6013623852
18.
5717976012
59559709
112
19.
20.
62710809
21.
64419552
22.
63367009
58232709
113
23.
24.
64933209
58067109
25.
26.
64868209
6291509
114
27.
659359 09
29.
28.
5913984312
30.
64413309
59570509
Figure 3.20: Incongruent Scalene tetrahedron with integer length
115
(Ten incongruent tetrahedra with integer length sides of partition type {2, 1, 1, 1, 1} of a possible
15.) For each case the value of the McCrea determinant is given.
1.
2.
D(S) = 28450506
D(S) = 11274106
3
D(S) =60241706
4.
D(S) = 35838706
116
5.
D(S) = 55811506
7.
D(S) = -33321306
6.
D(S) = 36759906
8.
D(S) = 10224306
117
9.
D(S) = 43534106
10.
D(S) =19630106
11.
D(S) = -27049207
12.
D(S) =-27734707
118
13.
14.
D(S) = -3076996
D(S) = -2298707
15.
D (S) = 1912036
Figure 3.21: {2, 1, 1, 1, 1} incongruent tetrahedra with integer length
119
Examples with negative McCrea determinants don’t exist. Which values from 0 to 15 for this
partition type are obtainable can serve as a project.
The monotone up and monotone down series generate tetrahedra from a given sextuples S by
increasing and decreasing the sequence of the given edge lengths as shown in the Figure 3.22.
For example S = (35, 36, 42, 28, 30, 45) goes next to S = (35, 36, 42, 30, 30, 45) in the
increasing sequence and S = (35, 36, 42, 28, 30, 42) in the decreasing sequence.
D(S) = 2341239800
120
Monotone up
D(S) = 3337587000
Monotone down
D(S) = 3578189792
{2, 1, 1, 1}
{2, 1, 1, 1}
D(S) = 8432093950
D(S) = 6098540000
{3, 1, 1, 1}
{3, 1, 1, 1}
121
D(S) = 9966159000
{4, 1, 1}
D(S) = 23340101400
{5, 1}
D(S) = 46699915550
{4, 1, 1}
D(S) = 2703859200
{5, 1}
122
D(S) = 2703859200
{6}
D(S) = 1927561216
{6}
Figure 3.22 a: monotone up and monotone down series
D(S) = 2250079808
123
Monotone up
D(S) = 3270405600
{2, 1, 1, 1, 1}
D(S) = 8432093950
{3, 1, 1, 1}
Monotone down
D(S) = 3578189792
{2, 1, 1, 1, 1}
D(S) = 4931436128
{3, 1, 1, 1}
124
D(S) = 99661590000
{4, 1, 1}
D(S) = 23340101400
{5, 1}
D(S) = 46699915550
{4, 1, 1}
D(S) = 27033859200
{5, 1}
125
D(S) = 33215062500
{6}
D(S) = 1927561216
{6}
Figure 3.22 b: Monotone up and Monotone down series
Many interesting research questions and projects grow out of these ideas.
126
Degenerate Tetrahedra
A sextuple tetrahedron S = {a, b, c, d, e, f} is a degenerate tetrahedron if and only if D(s) is zero
and S satisfies the triangular inequality (the sum of two sides of a triangle is greater or equal to
the third side). From the sextuples of degenerate tetrahedra one can try to generate different types
of quadrilaterals with integer lengths.
The partition of {6} does not exist in the plane with integer or real number lengths. For the other
partitions of 6, namely {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2},
{2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1} one can find a quadrilateral with real number
lengths in the plane which can be interpreted as a degenerate tetrahedron.
The degenerate tetrahedron
(In this section the first diagram shows a degenerate tetrahedron and the second shows a
polygon)
{6} don’t exist as a degenerate tetrahedron in the plane.
127
{5, 1}
D(S) = 0
AD = 3 ⊕ 5 = 8.66
The complete quadrilateral above arises from the metric rhombus shown here.
Rhombus
128
Note that it can’t be realized with integer length.
This follows from the fact that a rhombus with equal diagonals must be a square.
{4, 2}
AB = CD =
Square
2 ⊕ 17 = 24.04
129
2-2. {4, 2}
AB = CD = 2 ⊕ 5 = 7.07
Square
130
{4, 1, 1]
Rhombus
131
3-2. {4, 1, 1}
Rhombus
132
4-1. {3,2, 1}
c
13
13
B
13
A
24.78
15
15
13
1.
u
A
1 !;
1.
D
Kite
133
4-2. {3, 2, 1}
This kite type can’t be realized using only integers.
Kite
134
5. {2, 2, 2}
Rectangle
135
6. {2, 2, 1, 1}
Trapezoid
136
7. {1, 1, 1, 1, 1, 1}
Figure 3.23a: Degenerate tetrahedra
To get another perspective on degenerate tetrahedra consider the following.
137
Figure 3.23c shows a "degenerate" tetrahedron because the lengths making up ∆ABC do not
obey the strict triangle inequality. This degenerate tetrahedron can be redrawn in the plane as
show in Figure 3.23b. In this diagram vertices A, B, C, and E are collinear.
Figure 3.23b: Degenerate tetrahedra
Figure 3.23c: Degenerate tetrahedra
If we consider triangle BCD in Figure 30 and flex it by moving vertices B and C towards each
other (bending the triangle along line segment AD) we can form a tetrahedron in 3-space.
We can think of this "flexing" as shortening the length of BC in Figure 3.23b. If we don't
shorten AD "too much" we will be able to get legal 3-dimensional tetrahedra for a range of
values where the length of AD is even an integer. Examples, of such tetrahedra with shorter
length values for BC are shown in Figure 3.24.
Labels now are for a “standard “labeled tetrahedron
138
Figure 3.24: labeled tetrahedron
If edge BC is shorted from 5 to 4, then again we will have a degenerate tetrahedron since the
triangle inequality again will no longer hold for ∆ABC.
139
CHAPTER IV
NETS OF TETRAHEDRA
Nets are obtained by cutting three edges of the tetrahedron at a vertex of the tetrahedron or along
a sequence of three edges that visit each vertex exactly once and flattening out the tetrahedron.
Thus, there are two types of nets: star nets and path nets
Star nets:
Star nets are obtained by cutting three edges of a tetrahedron at one vertex of the tetrahedron.
Figure 4.1: spanning cut tree and their nets
Path nets:
Path nets are obtained by cutting three edges of a tetrahedron along a sequence of three edges of
a tetrahedron that form a path.
140
Figure 4.2: spanning cut tree and their nets
The dark (Thickened) edges in Figures 4.1and 4.2, indicate the edges of a spanning tree which
are to be cut to obtain a net. A spanning tree is a sub graph of a graph which is a tree and
includes all the vertices.
Figure 4.3: spanning cut tree
141
There are potentially 16 distinct edges unfolding of a tetrahedron of which four are star nets and
twelve are path nets.
For example for the {5, 1} partition, we have 4 spanning cut trees leading to star nets and 12
spanning cut trees for path nets, some of which due to symmetry may yield the same net.
Figure 4.4: {5, 1} tetrahedron
Star nets:
For {5,1}, when we cut along the 4 spanning trees of a tetrahedron, which cut all of the three
edges at a single vertex, one will find 2 different star nets.
142
Figure 4.5a: star net for {5, 1}
Path nets
For a {5, 1} partition the 12 possible path nets give rise to three different path types: a, a, a, a,
b, a and a, a, b. Note b, a, a gives rise to the same net as a, a, b. Also for a {5, 1} partition there
are two types of a edges: an a edge with two a edges at each end and an a edge with two a’s at
one end and a and b at the other.
143
c
D
c
c
8
D
c
D
A
-- --- -A-
B
c
c
Figure 4.5b: path net for {5, 1}
144
Among the 16 spanning cut trees of a tetrahedron for partition type {5, 1} one can find 5 distinct
nets (Figure 5a and Figure 5b).
Partition type {2, 1, 1, 1, 1} has two “sub-types “when the two equal length edges form a path or
form a matching .We know that there can be a maximum of 4 star nets and 12 path nets for a
fixed tetrahedron. Fixed in the sense that one does not move around on the tetrahedron the edge
lengths (because when one does that for{2, 1, 1, 1, 1} there may be ten in congruent tetrahedra).
Type 1
Figure 4.6a :{2, 1, 1, 1, 1} type1 tetrahedron
Type 2
Figure 4.6b :{2, 1, 1, 1, 1} type 2 tetrahedron
1. For partition {2, 1, 1, 1, 1} where the two equal edges are a matching we have at most 4
spanning cut trees at vertices (star nets) and at most 12 spanning cut trees for a path nets.
145
Type 1
Figure 4.6a :{ 2, 1, 1, 1, 1} type 1 tetrahedron
Star Nets and Path Nets
Star nets
When we cut along the 4 spanning trees of a tetrahedron which cut all of the three edges at a
single vertex one can find 4 different star nets.
146
Figure 4.7a: Star net for {2, 1, 1, 1, 1} type 1 tetrahedron
147
Path nets
For ( 2, I, I, I, I } the 12 possible path nets give rise to 10 different path types: we have 10
different types of path nets (a, b, a), (a, c, a), (a, d, a), (a, e, a), (b, a, d), (b, c, d), (b, e, d), (c, a,
e), (c, b, e) and (c, d, e).
D
D
B
D
D
c
c
B
148
D
D
8
D
C
b
A
D
•J
0
149
8
D
D
c
e
b
A
D
b
A
150
D
A
D
D
Figure 4.7b: Path net for {2, 1, 1, 1, 1} type 1 tetrahedron
151
Among the 16 potential nets for a tetrahedron for partition of type {2, 1, 1, 1, 1} with two equal
edges in a matching one can find 14 distinct nets.
2. Partition type {2, 1, 1, 1, 1} where the two equal edges are a path.
We have 4 vertices spanning cut trees and 12 spanning cut trees for a path net.
Type 2
Figure 4.6b :{2, 1, 1, 1, 1} type 2 tetrahedron
Star nets
When we cut along the 4 spanning trees of a tetrahedron which cut all of the three edges at a
single vertex one can find 4 different star nets.
152
D
c
8
D
D
153
Figure 4.8a: Star net for {2, 1, 1, 1, 1} type 2 tetrahedron
Path nets
For {2, 1, 1, 1, 1} the 12 possible path nets give rise to 12different path types : ( a, a, d), (a, e, d),
(b, a, e), (b, c, e), (b, d, e), (c, a, a), (c, b, a), (c, d, a), (c, e, a), (d, b, a), (e, a, b) and (d, c, a) .
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D
D
C
D
B
155
A
D
156
D
c
A
c
8
8
c
157
Figure 4.8b: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron
Among the 16 potential nets for a tetrahedron for partition of type {2, 1, 1, 1, 1} with two equal
edges form a path of length two ) one can find 16 distinct nets .
For one tetrahedron with two paths with the same letters, the two paths can give rise to different
nets. Paths with two different letter patterns must give rise to different nets.
If we consider a tetrahedron with partition {2, 1, 1, 1, 1} type 2 with the same letters on different
paths (path nets b, a, e) they can have different nets. They both have hexagons with the same
pattern of edges.
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Figure 4.9a: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron
Figure 4.9b: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron
Could one argue that these two tetrahedron are the “same “because they had the same path net?
These two path nets if congruent would fold to the same tetrahedron but the tetrahedron are not
congruent and the complements of the path nets are not the same Type -1 (b, a, e; a, c, d) and
Type-2 (b, a, e; a, d, c). On combinatorial grounds these two tetrahedra can’t be the same.
If we consider Type 1and 2 partition of {2, 1, 1, 1, 1} tetrahedron with different letters of the
same path have different nets
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Figure 4.10a: Path net for {2, 1, 1, 1, 1} type 2 tetrahedron
Figure 4.10b: Path net for {2, 1, 1, 1, 1} type 1 tetrahedron
Theorem: If two paths have different letters for the same tetrahedron they can’t give rise to the
same net but two paths with the same letters can give rise to different nets. (See Figure 4.9)
If we consider partition type {2, 1, 1, 1, 1} tetrahedra with the same path and switch one pair of
edges in a matching edge one gets the dual tetrahedron and then their nets are the same.
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Figure 4.11: Dual tetrahedron with the same path for {2, 1, 1, 1, 1} type
From the above fact, the dual tetrahedron with the same path has the same net. The faces of their
nets are the same with different patterns of cut edges.
If we switches edge length bd (instead of edge length ce) in a matching edge one gets the dual
tetrahedron and their nets are the same.
161
Figure 4.12: Dual tetrahedra with a matching edge for {2, 1, 1, 1, 1} and their nets
For a tetrahedron if one takes a star net, one can always unfold to a non-overlapping polygon.
However for a path net there is no guarantee that one can unfold to a non-overlapping polygon.
There might be a spanning tree for a tetrahedron that when cut will lead to a way to open up the
tetrahedron which will overlap.
Can any tetrahedron be cut along a path and unfolded into the plane without overlap?
There is a conjecture (Fukuda) that for every tetrahedron the cut tree yields a simple unfolding.
In response to this conjecture that overlap couldn’t occur by F. Fukuda, M. Namiki found a
counterexample. An explicit example of this phenomenon was published by Branko Grünbaum
inspired by Namiki’s work.
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Recall that, according to Demaine and O’Rourke an edge unfolding of a polyhedron is a cutting
of the surface along its edges that unfolds the surface to a single, non-overlapping piece in the
plane. It has three characteristics:
* The unfolding is a simply connected piece.
* The boundary of the unfolding is composed of edges of the polyhedron.
* The unfolding doesn’t overlap.
Which partition types for tetrahedra admit spanning tree cuts which lead to overlap?
Grünbaum’s example is a type {1, 1, 1, 1, 1, 1}
There exists a {2, 1, 1, 1, 1} tetrahedral which can be cut along its edges DABC and has an
overlapping unfolding.
Figure 4.13a: {2, 1, 1, 1, 1} tetrahedron
163
I found an overlapping net for a tetrahedron with partition type {2, 1, 1, 1, 1}
Figure 4.13b :{ 2, 1, 1, 1, 1} overlapping tetrahedron
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CHAPTER V
SUMMERY, CONCLUSION AND RECOMMENDATION
A polyhedron is a 3-dimensional object that contains flat faces and straight edges. Examples
include the Platonic solids and tetrahedra in general, combinatorial cubes, and other convex
polyhedra. The tetrahedron is the simplest three dimensional solid, having four vertices, four
faces (all triangles) and six edges. This thesis investigates new mathematical properties of
tetrahedra motivated by showing novel approaches to traditional questions by mathematics
educators interested in geometry and how geometry is connected to other parts of mathematics.
New theorems about tetrahedra are developed with these educational goals as motivation.
At the start of this dissertation a variety of mathematical and mathematics education issues were
raised .Here I wish to address how these issues were resolved and offer ways to connect up the
new mathematical ideas with the educational issues implicit in them .Referring to the need and
purpose of this study, I have developed new mathematical results which connect up also the
knowledge base of students with the definitions and ideas of geometry acquired through high
schools and middle schools, with the goals:
1. Showing ways to transition from geometrical problems arising in 2-dimensions (2D) to ones in
3-dimensions (3D)
2. Showing connections between different mathematics topics (algebra, geometry,
combinatorics, etc.) that should be treated in a more integrated way
3. Enhancing sense making for the role of definitions in geometry and to design activities
containing exploration problems and classroom activities that enable students to study geometry
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by investigating, exploring, and establishing geometric conjectures, which eventually will
develop into proofs.
4. Showing that there are new mathematical questions about polygons and tetrahedra accessible
at the college and pre-college level.
With regard to the above goal I introduce Partitions of tetrahedron and the notion of a net for a
tetrahedron (polygon).Student rarely see this approach to polyhedra.
I discuss that nets offer an alternative way of approaching the kinds of geometrical question
raised about polyhedr in 3D, in the plane 2D. Sample ideas for implementing this appear in the
Appendix .Partition type of tetrahedra offers a way to classify tetrahedra, and to address
questions of research interest to mathematics education researchers and mathematicians while
offering K-12 students a “laboratory” to make models, and get insight into representation of
geometrical objects, visualizing these objects, and telling when the objects are the same or
different. New theorems about tetrahedra are developed with these educational goals as
motivation.
The following answers have been given to the questions raised at the start
1. How to classify tetrahedron into different types based on the notion of a partition.
I have shown the following new results. There are 11 types of tetrahedra based on the partition of
6 (Figure 3.14) and the refined approach taking some geometric information (but not relative size
of edges) into account leads to potentially 25 classes (Figure 3.15)
Theorem: There are 25 different partition classes of tetrahedra taking into account graph
theoretical aspects of the position of the edges, and all 25 types exist.
166
We can use partition notion based on the congruence property by face and vertex and congruent
face / vertex partitions with edge partition behavior (Table3.1, Table 3.2, and Table 3.3).
Theorem1: A tetrahedron T cannot have exactly three congruent scalene triangles and one
equilateral triangle. Having (1E, 3S) tetrahedron is impossible.
Theorem 2: A tetrahedron T cannot have two faces which are each equilateral but have different
side lengths.
Theorem 3: A tetrahedron T cannot have exactly three equilateral triangles as faces.
Theorem 4: A tetrahedron T cannot have two congruent equilateral triangles and 2 scalene
triangles as faces.
2. Existence questions for tetrahedron with integer lengths.
For all 11 types these types can be realized with integers. This extends to the refined 25 classes.
If S is a sextuple for (a potential) tetrahedron T, S = (a, b, c, d, e, f) then T has faces a, b, c; a, e,
f; b, d, f and c, d, e and the edges at the vertices has the pattern a, b, f; a, c, e; b, c, d and d, e, f .
If we have a potential tetrahedron T ' and where the pattern of faces and vertices is interchanged
then T ' is called the dual of tetrahedron T.
Theorem: If the potential tetrahedron T has sextuple S = (a, b, c, d, e, f) then the sextuple ( f, e, d,
c, b, a ) gives rise to the potential dual tetrahedron.
3.Relating the existence of nets of tetrahedra to the type of spanning tree that is used to generate
the net, and under what circumstances attempts to generate a net for a tetrahedra fail due to
overlap of the faces when the tetrahedron is flattened into the plane.
167
Nets are obtained by cutting three edges of the tetrahedron at a vertex of the
tetrahedron or along a sequence of three edges that visit each vertex exactly once
and flattening out the tetrahedron. Star nets can’t result in overlaps; only for path
nets can this phenomenon occur.
4. Which partition types of tetrahedra admit spanning tree with cuts that lead
overlaps?
Grunbaum, in my notation showed there is a {1, 1, 1, 1, 1, 1} tetrahedron that
leads to an overlapping “net “when a path is cut. I have shown there also exists a
{2, 1, 1, 1, 1} tetrahedron .
The {6} partition can’t lead to an overlap.
There exists a {2, 1, 1, 1, 1} tetrahedra (Figure 4.13a) which can be cut along its
edges and has an overlapping unfolding (Figure 4.13b).
Recommendation
There is still the need to extend the pedagogical examples to support the evolving changes in
geometry occasioned by the Common Core State Standards in Mathematics. Additional ideas
related to those developed in the appendix can be used to show that mathematics, geometry in
particular, is a dynamic subject where one can get new insight into well studied objects
( tetrahedra ) as well as directions for further insights (nets). Thus educators can use the study
using the mathematics provided and the associated material in the Appendix.
An important open problem that is accessible to researchers, mathematics educators,
mathematician, and undergraduates or high school students is to determine for each of the 25
partition classes what are the admissible numbers of incongruent tetrahedra for the given
class?
168
For {1, 1, 1, 1, 1, 1} the answer includes 0 and 30 but is there a set of lengths with exactly say,
23 incongruent tetrahedra?
An interesting research question is to see what other partition types of tetrahedra allow cut trees
that lead to overlapping faces.
169
REFERENCES
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Approaches to Classifying triangles and Quadrilaterals (Doctoral dissertation, Teachers College,
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Alonso, Orlando B., & Joseph Malkevitch. “ Enumeration via partitions." Consortium 98 (2010):
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Benton, A., & O’Rourke, J. (2008).A class of convex polyhedra with few edge un folding. arXiv
preprint arXiv: 0801.4019.
Board of Regents of the University of the State of New York. (2005). NYS Mathematics Core
Curriculum, MST Standard 3. Retrieved March 25, 2009.
http://www.emsc.nysed.gov/ciai/pub/pubmath.html.
Dekster, BV &Wilker, JB, ‘Edge-lengths guaranteed to form a Simplex’, Arch. Math. 49 (1987),
351- 366.
De Villier, M.D. (1996). Some adventures in Euclidean geometry, Durban, South Africa:
University of Durban West Ville.
Demaine E &O’Rourke,J. (2007). Geometric Folding algorithms: linkages, origami, polyhedra.
(Cambridge University Press, New York).
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Demaine, E. D., Demaine, M. L., Itoh, J.I.,Lubiw, A., Nara, C., & O’Rourke, J. (2012, March).
Refold Rigidity of Convex Polyhedra. In 28th European workshop comput. Geom. (EuroCG) (pp
273-276).
Dressler, I. (1973). Geometry review guide. Amsco school pubns, INC. New York.
Grünbaum, B., & Johnson, N.W. (1965). The faces of a regular-faced polyhedron. Journal of the
London Mathematical society, 1(1), 577-586.
Grünbaum, B. (2009). Configurations of points and lines (vol. 103).American Mathematical
Society.
Hilbert, D. (1950).The foundations of Geometry, translation by e. J. Townsend. Reprint.
Miller, J.B. (2007).Plane Quadrilaterals. Australia Mathematical Society. Gazette.
O’Rourke. J. (2011). Common edge-un zipping for Tetrahedra.arXiv preprint arXiv: 1105.5401.
Prenowitz, W., & Jordan, M. (1989).Basic concepts of geometry.Rowman& Little field
publishers.
Tucker, A. (2007). Applied Combinatories , Danvers, MA .John Wiley & Sons, Inc.
Usiskin, Z & Griffin, J. (2007). The classification of Quadrilaterials: A study of Definition (
Research in Mathematics education ).Information Age Publishing ,Inc.
Wirth, K., & Dreiding, S. (2009). Edge length determining tetrahedron. Elementeder
Mathematik, 64(4), 160-170.
William H. McCrea. (2006). Analytical geometry of three dimensions.(Courier Dover
Publications).
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Zalgaller, V. A. (1969). Convex polyhedra with regular faces. Zapiski Nauchnykh Seminarov
POMJ, 2, 5-221.
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APPENDICES
NEW IDEAS TO AUGMENT TRADIONAL TOPICS AND PEDAGOGY RELATED TO
POLYGONS AND POLYHEDRA (TETRAHEDRA)
Appendix A
Geometry in School
The new mathematics developed in chapters 1-4 has implications for teaching geometry in the era
of the common core state standards in Mathematics –CCSS-M. I will be concerned with the
important ideas touched on in the CCSS-M that involve geometry and connections between
geometry and other parts of the middle and high school curriculum. In particular, I will also treat
the way one moves from the simplest polygon, a triangle, to more complex polygons like
quadrilaterals and n-gons. This is a change within the dimension of the object (2D). I will also be
concerned with the transition between 2D and 3D. Thus, a triangle is a 2D idea, while its natural
generalization to 3D is the tetrahedron. Tetrahedra are a special case of a more general kind of
3D object, namely polyhedra. Let me give background for these issues by giving a brief account
of the history of geometry, in particular, as the history relates to the concepts of polygon and
polyhedra and what students are encouraged to learn about these geometrical objects.
For thousands of years civilized humanity has needed to know how to work with the size,
shapes, or position of things in order to help him solve many of the practical problems of the day.
We devised methods of measuring line segments, angles and surfaces. (Dressler)
Geometry begins with a practical need to measure shapes. The word geometry means to measure the earth
and has come to concern itself with the science of shape and size of things, as well as visual phenomena.
Around 2000 BC the first Egyptian pyramid was constructed. Knowledge of geometry was essential for
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building pyramids which consisted of a square base and triangular faces. Actually, these pyramids have
steps. The triangular faces are not flat. They only look that way from a distance. The earliest record of a
formula for calculating the area of a triangle also dates back to 2000 BC.
The Egyptians (5000-500 BC) and the Babylonians (4000-500 BC) developed practical geometry to solve
everyday problems but there is no evidence that they logically deduced geometric facts from basic
principles.
It was the early Greeks (600 BC – 400AD) who developed the principles of modern geometry.
Noteworthy, is the work of Thales of Miletus (624-547BC). Thales is credited with bringing the
science of geometry from Egypt to Greece. Pythagoras (569-475BC) was a Greek geometer and
is regarded as the first pure mathematician to logically deduce geometric facts from basic
principles.
Euclid of Alexandria (325-265BC) was one of the greatest of all the Greek geometers and is
considered by many to be the father of modern geometry. His book Elements is one of the most
important works in history and had a profound impact on the development western civilization.
Euclidean geometry today is a study of geometry based on definitions, undefined terms (points,
lines and planes) and the assumptions (axioms) stated. However, it is widely viewed that rather
than being a fully axiomatic system in the modern sense, that Euclid was codifying the geometry
of the space that one sees around one in daily life. One can view geometry as either a branch of
mathematics or a branch of physics. Euclidean geometry as presented in the Elements is based on
five basic notations and five fundamental axioms. However, contrary to what would be done
today, Euclid defines such words as point and line. His axioms are an attempt to assert as little as
possible without proof and everything else results by deductive reasoning from the rules of logic.
Euclidean Geometry enables us to visualize space and provide the tools often used to understand
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the space we live in. However, physicists are still unsure of the exact nature of the space we live
in. Thus, we may, in fact, live in a non-Euclidean space even though in most applied settings we
can use results from Euclidean geometry to build skyscrapers and bridges.
The next great development in geometry came with the development of non-Euclidean Geometry
however almost nothing is done with non-Euclidean Geometry in K-12.
The importance of geometry, specifically three dimensional / space geometry, will give an
enormous amount of attentions in the school system to graphical representation and mental
visualization in terms of developing an understanding of geometric relationships in a plane and
in space and the ability to think creatively and critically in both mathematical and nonmathematical situations.
Carl Fredrick Gauss (1777-1855) who along with Archimedes and Newton is considered to be
one of the three greatest mathematicians of all time, invented non-Euclidean geometry prior to
the independent work of Janos Bolyai (1802-1860) and Nikolai Lobachevski (1792- 1858) .
Geometry in the Era of the CCSS-M
From the NCTM and of Regents of the University of the State of New York, students need to
develop a better way of understanding of the definitions, properties and relations of geometrical
objects in a plane and in space and integrate geometry with arithmetic, algebra, and
trigonometry. The current New York State curriculum documents emphasize the use of terms
such as investigate, explore, discover, conjecture, reasoning, arguments, justify, explain, proof,
and apply as performance indicators in developing students mathematical reasoning ability in
connection with the necessity of the development of students skill in relation with the core
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curriculum process strands: representation, connections, communication, reasoning and proof,
and problem solving.
The study of geometry is categorized in terms of shapes (circles, triangles, and squares), figures
and positions in space and measurements and comparison of lines, angles, points, planes and
surface, and solids composed of combinations of these. A shape is the form of an object or figure
such as a circle, triangles, squares, rectangles, parallelogram, trapezoid, rhombus, octagon,
pentagon and hexagon. For the most part polygons will be thought of as "rod structures" and
when drawn in the plane. A polygon will be the boundary between two regions, the inside of the
polygon (a bounded region) and the exterior of the polygon which is an unbounded region. (This
is in essence what is known as the Jordan Curve Theorem for polygons.) A solid is a three
dimensional figure such as cube, cylinder, cone, prism or pyramid; other solid shapes include the
tetrahedron, octahedron and dodecahedron.
Below are indicated pedagogical and mathematical approaches to the study of geometrical objects
and their classifications using the tetrahedron as a specific example in the school . The activities
are organized within four units. Most of the activities are preceded with the background
information to guarantee students can have active and successful participation in the solution of
problems with different levels of difficulty through cooperative learning work.
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Appendix B
Background information
Unit I: Points, lines and polygons
A point is represented by a dot and capital letters will be used to denote points. A point has no
length, width or thickness. It only has location.
Figure 1.1: A point
A line is set points which is “straight “rather than curve. A line has infinite length.
Figure1. 2: A line
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A line segment is a set of all points on the line between two end points.
Figure 1.3: Line segments
A plane is a set of points that form a completely flat surface extending indefinitely in all
directions. Planes are denoted by Greek letters.
Figure 1.4: A plane
Collinear points: a set of points which lies on the same line.
Figure 1.5: Three collinear points
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If A, B, and C are collinear then AC + BC = AB.
XY Will denote both the segment XY and its length.
Coplanar points: a set of points which lies on the same plane
Figure1. 6: Coplanar points
A Ray is a set of all points in a half line. It begins at a point and extends infinitely in one
direction. AB and AC are examples of a ray.
Figure 1.7: Rays
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Congruent segments are segments that have equal measure length.
AB 〈 CD
Figure 1.8: Two congruent segments
The midpoint of a line segment is the point of a line segment which divides the segment into two
congruent segments.
AC 〈 AB
Figure 1.9: Mid point of line segment
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An angle is a set of points which is formed by two rays that share a common end point.
Figure 1.10: Angle
A right angle is an angle whose measure is 90 degrees.
B is a right angle.
Figure1.11: Right angle
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An acute angle is an angle whose measure is greater than 0 and less than 90 degrees. Figure 1.12
has acute angles at A, B and C.
Figure 1.12: Acute angle
An obtuse angle is an angle whose measure is greater than 90 and less than 180. Figure 1.13 has
an obtuse angle at B.
Figure 1.13: obtuse angle
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A reflex angle is an angle whose measure is greater than 180 and less than 360 degree. Figure
1.14 has a reflex angle at B.
Figure 1.14: Reflex angle
Two angles are adjacent if they have a common vertex and common side but don’t have any
interior points in common. In Figure 1.15 <ABC and <CBD are adjacent angles .
Figure 1.15:Adjacent
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Two angles which have vertical angles are a common vertex and whose sides are two pairs of
straight lines. EA, ED, EC and EB are rays. AB and CD are two straight lines.
Figure 1.16: Vertical angles
Complementary angles are two angles whose measure sum to 90 degree.<ABC and <CBD are
complementary when B is a right angle .
<ABD = <ABC + <CBD
Figure 1.17: Complementary angles
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Supplementary angles are two angles whose measures sum to 180 degrees.
<ACD + <BCD = 180 degree .
Figure 1.18: Supplementary angles
Activity:
1. Which one represents a point, a line and a plane?
Figure 1.19: Point, line, and Plane
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2. Use your ruler to find the length of each line segment in Figure 20 and write your answer
Figure 1.20: Line segment
3. Are segments AB and CD in Figure 20 congruent?
4. Draw a line segment AB and mark C as a midpoint of AB such that AC 〈 CB
5. Draw a plane containing five points A, B, C, D and E with exactly three of the points A, C and
D on a line. What is relative position of the these points you need a diagram
6. List all the different segments with
a) 3 collinear points
b) 4 collinear points
c) 5 collinear points
e) 7 collinear points
f) Can you create a formula for the number of different segments?
7. If <ABC bisects < ABD then draw the diagram <ABD with angle bisector BC.
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8. Given the polygon
Figure 1.21a: Polygons
List the vertices of the polygon in clockwise consecutive order starting at vertex B.
9. Which angles in Figure 21 are acute? Obtuse? Reflex?
A polygon is a closed figure in a plane formed by connecting line segments.
The line segments are the sides of the polygon. The end points of the line segments are vertices
of the polygon.
187
Figure 1.21b: Polygons
A diagonal of a polygon is a line segment that connects two nonconsecutive vertices of the
polygon. AC and BD are diagonals of the polygon ABCD.
Figure 1.22: Diagonal of Polygons
188
Note that a convex polygon is a polygon each of whose interior angle measure less than 180
degrees.
A non-convex polygon is a polygon which has at least one interior angle that measure more than
180 degree. Note there is a reflex angle at A.
Figure 1.23: Non Convex polygon
An equilateral polygon is a polygon that has congruent sides.
Figure 1.24: Equilaterial Polygon
189
An equiangular polygon is a polygon that has congruent angles.
Figure 1.25: Equiangular polygon
A regular polygon is a polygon that has congruent angles and congruent sides.
Figure 1.26: Regular polygon
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A triangle is a polygon that has three sides.
Figure 1.27: Triangle
A, B, C are vertices of a triangle. AB, BC, AC are sides of a triangle.
The altitude of a triangle is a line segment which is drawn from any vertex of the triangle
perpendicular to one of the opposite sides.
191
Properties of Quadrilateral
Kite
Figure 1.28: Kite
DC=BC, AB=AD, AO ∑ OC if BD ) AC = O then the quadrilateral is convex .
A=
d1 d 2
2
d
1
and
d
2
are the diagonals.
192
Isosceles trapezoid
Figure 1.29: Isosceles trapezoid
AB
CD, <D= <C AD=BC, AC =BD, MN is the line of symmetry through midpoint of its base
(M and N are a line segment between AB and CD), C and D, and A and B are reflection images
each other.
A= ½ h ( b1 + b2 ) where AB = b1 and CD = b2
Rhombus
Figure 1.30: Rhombus
193
AB
CD and AD
BC, <B=<D and <A = <C AC BD and BD bisect <D and <B,
AD=BC=AB=DC
A= bh
(AC=BD for a square only).
Parallelogram
Figure 1.31: Parallelogram
AB
A=bh
CD and AD
BC ABD 〈 CDB, <D+<C=180 0 , AD=BC, AB=CD
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Rectangle
Figure 1.32: Rectangle
AB
CD and AD
BC, AB=CD and AD=BC, <A=<B=<C=<D=<90, AC =BD .
A = LW where L is length and w is width
Square
Figure 1.33: Square
AB=DC=AD=BC,<A=<B=<C=<D=<90
A= S x S where S is a side of the square.
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Students can look at it the issue of the existence and properties of triangles with specified edge
lengths, area formulas for triangles and quadrilaterals and integer realization of triangles
(including Pythagorean triples) and quadrilaterals.
If A, B, C are distinct and non collinear points the set union AB BC AC A B C is
ABC ) . A, B, C are the vertices of a triangle ABC and, AB, BC, AC are
called triangle ABC ( 
sides of the triangle. Furthermore <ABC, <BCA and <CAB are angles of the triangle.
Figure 1.34: Triangle
Triangles are classified into three types in terms of sides and angles (Figure 1. 35)
Scalene triangle: a triangle that has no congruent sides.
Isosceles triangle: a triangle that has at least two congruent sides.
Equilateral triangle: a triangle that has three congruent sides.
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Figure 1.35: Equilateral, Isosceles, and Scalene triangles
Note that under this definition a triangle that is equilateral is also isosceles.
Inequalities in a triangle:
Theorem: The sum of the lengths of two sides of a triangle is greater than the length of the third
side.
AB +BC > AC, BC+AC>AB and AC+AB>BC
Theorem: If two sides of a triangle are unequal, the angles opposite these are unequal and the
greater angle lies opposite the greater side.
Figure 1.36: Triangle
If AC is greater than BC then angle ABC > angle BAC
Theorem: If two angles of a triangle are unequal, the sides opposite these angles are unequal and
the greater side lies opposite the greater angle.
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In ABC, if BAC > ABC ABC then the measure of the side opposite < BAC is greater than
the measure of the side opposite <ABC or BC > AC
Congruent Triangles: Two polygons are congruent if there is a one to one correspondence
between their angles and sides.
. All pairs of corresponding angles are congruent
. All pairs of corresponding sides are congruent
Note: Triangles that have equal angles need not be congruent.
In ABC and DEF if
Figure 1.37a: Congruent triangles
1) ⊕ D , B ⊕ E and C ⊕ F
2) ⊕ DE, BC ⊕ EF and AC ⊕ DF , then ABC ⊕ DEF
Postulate (SAS): Two triangles are congruent if two sides and the included angle of one triangle
are congruent respectively to two sides and the included angle of the other.
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Figure 1.37b: Congruent triangles
AB ⊕ AC, AD ⊕ AD, BAD ⊕
CAD and then ABD ⊕ ACD
Postulate (ASA): Two triangles are congruent if two angles and the included side of one triangle
are congruent respectively to two angles and the included side of the other.
Figure 1.38a: Congruent triangles
ACD ⊕ BCD, CD ⊕ CD and ADC ⊕ BDC then 
ACD ⊕ BCD
Postulate (SSS): Two triangles are congruent if the three sides of one triangle are congruent
respectively to the three sides of the other.
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Figure 1.38b: Congruent triangles
AD ⊕ BD, AC ⊕ AB and CD ⊕ CD then ACD ⊕ BCD
Note SSA does not guarantee congruence.
1
The area of a triangle (figure 1.39) A= bh where are A is the area, b is base of the triangle and
2
h is the height of the triangle
Figure 1.39: Right angle triangle
A Heron’s triangle is a triangle whose side lengths and area are all rational numbers. The Greek
mathematician named Hero found a formula for the area of a triangle:
200
s(s a)(s b)(s c) and s= a + b + c where A is the area of the triangle a, b, and c are
2
the lengths of the three sides of the triangle, and s is the semi-perimeter.
A=
Theorem: An equilateral triangle with rational side length can’t have rational area. No equilateral
triangle is Heronian.
Figure 1.40: Heronian triangles
x2 + y 2 =
A=
(x a )2 + y 2
=a
s(s a)(s b)(s c)
3a  a  a  a a 2 3
=
= 〉 〉 〉 〉  =
4
∫ 2 ∫ 2 ∫ 2 ∫ 2 
Pythagorean Theorem: For the three sides of a right triangle the square of the hypotenuse is equal
to the sum of the squares of the two sides (legs). The theorem can be written in equation form
a2 +b 2  c 2 where a and b are the length of the two sides and c is the length of the hypotenuse.
Any triangle whose side lengths for a Pythagorean triple is Heronian.
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Pythagorean Triple: Suppose that m and n are two positive integers with m < n then
a = n 2 m 2 , b = 2mn, c = n 2 + m 2 is a Pythagorean triple. One can simply generate the
Pythagorean triple.
For example n= 4 and m =3 then n 2 m 2 = 16- 9 = 7, 2mn = 24 and n 2 + m 2 = 25 and
7 2 + 24 2 = 25 2
Activity
1. Which of the following number triples may be used as the lengths of the sides of a triangle a)
(7, 4, 2) b) (8, 5, 3) c) (9, 6, 4) d) (16, 12, 3) e) (16, 14, 12)
2. Find the area of a triangle whose base and altitude are 12 inches and 8 inches.
3. If the area of a triangle is 12 sq inches and if one side of a triangle is 6 inches then find the
length of the altitude to that side.
4. Find the area of an isosceles right triangle whose hypotenuse is 4 2 .
5. If the sides of a triangle have lengths of 3, 4, and 5, find the area of the triangle.
6. In ABC, <ACB = 90 0 and <ABC=40 0 . Name the shortest side of this triangle.
7. In ABC, <BAC = 84 0 and <ABC = 48 0 , which is the longest side of the triangle.
8. If the length of the sides of ABC are AB = 15, BC =9 and AC = 16 then find the area of 
AB.
9. In ABC, AB =10, BC =12 and AC=16, name the largest angle of a ABC, and find the
area of the triangle.
10. If the coordinates of the points are given by P(0,0) , Q(3,0) and R(3,4) then find the area of 
ABC using Heron’s formula?
11. Can the integer side length of Heron’s triangle always have integer coordinate?
Here I develop some ideas to set forth the properties of quadrilaterals in order to set the stage for
comparing and contrasting the behavior of polygons (2D) with polyhedra (3D).
A quadrilateral (quad-four and lateral – side) is a polygon with four sides (or edges) and four
vertices or corners. If we consider a quadrilateral ABCD (Figure 44 ) we can (sometimes ) flex it
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to a tetrahedron along AD or BC or sometimes it becomes a degenerate tetrahedron , when we
add the segments AD and BC .
Figure 1.41a: Quadrilaterals
Intuitively, a quadrilateral is just any four sided, shape. There are many ways to classify
quadrilaterals. One approach due for Miller is: Plane quadrilaterals can be classified into types as
follows:
Convex in which the two diagonals are internal and intersect
Non-Convex and not self-Intersecting (‘dart’) in which one diagonal is internal and one
external, the two not intersecting the external diagonal spanning the “concavity.”
Self-intersecting (Zigzag) in which one pair of opposite sides intersect and both diagonals
are” external “and don’t intersect.
* (Partially) degenerate, in which a particular two adjacent side lie in the same line.
- A flag, where one vertex is an internal point of a side and two sides overlap.
- A triangle where two adjacent sides are in one straight line but not overlapping.
- A bent line where two opposite vertices coincides.
Fully degenerate, in which the whole figure is contained in one dimension.
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Convex
dart
Zigzag
Flag
bentline
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fullydegenerate
Figure 1.41b:Quadrilaterials
Quadrilaterals with self-intersection are called complex while those with no self-intersection are
called simple. Simple quadrilaterals are either convex or concave and their interior angles are
add up to 360 0 (angle sum formula (n-2)180).
Quadrilaterals are classified into eight types in school geometry by their properties:
Parallelogram
Area=base x height
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Rectangle
Square
Area = Length X width
Area = Side X Side
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Rhombus
Kite
Area = base X height
Area =
d1d 2
2
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Isosceles trapezoid
Area = ½ h ( b1 + b2 ) where
b1 = BV and b2 = ZW
Figure 1.41c: Quadrilaterials
According to Usiskin & Griffin : Hierarchies of Quadrilaterals .
Figure 1.42: Hierarchies of quadrilaterals
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A quadrilateral is a parallelogram if and only if both pairs of opposite sides have the same length
and are parallel
– Both pairs of opposite angles have the same measure
–Its diagonals have the same midpoint (bisect each other)
- It possess rotation symmetry
-Two consecutive angles are supplementary
A quadrilateral is an trapezoid if one pair of opposite sides is parallel
A quadrilateral is Isosceles Trapezoid if the non parallel sides are congruent and their base
angles are equal measure
A quadrilateral is rectangle if it has all the properties of a parallelogram
–If it contains four right angles
–The diagonals of a rectangle are congruent
A quadrilateral is rhombus if it has all the properties of a parallelogram,
It is an equilateral
- The diagonals are perpendicular to each other
-Their diagonals are bisecting its angle.
A quadrilateral is Square if it has all the properties of a Rectangle
– If it has all the properties of a rhombus
A quadrilateral is kite if two pairs of adjacent sides are of equal length.
Theorem: a quadrilateral is cyclic If and only if the perpendicular bisectors of all its sides are
concurrent.
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Cyclic Quadrilateral is a cyclic Quadrilateral is a Quadrilateral whose vertices all lie on a single
circle
Figure 1.43a: Cyclic quadrilaterals
According to Usiskin and Griffin: A cyclic quadrilateral possesses two important properties,
Ptolemy’s theorem and Brahmagupta’s Theorem.
Ptolemy’s theorem: In the cyclic quadrilateral ABCD, AC x BD=AB.CD+AD.BC (In a cyclic
quadrilateral, the product of the lengths of the diagonals equals the sum of the products of the
lengths of its opposite sides)
Brahmagupta’s Theorem: The area of a cyclic quadrilateral with sides a, b, c, and d and semi
perimeter s is
( s a)(s b)(s c)(s d ) where s =
a+b+c+d
2
S is the semi perimeter.
According to Miller’s for a given ordered tuple ( a, b, c, d ) of four positive numbers a
necessary and sufficient condition for the existence of a convex Quadrilaterals having those side
lengths is each variable is less than the sum of the remaining three.
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Brahmagupta’s Theorem is a generalization of the better known Heron’s formula for the area of
a triangle that is often found in high school geometry texts. Let d=0 in Brahmagupta’s formula
by bringing two vertices of the cyclic quadrilateral together and Heron’s formula appears. So a
triangle can be thought of as a degenerate cyclic quadrilateral.
Figure 1.43b: Cyclic quadrilaterals
Using Ptolemy’s theorem, Pythagorean Theorem and Heron’s formula one can generate the
length of the quadrilateral and be able to find the area and perimeter of the quadrilateral.
Activity:
1. Find the area of a rectangle whose width is 4inch and whose length 5 inch.
2. Find the base of a rectangle whose area is 20 square inch and whose altitude is 5 inch.
3. The perimeter of a rectangle is 24 inch. Find the area of rectangle if one of the sides is 6 inch.
4. Find the area of a square whose side is 8 inches.
5. Find the area of a square whose perimeter is 36 inches.
6. Find the area of the parallelogram whose base and altitude are 4 inches and 6 inches.
7. The area of a parallelogram is 30 square inches and the base is 10 inches.
Find the altitude.
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8. Find the area of a rhombus one of whose angles is 60 degree and whose shorter diagonal is 6
inches.
9. Find the area of trapezoid whose bases 8 inch and 4 inches and height is 6 inches
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Appendix C
Unit II: Classification of degenerate tetrahedra in the Euclidean plane
A partition of a positive integer n is a way of writing n as a sum of positive integers and is
denoted by the list of positive integers in the sum.
For example {2, 1} is a partition of 3.The partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1},
{1, 1, 1, 1}.
Activities
1. List the partition of 3.
2. If one interprets the partition {1, 1, 1} as a triangle with unequal length sides (scalene
triangles) what type of triangle do the other partitions of 3 represent?
Nets of a tetrahedron are intuitively two dimensional patterns that can be folded to form a three
dimensional figure. We can study nets of polygons, one dimension down from the usual
discussion.
Figure 1.44:Traiangle
If i cut at A we will get ABCA
If i cut at B we will get BCAB
if If i cut at C we will get CBAC
3, 2, 4 = 4, 2, 3 = ACBA
2, 4, 3 = 3, 4, 2 = BACB
2, 3, 4 = 4, 3, 2 = CABC
Thus , a triangle which is scalene will have three different “segment” nets .
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3. Is it possible to have six stick lengths each triple of which obeys the strict triangles inequality
but for which one can’t make a tetrahedron?
4. If we cut ABC at each vertex, how many “nets” of a triangle are there? If the length of three
sides are different, if the length of two sides are equal and the third side is different, if the length
of three sides are equal. List all of them.
A sextuple S= (a, b, c, d, e, f) is a degenerate tetrahedron if and only if the McCrean determinant
of S is zero and S satisfies the triangular inequality (the sum of the two sides of a triangle is
greater or equal to the third side). Degenerate tetrahedra exist for each of the partition {5, 1},
{4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1}.
Activities
1. How many partitions are there?
a) For {6}
b) For {4}
c) List all partitions of 6 and 4 numbers
2. How many different subsets of size three are there for six different numbers? What happen
when repeats are allowed?
3. If one has four six sticks of different lengths and two lengths the same, how many different
triples can one form?
4. If one has three of six sticks of different lengths and three lengths are the same, how many
triples can one form?
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5. Four points determine six different Euclidean distances. For partitions of six can we find a
collection of four points (no three of them are collinear) forming a quadrilateral in a plane which
corresponds to the partitions. Draw a quadrilateral for each partition which can be exists?
6. Explain how one can interpret a plane quadrilateral as a degenerate tetrahedron.
7. Give example of convex quadrilateral for each partition type of with integer lengths are there.
8. What are the non-congruent two distinct distances sets formed by 4 points in the Euclidean
plane?
Examples:
2 distance set
2 distance set
2 distance set
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3 distance set
3 distance
Figure 1.45: Degenerate tetrahedra
216
Appendix D
Unit III: Nets of tetrahedra
Nets of tetrahedron are obtained by cutting three edges of the tetrahedron at a vertex of the
tetrahedron or along a sequence of three edges that visit each vertex exactly once and flattening
out the tetrahedron.
Activities
1. How many types of nets can one find for a given tetrahedron?
2. How many nets can one find for partition {5, 1}? List the nets.
3. How many nets of tetrahedron are there for partition (4, 2)? List all the nets?
4. If we switch a pair of edges in a matching edge one gets a dual tetrahedron. Do these
tetrahedra have the same nets? List the net?
5. If two paths of a tetrahedron have the same letters then do they have the same /different path?
Can you give an example for partition {2, 1, 1, 1, 1}.
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Appendix E
Unit IV: Classification of tetrahedron in 3D
Tetrahedron lie in one of 11 classes and these classes exist as 3D type but not all as degenerate
2D types. Tetrahedra (6 edges ) can be classified into eleven classes by partition type in terms of
their edges {6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1},
{2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}.
Activity
1. List all partitions of the number six.
2. Draw a diagram to represent a tetrahedron in a plane.
3. How many faces, edges and vertices are there for a given tetrahedron?
List all the edges faces and vertices of the tetrahedron.
4. a) What is the smallest series of consecutive positive integers where every triple satisfies the
strict triangle inequality interpreting the integers as side lengths?
b) Does every triple from {1, 2, 3, 4} obey strict triangle inequality?
c) Which triple from {3, 4, 5, 6, 7, 8} don’t obey the strict triangle inequality ?
5. How can one tell if two tetrahedron with the same length of six edges are congruent or
different?
6. Can one may be a tetrahedron with the edge lengths {1, 2, 3, 4, 5, 6} for partition
{1, 1, 1, 1, 1, 1}? Can one label a plane drawing of a tetrahedron with these numbers so that the
faces obey the triangular inequality?
7. What is the significance of a negative value of the McCrea determinant of a tetrahedron with a
given set of edge lengths?
8. For each partition type of tetrahedron how many equilateral, isosceles and scalene triangles are
there? List all the partitions and types triangles.
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9. If S is a sextuple for which a tetrahedron does exist when does the monotone up and monotone
down series exist with integer length? Take one tetrahedron with integer length and list all the
monotone up and monotone down series.
10. Does the set S = {6, 7, 8, 9, 10, 11} generate all 30 scalene tetrahedra?