TOPIC 5
VISUALIZATION AND LOCUS
One of the important skills for the study of geometry is the ability to visualize the relative
positions of objects in two and three dimensions.
There are some assumptions that we can make about a drawing that are valid, but there are other
aspects of a drawing which we cannot assume.
What we can assume:


Collinearity: If points appear to be collinear, we can assume they are, unless there
is contrary information provided.
Between-ness: If one point looks to be between two others, we can assume that
the idea of between-ness is true.
What we cannot assume to be true, even though they may look to be:






Congruence of segments: If two segments appear to be congruent, we cannot
assume that that is true without additional information.
Midpoints of segments
Congruence of angles
Size of angles: An angle which appears to be acute is not necessarily acute. The
same is true about right angles and obtuse angles.
Parallel lines and parallel planes
Perpendicular lines
We will start to use some visualization of figures with some three dimensional figures called
solids. This first investigation asks you to count features of the figures. This is a combinatorial
exercise (compared with a metric exercise later, where you will learn to measure aspects of a
figure, such as length, perimeter, area, or volume).
1.
If a 3 x 3 cube (such as Rubric’s Cube) is immersed in paint, how many of the cubes
have 0 faces painted? 1 face painted? 2 faces painted?
TOPIC 5: Visualization and Locus
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It may be helpful to be able to see what these solids look
like. One of geometry’s skills is visualization. We begin
here with the idea of an isometric drawing to help answer
the questions above (and questions which extend it).
We will draw a 3 x 3 cube here.
2.
If a 4 x 4 cube is immersed in paint, how cubes faces have 0, 1, 2, … faces painted?
Draw a 4 x 4 cube here …
Now we will generalize the question, and ask if there is a way to predict how many cubes have a
(“a” is a variable) faces painted on a n by n (“n” is a variable) cube which is immersed in paint.
So we want to know how the number of faces painted depends on the size of the original solid.
TOPIC 5: Visualization and Locus
You are using some algebra, some visualization, and some inductive thinking.
number of painted faces
n (for n by n cube)
0
1
2
3
4
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2
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3
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4
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5
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6
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n (general)
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Our next skill is to make some isometric drawings.
Copy the drawings on the isometric paper next to each drawing.
3.
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4.
5.
For these solids, draw the top view, he left side view, and the back view. Then re-draw the solid,
showing the right side instead of the left side.
Original Solid
Top View
TOPIC 5: Visualization and Locus
Left Side View
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Back View
Here is another one. Do the same three views.
Original Solid
Left Side View
Top View
Back View
TOPIC 5: Visualization and Locus
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Flip the given solid over a vertical plane which is placed to the left of the original. Re-draw it
on the second graph paper.
With the plastic cubes, practice building some of the solids whose drawings have been provided
on pages 3 through 6.
Cube Nets
A net is a two-dimensional figure that can be folded into a three-dimensional object. Which of
the nets below will form a cube? (This is some more work with visualization.)
Use the website for Illuminations from NCTM:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=84
or http://illuminations.nctm.org/activity.aspx?id=3544
Instructions
Click on any net. Nets that are able to be folded into a cube will change color.
If you click on a net that cannot be folded into a cube, an explanation will be provided.
Exploration
The net below can be folded into a cube. Do you see it? In your mind, try to
figure out how it happens. Then, click on the image to watch the net fold into a
cube.
TOPIC 5: Visualization and Locus
There are exactly eleven nets that will form a cube. Which of the figures below
can be folded into a cube? Click on the figures to find out (using the website
above). Those that form a cube will change colors. For the others, you will be
told why they won't work.
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Now, here are a couple of definitions that pertain to these drawings. We will learn in a day or
two how to do a much better job with “definitions”, but these will serve for an introduction for
now.
A vertex (plural = vertices) is a corner of a solid. A face is a flat surface. An edge is a line
segment where faces meet. We will count the number of faces (f), vertices (v), and edges (e).
6.
In the adjacent drawing, the cube has 6
faces, 12 edges, and 8 vertices. Make sure
that you can see them.
We want to see if there is a pattern in the counts of the vertices, faces, and edges, just as we were
looking for patterns in the number of painted faces on the solids at the beginning of this packet.
7.
Count the v, e, and f for each of these.
For each of the following, count the number of vertices, edges, and faces. Record the results in
the table that follows. (We will also record the previous results, so that we can search to see
what all of these counts have in common, if anything.)
8.
rectangular prism
TOPIC 5: Visualization and Locus
9.
hexagonal prism
10.
square pyramid
11.
hexagonal pyramid
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TOPIC 5: Visualization and Locus
12.
Do a count of vertices, edges, and faces for each of the Archimedean solids
Do the count for one tetrahedron. It is very possible that you do not know what a
tetrahedron looks like. So this could be an exercise in visualization
as well, where
visualization is another goal of a geometry class.
Here is a picture of a tetrahedron, but there is another method to
figure out what is going on with it (and other solids).
v=
e=
f=
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Without a picture, you know that “tetra” is a Greek prefix for “four”. If you also know
that the solid is composed only of equilateral triangles, you can do the following analysis.
a)
Since there are 4 triangles, then there would be (4)(3) = 12 edges, if the triangles
were not attached. When a solid is built, only two edges can coincide. So instead
of 12 edges, there will be 12/2 = 6 edges in the tetrahedron.
b)
If you also know that 3 triangles meet at a vertex, then the (4)(3) = 12 vertices (if
the triangles were separate) amounts to 12/3 (because 3 triangles meet) = 4
vertices.
c)
“Tetra” means “4”, so there are 4 faces.
Thus v = 4 , e = 6, and f = 4
Do the count for an octahedron. An octahedron is made up of equilateral triangles, and 4
triangles meet at a vertex.
Do the count for a dodecahedron. A dodecahedron is made up of equilateral pentagons, and 3
pentagons meet at a vertex.
Do the count for an icosahedron. An icosahedron is made up of equilateral triangles, and 5
triangles meet at a vertex.
“It has been suggested that the regular icosahedron, not being found in nature, is
the first example of a geometrical object that is the free creation of human thought.”
Special Session on History and Philosophy of Mathematics, 2009 Fall Western Section
Meeting of the AMS, UC Riverside, November 7, 2009
(http://www.math.ucr.edu/home/baez/icosahedron/)
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Now, let’s collect the information in a table. (The numbers are from the numbered drawings
above.)
# of vertices
# of edges
# of faces
#6
#7
#8 rectangular prism
#9 hexagonal prism
#10 square pyramid
#11 hexagonal pyramid
#12 tetrahedron
#12 octahedron
#12 cube
#12 icosahedron
#12 dodecahedron
To do some inductive thinking, we want to look at the data from several examples, which the
table contains for us. Let’s ask this question:
If I know any two of the three counts (among v, e, and f), how could I know the
correct count for the third?
Investigate the data to see what conjecture you can come up with. A good conjecture should be
able to be verified with all of the data. A counterexample is a set of data which does not follow
your conjecture.
We did all of these examples in three dimensions. Does your conjecture work in a planar figure,
too? Let’s investigate.
For each of these figures, count the vertices, edges, and faces (polygons). Record you count for
each.
TOPIC 5: Visualization and Locus
13.
v=
e=
f=
14.
v=
e=
f=
15.
v=
e=
f=
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TOPIC 5: Visualization and Locus
16.
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v=
e=
f=
17.
Make your own drawing. (You could even draw a figure where the connections between
points are NOT straight line segments!)
Does your earlier conjecture still hold? If not, revise your conjecture for one that will work in
two dimensions.
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LOCUS of a point(s)
In some contexts, locus means “position”. In its use in geometry, it refers to all of the locations
where a point(s) could be under a given set of conditions. When all of these possible points are
taken together, we can describe a geometric shape or item that is the collection of those points,
and that is the locus of the point.
Example:
What is the locus of a point, in a plane, that is 4 inches from a point P?
Answer: The locus is a circle of radius 4 inches which center P.
P
Example:
What is the locus of a point, in a plane, that is 5 inches from a line, m?
Answer: The locus is two parallel lines, one on each side of m and each 5 inches from
m.
m
Exercises with locus. Make a drawing, and provide a sentence to describe the locus.
18.
What is the locus of a point that is 4 inches from a point P? (Notice that the
phrase “in a plane” has disappeared from the condition. So, you should now
think in 3D.)
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19.
What is the locus of points, in a plane, which are equidistant from two parallel lines?
20.
What is the locus of points which are equidistant from two parallel lines?
21.
When two conditions are listed, then there are two locus problems involved. The plural
of “locus” is “loci”. You need to consider each locus separately, and then decide what
the intersection of those individual locus results could be. There could be multiple
possible answers, so you would list each of them.
What is the loci of points in a plane which are 4 inches from a point, A, and equidistant
from two parallel lines, p and q.
22.
What is the loci of points in a plane which are 6 cm from point A and 9 cm from point
B?
TOPIC 5: Visualization and Locus
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23.
What is the loci of points which are 6 cm from point A and 9 cm from point B?
24.
What is the locus of points in a plane which are equidistant from the sides of an angle,
<RKS?
25.
What is the locus of points which are equidistant from two parallel planes?
26.
What is the locus of points which are equidistant from two perpendicular planes?
27.
What is the locus of points in a plane which are equidistant from three distinct points, J,
K, and L?