McGILL UNIVERSITY
FACULTY OF SCIENCE
FINAL EXAMINATION
MATHEMATICS 348
TOPICS IN GEOMETRY
EXAMINER: Professor W. G. Brown
DATE: Wednesday, December 13th, 2006
ASSOCIATE EXAMINER: Prof. Daniel Wise
TIME: 14:00 – 17:00 hours
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
Instructions
1. Fill in the above clearly. This question book, and all your other writing — even
rough work — must be handed in.
2. This examination booklet consists of this cover, Pages 1 through 4 containing
questions; and Page 5, which contains a list of propositions.
3. This is a CLOSED BOOK examination. CALCULATORS ARE NOT PERMITTED. Translation dictionaries are permitted; no other dictionaries are permitted.
Final Examination — Mathematics 348 2006 09
1
1. (a) [3 MARK] State a set of axioms for a projective plane.
(b) [6 MARKS] Using homogeneous coordinates, list all the points of the
projective plane constructed using the field F2 of residues
modulo 2. List the equations in homogeneous coordinates for each of the
lines.
(c) [6 MARKS] Indicate precisely which points lie on which of your lines. Make a
“sketch” of the projective plane referred to in part 1b, showing the coordinates
of the points and the equations of the lines through them.
2. [17 MARKS] Answer the following questions for each of the following strip patterns
composed of equally spaced characters:
I.
= X
× X
= X
× X
= X
×
X
...
...
×
II.
=
×
=
×
=
= X
× X
= X
× X
= X
×
X
...
...
=
×
=
×
=
×
III. . . . VΛZVΛZVΛZVΛZVΛZVΛZVΛZVΛZVΛZV. . .
(a) [4 MARKS ALTOGETHER] If the pattern has reflective symmetry in some
vertical mirror, describe all locations for vertical mirrors that are possible.
(Variations in the thicknesses of the symbols are not intentional.)
(b) [4 MARKS ALTOGETHER] If the pattern has rotational symmetry under a
half-turn, describe the locations of the centres of all such half-turn(s).
(c) [3 MARKS ALTOGETHER] If the pattern has symmetry under a translation
which is not the identity, describe an interval of minimum length whose images
under iterations of one translation cover the entire strip; e.g., for the pattern
...UVUVUVUVUV... one such interval would be UV, another would be
VU; but UVUV would not be acceptable, because it is not as short as
possible.
(d) [3 MARKS ALTOGETHER] Does the pattern have symmetry under a horizontal reflection? You are expected to prove your claims.
(e) [3 MARKS ALTOGETHER] If the pattern has symmetry under glide reflections, describe precisely how each type of character in the pattern is mapped
under a glide reflection of your choice; e.g., in · · · VΛVΛV· · · there is a glide
reflection under which any consecutive pair of characters VΛ is mapped on
to a consecutive pair ΛV.
Final Examination — Mathematics 348 2006 09
2
3. [17 MARKS] Suppose that 4ABC is given with |AB|=|AC|, and that D, E, F
are respectively the midpoints of sides BC, CA, AB. Prove carefully that 4DEF
is isosceles. You are expected to set this theorem up in the style of the proofs
of Euclid’s propositions. Every step in your proof should be carefully justified,
referring to the Propositions you are using by number (see page 5). You are not
to use any propositions other than those listed. (Hint: Don’t make this problem
more difficult than it needs to be.)
4. (a)
i. [3 MARKS] Define what is meant by a metric.
ii. [1 MARKS] Define what is meant by the Manhattan metric on the set
R × R of ordered pairs of real numbers.
iii. [2 MARKS] Consider the triangles ABC, DEF in R2 whose vertices have
the following coordinates:
{A(6, 0), B(0, 0), C(0, 6)} and {D(5, 5), E(8, 8), F (5, 11)}.
Where ρ is the Manhattan metric, show that
ρ(A, B) = ρ(D, E)
ρ(B, C) = ρ(E, F ) .
Observe that, in these triangles,
∠ABC = ∠DEF ,
but that the triangles are not congruent, since the corresponding third
sides have different lengths.
iv. [1 MARKS] One of the theorems you have studied in Euclid appears to
be contradicted by this example. State which theorem that is. (You
needn’t attempt to explain this phenomenon: Euclid’s theorems apply to
the plane endowed with the Euclidean metric, not with the Manhattan
metric.)
(b) [10 MARKS] Using a square with vertices, in cyclic order, P, Q, R, S, give a
Cayley (composition) table for
¶ of its symmetries, using for
µ the group of all
P Q R S
used in the lectures, where
the symmetries the notation
A B C D
A, B, C, D are each one of P, Q, R, S.
Final Examination — Mathematics 348 2006 09
3
5. The Escher graphic in Figure 1 shows birds and fish in motion. Assume that
the graphic extends to infinity both horizontally and vertically in both directions.
Do not attach any significance to the shadings of two figures that are otherwise
identical.
Figure 1: A graphic design by M. C. Escher
(a) [14 MARKS] Describe 2 different isometries that you find in this graphic,
where neither of the isometries you choose is a power of the other. You may
wish to write marks on or beside the figure for this purpose. For each type of
isometry F that you discuss, provide the following details:
i. [1 MARK] Give the number and location of the fixed points of F . If the
number is infinite, explain.
ii. [1 MARK] Does there exist a positive integer n such that the nth power
of F is equal to the identity? If there does exist such an integer, give the
smallest with that property for each isometry you discuss.
iii. [5 MARK] Express the isometry precisely as a composition of reflections.
(b) [3 MARKS] Describe precisely the action of the isometry which is the product
(in either order) of the two isometries that you have chosen above.
Final Examination — Mathematics 348 2006 09
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Figure 2: Problem 6
6. In Figure 2 a portion of a tiling of the plane by 2 × 1 bricks is marked by a
parallelogram. You are to take this parallelogram as defining a region that is to
become a torus: sides AB and DC are to be identified, and similarly sides AD and
BC, in the given orders. In this way a finite map M on the torus is created. Each
of the bricks is to be considered as having 6 vertices and 6 sides. Note that the
vertices of the defining parallelogram become a vertex of the resulting map.
(a) [6 MARKS] Describe the structure of the graph of the resulting map on the
torus: how many vertices and edges does it have? What are their degrees.
Discuss the presence of loops and/or multiple edges, if any.
(b) [6 MARKS] Determine the numbers of vertices, edges, and faces of the map
M . Use these data to determine the Euler-Poincaré characteristic of the torus.
(c) [5 MARKS] Determine the structure of the graph of vertices and edges of the
dual map of M : determine its numbers of vertices and edges, and discuss the
presence of loops and/or parallel edges.
Final Examination — Mathematics 348 2006 09
5
First Propositions in Book I of Euclid’s Elements
Proposition I.1 On a given finite straight line, to construct an equilateral triangle.
Proposition I.2 To place at a given point (as an extremity) a (straight) line segment
whose length is equal to that of a given (straight) line segment.
Proposition I.3 Given two unequal line segments, to cut off from the greater a line
segment whose length is equal to the less.
Proposition I.4 If two triangles have two sides equal to two sides respectively, and have
the angles contained by the equal line segments equal, they will also have the base equal
to the base, the triangle will be equal to the triangle, and the remaining angles will be
equal to the remaining angles respectively, namely those which the equal sides subtend.
Proposition I.5 In isosceles triangles the angles at the base are equal to one another;
and, if the equal sides be extended below the base, the angles under the base will also
be equal to one another.
Proposition I.6 If, in a triangle, two angles be equal to one another, the sides which
subtend the equal angles will also be equal to one another.
Proposition I.7 Given two line segments, joining the ends of a line segment to a common point, there cannot be constructed from the ends of the same line segment and on
the same side of it, two other line segments meeting in a different point, respectively
of the same lengths as the former two line segments, namely each equal to that which
emanates from the same end of the fixed line segment.
Proposition I.8 If two triangles have two sides equal in length to two sides respectively,
and also have the bases equal in length, they will also have the angles equal which are
contained by the equal sides.
Proposition I.9 To bisect a given angle.
Proposition I.10 To bisect a given line segment.
Proposition I.11 To draw a straight line at right angles to a given line segment, from
a given point on it.
Proposition I.12 To a given straight line, from a given point which is not on it, to
draw a perpendicular straight line.
Proposition I.13 A line segment set up on a line makes angles with that line which
sum to two right angles.
Proposition I.14 If at any point on a line segment, two line segments not lying on the
same side make the adjacent angles equal to two right angles, the two line segments will
be in a straight line with one another.
Proposition I.15 If two lines cut one another, they make the vertical angles equal.