The Shape Of Things

About The Course

This is a course offered through the University of Toronto Department of Mathematics for mathematically interested and capable high school students, particularly those in grades 10 and 11. See the announcement for details, or you may browse week-by-week details that will be posted below. You may also contact the instructor, Peter M. Garfield.


The next and last meeting of the course will be Wednesday, February 25th. We will not meet Wednesday, February 18th, as that is during the University's Reading Week.

Week By Week Class Summaries

Week Twelve: From Two To Three Dimensions

We began the day trying to make sense of the Gauss-Bonnet Theorem. This, in a form more general than last week, says:

If E is a surface, then (Average Gaussian Curvature) × Area(E) = 2π χ(E).

We discussed what this means for surfaces that are topologically the same as our familiar friends S2 and T2.

We then tried to understand the notion that every surface can be given a spherical, Euclidean, or hyperbolic geometry. We saw that we can't give the torus a hyperbolic geometry because this produces cone points, and we can't give it a spherical geometry because it produces inverse-of-a-cone points. The next step was to identify how to give a hyperbolic structure to the two-holed torus T2#T2. (We know it must have a hyperbolic, rather than spherical or Euclidean, because last week we figured out that it had negative Euler number.) This came down to chopping the surface up into four hexagons that meet at each vertex in groups of four. This means that we have to shrink the interior angles of the vertices from 120° to 120° by putting a hyperbolic geometry on it. We discussed how to extend this to the n-holed torus as well.

As a transition to the geometry and topology of three-dimensional surfaces (which we'll call 3-manifolds), we discussed the notion of the product of two spaces. After warming up on the product S1 × I of the circle and an interval (which is a cylinder), the product S1×S1 of two circles (which is the torus T2), and several others, we got to the three-dimensional versions. The product S1× S1× S1 of three circles is the three-torus T3, and the product S2×S1 of the sphere and the circle was thought-provoking.

Finally, to end the day we looked at some 3-manifolds obtained by identifying faces on (solid) Platonic solids. We'll begin next week with more discussions of these solids.

Homework Twelve

No homework this week.

Week Eleven: From Plato to Euler and Gauss-Bonnet

As promised, we began our day with the Platonic solids. These are the five polyhedra whose sides are all the same regular polygon, and whose vertices are all surrounded by the same number of polygons. We discussed several things: they all have Euler number χ = v - e + f = 2 and there is a duality between them. (The cube and octahedron are duals, as are the icosahedron and dodecahedron. The tetrahedron is dual to itself.) Finally, we proved (using graph theory) that there are only these five Platonic solids. (We skipped a small algebraic step; see the homework for the details.)

We then proceeded to a discussion of the Euler number (or Euler characteristic) on surfaces. This requires us to first introduce a cell division of the surface: basically, we cut the surface into polygons. We then proved the following (simplified version of the) Gauss-Bonnet Theorem:

If E is a surface with constant Gaussian curvature K and area A, then K A = 2π χ(E).

We actually only proved this when K was +1, 0, or -1, but this seemed like enough.

The rest of class was spent actually computing Euler numbers. We computed the following:

Euler Orientable Non-orientable
Number Surface Surface
2 S2
1 P2
0 T2 K2=P2#P2
-1 P2#P2#P2
-2 T2#T2 P2#P2#P2#P2

We concluded, but did not prove (see the homework), that this pattern continues. Finally, we formalized the idea of ``number of holes'' by defining the genus g of an oriented surface E to be defined by χ(E) = 2-2g, so that the n-holed torus T2# ··· # T2 (n summands) has genus n.

Homework Eleven

PDF or web.

Week Ten: Topological Games and Graphs

We began the day with a bunch of games and puzzles. The two big games of the day were hex and gale. We produced an argument that hex should be won by the first player, and that hex should always produce precisely one winner. The first argument, at least, applied to gale as well. (We said that the hex theorem was equivalent to the Brouwer Fixed Point Theorem. See links to this theorem at Harvey Mudd College and Wikipedia. It's a little over our heads, but interesting. The proof of the equivalence of these two theorems I read from David Gale's paper ``The Game of Hex and the Brouwer Fixed-Point Theorem,'' published in the American Mathematical Monthly, Vol. 86, No. 10 (Dec. 1979).)

The most popular (that is, vexing) of the remaining puzzles was that of the monkey and the coconuts. The answer I gave was assuming that the monkey got one coconut in the morning – not the way I originally stated it. (The smallest positive solution for the way I originally stated it is 3,121=54-4.)

Most of the remaining problems were related (somehow) to a field of math called graph theory. The purpose of bringing graph theory into our course (besides being able to use it to solve lots of puzzles) is to give a proof that there are precisely five Platonic solids. We'll do this at the beginning of next week's class.

Homework Ten

For next time, please read the article I passed out in class (unavailable on the web) and bring the five Platonic solids to class. (You can build your own or find examples. See the bibliography, below, for links to some model-building sites.)

Week Nine: Cutting Tori & More On The Projective Plane

After last week's foray into the world of two-dimensional topology, today's class was intended to get you more comfortable with two of the surfaces from last week: the torus (or donut-shape) and the projective plane.

We began with the torus. After a few minutes on cutting 3-space with n tori (which produces a maximum of (n3+5n+6)/6 regions), we started cutting our tori (courtesy of Tim Horton's) with three planes. The largest number of pieces anyone came up with was 12 (at least, honest pieces, not counting biting or crumbling). The solution is 13 for 3 planes, or (n3+3n2+8n)/6 for n cuts.

As an interlude we had a brief introduction to the mathematical theory of induction, proving that 1+2+···+n = n(n+1)/2 and 1+3+···+(2n-1) = n2. We also managed a false proof that all cows (rather than horses for some reason) have the same colour eyes.

The last half of the class was spent on projective geometry. We gave several approaches to P2 as (a) the plane with a ``line at infinity'' (one point for each slope, including infinite slope), (b) the set of lines through the origin in 3-space, and (c) using the ``homogeneous coordinates'' [x : y : z ] (really just a represenation of version (b) in a coordinate system). These all generalized to at least P1 (and some to Pn as well, although that's beyond us at this point).

Finally, we ended the class with finite projective planes. We assumed that a finite projective plane of order n satisfied the following requirements:

  1. Through any two points passes a unique line.
  2. Any two lines intersect in a unique point.
  3. Every point lies on n+1 lines.
  4. Every line consists of n+1 points.

(Notice that these come in pairs, and that the notion of ``point'' and ``line'' are more or less interchangable.) We constructed the n=2 ``Fano plane'' and derived the 13-point, 13-line picture for n=3. We claimed, but did not prove, the following: first, that there are n2+n+1 points on the order n plane; second, that such configurations exist whenever n=pk (and p is prime); and third, that no such configuration exists for n=6 or n=10.

Notice that we will not be having class next week (Wednesday, January 21, 2004) because of exams. We will be starting up again on January 28th.

Homework Nine

No homework this week

Week Eight: Topology of Surfaces

Today we focuesed on the topology of surfaces. This means several things: first, we considered only finite two-dimensional surfaces (like the sphere S2, torus T2, or Klein bottle K2 but not the Euclidean plane E2); second, we only are interested in surfaces without boundary (this excludes the disc D2 and the Möbius strip M).

We first recalled the construction of gluing that we introduced way back in week one (remember tic-tac-toe?). By gluing opposite sides of squares (or rectangles) we constructed the cylinder, the torus T2, and the Klein bottle K2. By gluing opposite semicircles of the boundary of the disc, we also constructed the sphere S2 and projective plane P2.

The other major construction tool of the day was the idea of a connected sum. The connected sum of two surfaces M and N is written M # N. This is constructed by removing a disc from each surface, then identifying the edges left by the removed discs. Our first examples were that T2 # T2 was a two-holed torus, and that T2 # S2 = T2. We later showed that P2 - D = M (the Möbius strip) and so P2 # P2 = K.

The main theorem of the day is that all connected surfaces are topologically a connected sum of some number of tori and projected planes. (This statement is only so simple when we agree to say that the sum of zero tori and zero projective planes is the sphere S2.) We then proved that T2 # P2 = K2 # P2, so (in fact) all connected surfaces are topologically the connected sum of either some tori or some projective planes. (That T2 # P2 = K2 # P2 follows from the fact that T2 # M = K2 # M since P2 is simply M with a disc attached along its edge. We proved that T2 # M = K2 # M through some carefully drawn pictures.)

Today's class followed pretty closely from a wonderful book by Jeffrey Weeks. Highly recommended reading.

Homework Eight

No homework this week

Week Seven: Fractal-Mania!

We began the day with a review of dimension, with an expansion on some of the examples of last time. We considered the Koch snowflake, the Sierpinski triangle or gasket, the Sierpinski carpet, and a natural-looking fractal image of a plant. This was all an excuse for some pretty pictures (see the bibliography) and some quick study of logarithms.

We then turned our attention to the study of iterates of the function f(x) = x2 - c. That is, we looked at what happens to f(n)(x) = f(f(f(...f(x)...))) (where there are n compositions of f(x)) when n gets large. We considered this function on the real line for one value of c.

We then turned to a quick study of the complex numbers (note: some of this handout should be considered well over our heads), and considered f(z) = z2 - c, where now z (and c) can be complex numbers. This led us to Julia sets and the Mandelbrot set (``the most complicated thing in mathematics,'' to quote someone), which was (again) just an excuse to look at pretty pictures.

By the way, Merriam-Webster says that it's spelled smidgen, with smidgeon as a variant. The OED says more or less the same thing (smidgen, also smidgeon). So: everyone's right!

Have a great holiday, and I'll see you January 7th!

Homework Seven

PDF or web.

Week Six: A Little Spherical Geometry; Sneaking Up On Fractals

We started the day with a proof of the area formula for triangles on a sphere of radius r. It turns out not to be difficult, involving only the fact that the area of a double lune with angle α (in radians) has area 4αr2. From this we proved that the area of a triangle on a sphere of radius r is (S(Δ)-π)r2, where S(Δ) is the angle sum of the triangle (in radians). One interesting consequence of this is that circles on spheres are smaller than circles (of the same radius) in the Euclidean plane.

We began our sneak attack on fractals by looking at the Cantor set C. This set is obtained by repeatedly removing the middle third of intervals from a starting interval (we started with [0,1], the interval from 0 to 1). We proved that C had infinitely many points, a total length of 0 (whatever length means in this context), and was totally disconnected (in the sense that between any two points of C was a point not in C).

We moved on to a seeming generalization called the Koch snowflake. Here we started with an equilateral triangle and, instead of removing a third of a side at each step, we replaced the middle third of an edge with an equilateral triangle. This quickly became very involved, but we showed that the area of the resulting snowflake was exactly 1.6 times the area of the original triangle, but the perimeter of the snowflake was infinite.

This led us to a discussion of dimension. What should dimension mean? We defined a similarity dimension, under which the Koch snowflake had dimension log(4)/log(3) ≈ 1.26. This only works for shapes with a precise similarity, so we tried defining a box dimension, under which the Cantor set C has dimension log(2)/log(3) ≈ 0.631. Next time we will do a few more examples, look at some pretty pictures, and try to make sense of these fractional dimensions.

Homework Six

PDF or web.

Week Five: The Last Stand: More Hyperbolic Geometry

We began our day with a proof of the claim from the end of last week: the three perpendicular bisectors of the sides of a triangle in hyperbolic space meet in a point that is real, ideal, or ultra-ideal. This led us to the definitions of horocycle (or limiting curve) and equidistant curve, which are generalizations of the idea of a circle. We saw that three non-collinear points in hyperbolic space determine one of a circle, a horocycle, or an equidistant curve.

We next discussed the possibility of tiling the hyperbolic plane with regular n-gons, as we did for the Euclidean plane in Week One. We soon discovered that it depended on what we meant by regular, but that with a reasonable definition it was possible for any n ≥ 3. Not only possible, but ready to produce many pretty pictures on the web. See the bibliography for some nice pictures on the web.

The Poincaré disk model of the hyperbolic plane has suited our needs, but it has a major drawback: distances are distorted. (Angles are preserved, so this model is called a conformal model.) We discussed a distance-preserving (isometric) model of (part of) the hyperbolic plane, namely the pseudosphere. We described this model as a rotation of a curve (called a tractrix) around an axis. [I will post a picture by next week.] This model distorts angles, but it preserves distance.

Finally, we discussed a theorem of Gauss that says that, for the geometries we've been looking at, there is a constant K (one per geometry) such that K A(Δ) = S(Δ) - π (where the angle sum is in radians). This holds for hyperbolic geometry with K=-1/k2 and spherical with K=1/r2. In Euclidean geometry, K=0, so this simply says the angle sum is always π. This followed a bit of discussion of spherical geometry, and led to some examples of other spaces with K=0 (the infinite cylinder and the flat torus, or donut).

Homework Five

PDF or web.

Week Four: Still More Hyperbolic Geometry

We began the day with a little bit of catching-up on formal axiom systems, from last week. I hope the notion of dependent and independent axioms make more sense, now that I've had the chance to explain them.

We then turned to the notion of area in hyperbolic space. After proving that the usual ``Area of a triangle equals half the base times the height'' can't work in hyperbolic geometry, we searched for a substitute. We decided that the area had to be positive (on non-empty polygons) and additive (so the area of a disjoint union is the sum of the areas), we didn't prove that the ratio of area to defect is constant. (See the homework for a detailed sketch of a proof.) This led to a nice formula for area: A(Δ)=k2D(Δ), with a conversion factor if we measure defect in degrees rather than radians. We also remarked that this constant k will appear elsewhere; for example, the angle of parallelism Π(d) satisfies tan(Π(d)/2) = e-d/k.

Next we turned our attention to the concept of an ideal point. This is a family of lines (a sheaf, which turns out to be Saskatchewan's ``basic symbol of the province's visual identity program'') all parallel to a given line in a given direction. We roughly thought of this as the point at infinity where all these lines meet. This gave us a way to think of ideal triangles; that is, triangles with two parallel sides that meet at an ideal point ``at infinity.'' We also discussed ultra-ideal points, the sheaf of lines all perpendicular to a given line (the axis or carrier). We ended by stating a theorem (and we'll pick up here next week): the three perpendicular bisectors of a triangle all meet in a point, an ideal point, or an ultra-ideal point. See a proof next week!

Here's a link to the lecture I announced in class (Bill Casselman of the University of British Columbia Math Department speaking on 4000 years of mathematics in images). I'll be going; let me know if you want to meet up.

Homework Four

PDF or web.

Week Three: More Hyperbolic Geometry

We began today with two more erroneous proofs of the (oh so false) ``fact'' that Euclid's fifth postulate follows from what we've been calling absolute geometry (Euclid's other postulates). The first proof hinged on being able to find a circle passing through three non-collinear points. (One can do this if and only if Euclid's parallel postulate holds, we proved.) The second incorrect proof led us to an investigation of the intersection of lines in hyperbolic space. From this we learned that parallel lines in hyperbolic space approach each other asymptotically (that is, get arbitrarily close) in one direction and grow arbitrarily far apart in the other direction. This made the models for hyperbolic space presented last time seem a lot more sensible, I hope.

Finally, after several weeks of playing with formal axiom systems, we did a brief little introduction. These systems consist of what we called undefined terms (like lines and points in our study of geometry) and axioms (accepted truths, like Euclid's postulates). We defined the notions of consistency and independence and gave several examples. These examples are continued in the homework.

Homework Three

PDF or web.

Week Two: From Euclid to Non-Euclidean Geometry

We began today looking at Euclid and his rightly famous exposition of classical geometry. We quickly took a turn for the unusual when we proved that all triangles must have angle sum less than or equal to 180°. This is correct, of course, but a little unexpected (what's that ``less than'' doing there?), but we were soon led down the path of Bolyai and Lobachevsky to hyperbolic geometry. We proved that a line has more than one parallel through any given point, and that in hyperbolic geomtry we have ``Angle-Angle-Angle'' (two triangles with congruent angles are congruent), and other surprising results. We ended the day with a discussion of three models of hyperbolic geometry: the Poincaré disk, the Klein-Beltrami disk, and the Poincaré upper half-plane.

Homework Two

PDF or web.

Solutions for Homework Two

Problem 1: PDF only

Week One: Tilings and Tic-Tac-Toe

We began the course with an exploration of tilings of the plane. We uncovered the eleven Archimedian (or uniform) tilings as well as a few two-uniform tilings. A tiling is k-uniform if there are k distinct vertices after identifying any two vertices that can be moved to each other by a rigid motion of the plane that preserves the entire tiling. (A uniform tiling is simply a one-uniform tiling.) The standard book on tiling is by Grunbaum and Shephard.

See the bibliography for some links to some cool pages (and computer programs) on tilings.

We also had a small tournament in tic-tac-toe. To make things interesting, we played these variants: C-tic-tac-toe, in which the top row is assumed to be adjacent to the bottom row; T-tic-tac-toe, in which the top and bottom rows are adjacent, as are the left and right columns; and (for the big finale), K-tic-tac-toe, in which the top and bottom rows are adjacent, and the left and right rows are adjacent but only after flipping! (After a bit of discussion, it came out that this is tic-tac-toe on a cylinder, torus, or Klein bottle (whatever that is). We'll discuss this more later in the course.)

Homework One

PDF or web.

Bibliography & On-Line Resources


Web Pages

On Building Geometric Models

On Topology

On Fractals

On Hyperbolic Tilings

On Hyperbolic Geometry

On Euclidean Geometry

On Tilings Of The Plane

Directories, Link Collections, and Other Groupings