2014 — Regular Convex Bodies

• Introduction — Plan of the progam.
• Session 1 — Warm-up exercises.
• Session 2 — (No notes.) Vector notation. Diameter of high dimensional convex bodies. Stereographic projection.
• Session 3 — Metric spaces. Path metrics on graphs. Path metrics on the edge graphs of Platonic solids.
• Session 4 — Tiling problems. Soma Cubes. Simplicial complexes.
• Session 5 — Euler’s Formula. There can be at most five Platonic Solids.
• Session 6 — The Szilassi Polyhderon. Euler characteristic. There are dodecahedra.
• Session 7 — Groups. Symmetries. (Examples missing due to $\TeX$nical problems.)
• Session 7 — Announcement of final project. Rotational symmetries of the cube. How to read a math paper systematically. (No notes.)

Introductory remarks
The plan for the semester is an ambitious one. We’re going to understand the structure of all the regular convex polytopes in all dimensions, and build up a intuition for dimensions greater than three. We’ll spend most of our time learning the tools we need to understand how a geometric object can be pieced together. These tools will include vectors, metric spaces, symmetry groups, and simplicial complexes. I’m taking a very broad view of what constitutes a tool and counts as information about a space. The final result of the project will be a poster presentation about the solids and some 3-dimensional “nets” of the 4-dimemsional solids (the simplex, cube, cross-polytope, and 24-cell).

I can’t resist saying things about hyperbolic geometry. Therefore, if we get to the end of the proposed project, we’ll take a stab at using out high brow high dimensional intuition to understand the Gieseking manifold. There is more than enough stuff to say about the platonic solids, so we’ll see how far we get.

Session 1

The plan for the semester is to:

• Classify the regular convex polyhedra in all dimensions.
• Learn about negatively curved three dimensional spaces.

This is a slightly edited version of a set of exercises written by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston in 1991 as part of their course Geometry and the Imagination available here: http://goo.gl/7PwbZ7.

I’ve added several homework to the end: ${\circ_{\triangle\bigcirc\square}, \circ_{15}, \square_2, \square_3, \square_4, \circ_?}$. The second last question ${\square_4}$ is difficult, please attempt it, and we’ll discuss your attempts next week. Resist the urge to try and look up solutions. The last question ${\circ_?}$ is quite open ended. Please come up will several ideas and we’ll investigate them next week.

• Cut off each corner of a square, as far as the midpoints of the edges. What shape is left over? How can you re-assemble the four corners to make another square?
• Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What do you get?
• Take two squares. Place the second square centered over the first square but at a forty-five degree angle. What is the intersection of the two squares?
• Mark the sides of a square into thirds, and cut off each of its corners back to the marks. What does it look like?
• How many edges does a cube have?
• Take a wire frame which forms the edges of a cube. Trace out a closed path which goes exactly once through each corner.
• Take a ${3 \times 4}$ rectangular array of dots in the plane, and connect the dots vertically and horizontally. How many squares are enclosed?
• Find a closed path along the edges of the diagram above which visits each vertex exactly once? Can you do it for a ${3 \times 3}$ array of dots?
• How many different colors are required to color the faces of a cube so that no two adjacent faces have the same color?
• A tetrahedron is a pyramid with a triangular base. How many faces does it have? How many edges? How many vertices? How many different colors are required to color the faces so that no two adjacent faces have the same color?
• An octahedron is the shape formed by gluing together equilateral triangles four to a vertex. Balance it on a corner, and slice it halfway up. What shape is the slice?
• How many colors are required to color the faces of an octahedron so that faces which share an edge have different colors?
• Rest a tetrahedron on its base, and cut it halfway up. What shape is the smaller piece? What shapes are the faces of the larger pieces?
• Rest a tetrahedron so that it is balanced on one edge, and slice it horizontally halfway between its lowest edge and its highest edge. What shape is the slice?
• Cut off the corners of an equilateral triangle as far as the midpoints of its edges. What is left over?
• Cut off the corners of a tetrahedron as far as the midpoints of the edges. What shape is left over?
• You see the silhouette of a cube, viewed from the corner. What does it look like?
• Imagine a wire is shaped to go up one inch, right one inch, back one inch, up one inch, right one inch, back one inch, …. What does it look like, viewed from different perspectives?
• The game of tetris has pieces whose shapes are all the possible ways that four squares can be glued together along edges. Left-handed and right-handed forms are distinguished. What are the shapes, and how many are there?
• Someone is designing a three-dimensional tetris, and wants to use all possible shapes formed by gluing four cubes together. What are the shapes, and how many are there?
• Rest an octahedron on a face, so that another face is on top. Slice it halfway up. What shape is the slice?
• Take a ${3 \times 3 \times 3}$ array of dots in space, and connect them by edges up-and-down, left-and-right, and forward-and-back. Can you find a closed path which visits every dot but one exactly once? Every dot?
• Do the same for a ${4 \times 4 \times 4}$ array of dots, finding a closed path that visits every dot exactly once.
• ${\circ_{\triangle\bigcirc\square}}$ What three-dimensional solid has circular profile viewed from above, a square profile viewed from the front, and a triangular profile viewed from the side? Do these three profiles determine the three-dimensional shape?
• ${\circ_{15}}$ Find a path through edges of the dodecahedron which visits each vertex exactly once.
• ${\square_2}$ Consider a square with unit side length. Draw circles of radius one half centered at each vertex. Now inscribe the largest circle possible ${C}$ centered at the center of the square and tangent to the previous circles. What is the area of ${C}$?
• ${\square_3}$ Consider a cube with unit side length. Draw spheres of radius one half centered at each vertex. Now inscribe the largest sphere possible ${C}$ centered at the center of the square. What is the volume of ${C}$?
• ${\square_4}$ Consider a cube in four dimensional space with unit side length. Draw spheres of radius one half centered at each vertex. Now inscribe the largest sphere possible ${C}$ centered at the center of the square. What is the radius of ${C}$? Is there anything counter-intuitive about the number you get?
• ${\circ_?}$ Suppose you have been shrunken down to the size of an ant and are trapped on the surface of a paper tetrahedron, octohedron, or icosahedron, near a vertex, with only a device that with measures area (planimeter). Suppose the surface is vastly larger than yourself. How can you tell which surface you’re solid you are on? What measurements can you make?

Session 2
No notes were distributed. Roughly, introduced vectors and curvature around vertices on polyhedra. We looked at the problem ${\square_4}$ from Session 1. Diameter of ${[0,1]^n}$. Stereographic projection.

Session 3 — Graphs as metric spaces Today we’re going to look at some different metric spaces, and see how they behave under various maps. This is tangentialy to our project of classifying solids, but it is important material.

Definition 1 A metric space ${(X,d)}$ is a set ${X}$ and a distance function ${d : X \times X \rightarrow \mathbf{R}}$ such that:

• Positivity ${d(x,y) \geq 0}$.
• Non-degeneracy ${d(x,y) = 0 \Rightarrow x=y}$ .
• Reflexivity ${d(x,x) = 0}$.
• Symmetry ${d(x,y) = d(y,x)}$.
• Triangle Inequality ${d(x,z) \leq d(x,y) + d(y,z)}$.

The notion of a metric space axiomatizes our notion of distances should be measured. For example symmetry says that the distance from ${x}$ to ${y}$ should be the same as a the distance from ${y}$ to ${x}$. Only the last two items of the definition really have any intuitive geometry content. The other three are there for purely technical reasons.It’s important to point out that there are other ways of formalizing distance where some of the properties of a metric fail. These are different kinds of spaces.

Here are several different structures on ${\mathbf{R}^n}$.

• ${d_1((x_i), (y_i)) = \sqrt{\sum (x_i - y_i)^2 }}$ is the standard Euclidean metric on ${\mathbf{R}^n}$.
• ${d_2((x_i), (y_i)) = \max \{ |x_i - y_i| \}}$
• ${d_3((x_i), (y_i)) = \sum |x_i - y_i|}$

Question 2 For any metric space ${(X,d)}$ we have a notion of balls and spheres. A metric sphere is ${S(x,r) = \{y : d(x,y) = r\}}$. A metric ball is ${B(x,r) = \{y : d(x,y) \leq r\}}$. Draw various metric balls in ${(\mathbf{R}^n, d_i)}$ for ${n = 2,3}$ and ${i = 1, 2, 3}$.

Definition 3 Given a graph ${G = (V,E)}$ we get a natural metric on ${G}$ as follows: A path in ${G}$ from a vertex ${v_1}$ to ${v_2}$ is a sequence of vertices ${\tilde{v}_1, \dots, \tilde{v}_n}$ such that ${v_1 = \tilde{v}_1}$, ${v_2 = \tilde{v}_n}$, and there is an edge from ${\tilde{v}_i}$ to ${\tilde{v}_{i+1}}$ for each ${i = 1, 2, \dots, n-1}$. The length of a path ${\tilde{v}_1, \dots, \tilde{v}_n}$ is ${n}$. The path metric on ${G}$ is given by: ${d(v_1,v_2)}$ is the minimal length of a path from ${v_1}$ to ${v_2}$.

Question 4 Draw some metric balls in the edge graphs of the platonic solids. Graph the number of vertices contained in a ball.

Aside: We noted that once can see the central symmetry of a platonic solid in the growth rate of its metric spheres.

Session 4

We constructed paper models of the dodecahedron and the tiling of planes by regular heptagons.

4.1. Soma Cube Structure

Question 8 (2D Warm-up #1) Consider a chess board with two non-adjacent corners removed. Can you cover the remaining spaces with 3-piece ${L}$-shaped dominoes?

Question 9 (2D Warm-up #2) Consider a chess board with two non-adjacent corners removed. Can you cover the remaining spaces with ${1 \times 2}$ dominoes?

Question 10 (2D Warm-up #3) Can you tile a standard chess board with ${1 \times 2}$-dominoes such that no ${2 \times 2}$-square consists of two dominoes?

Problem #2 is due to Martin Gardner, who inspired countless youth to pursue mathematics through his popular column in Scientific American.

We’ll now explore some of the mathematics of the Soma Cube. We’ll use techniques similar to the colouring we used in the second warm-up exercise.

We’ll use the labellings of the pieces of the cube given here. We’ll name the pieces ${\mathbf{1}, \dots, \mathbf{7}}$.

Note that pieces ${\mathbf{2}}$ and ${\mathbf{3}}$ are somewhat special in that they contain a ${1 \times 1 \times 3}$ bar. This means that they can occupy either ${0,1,2}$ corners of the cube each.

Suppose piece ${\mathbf{2}}$ occupies no corners. Each piece not equal to ${\mathbf{3}}$ can occupy at most one corner, so we can occupy at most ${2 + 5 = 7}$ corners. Thus, ${\mathbf{2}}$ must occupy a corner.

Question 11 Perform a similar analysis to show where ${\mathbf{2}}$ must wind up. You will get, taking in to account the symmetries of the cube, two positions.

4.2. Simplicial complexes

A simplicial complex is the “higher dimensional” version of a graph. They’ll help us to rigorously describe the structure of the high dimensional platonic solids.

Definition 12 A simplicial complex ${X}$ is a set of subsets of ${V}$ such that: if ${A \subseteq B}$ and ${B \in X}$ then ${A \in X}$. The set ${V}$ consists of the vertices of ${X}$, the elements of ${X}$ are the faces of ${X}$, and the dimension of ${B \in X}$ is ${|B|-1}$. An ${n}$-face of ${X}$ is an ${n}$-dimensional face.

The key property of a simplicial complex is that the intersection of two faces of a simplicial complex is again a face.

Example 1 (Graphs)

$\displaystyle \triangle = \left\{ \emptyset, \{1\}, \{2\}, \{3\},\{1,2\}, \{2,3\},\{1,3\} \right\}$

$\displaystyle \square = \left\{ \emptyset, \{1\}, \{2\}, \{3\},\{4\}, \{1,2\}, \{2,3\},\{3,4\}, \{4,1\} \right\}$

Definition 13 Recall the metrics,

• ${d_2((x_i), (y_i)) = \max \{ |x_i - y_i| \}}$
• ${d_3((x_i), (y_i)) = \sum |x_i - y_i|}$

From now on, ${B(0,1,d_3) \subset {\mathbb R}^n}$ will be called the cube, and we will use the notation: ${\square^{n}}$. Similarly, ${B(0,1,d_3) \subset {\mathbb R}^n}$ will be called the cross-polytope, with notation ${\Diamond^{n}}$. We will also use the notation ${\square^n = [0,1]^n}$. Warning: This notation is non-standard.

Question 14 Give a description of the face adjacency structure of the tetrahedron, octohedron, and icosahedron.

Question 15 Give a description of the face adjacency structure of ${\Diamond^4}$.

Session 5 — Euler’s Formula and the Platonic Solids

Today we’re going to proceed to the classification of regular convex polyhedra in two dimension. We’re going to use this famous theorem of Euler:

Theorem 16 For any connected planar graph ${G = (V,E)}$ we have:

$\displaystyle |V| - |E| + |F| = 2$

where ${|V|, |E|, |F|}$ denote the numbers of vertices, edges, and faces of ${G}$ when ${G}$ is drawn in the plane. Note that we include the exterior’ face of ${G}$.

Below we write ${V, E, F}$ for ${|V|, |E|, |F|}$.

There are many proofs of this fact. See David Eppstein‘s collection of proofs here.

Note: There are well understood philosophical issues with this proof of Euler’s formula. See Lakatos. I feel that this is a very intuitive proof. You might disagree.

First embed ${G}$ in the plane. During the proof, we will perform a number of operations on ${G}$ which will: preserve the quantity ${V - E + F}$ and eventually lead to a familiar graph for which ${V-E+F}$ is easy to compute.

We will abuse notation and write ${V,E,F}$ for the vertices, edges, and faces of any graph that appears during our proof.

We proceed as follows:

Does ${G}$ have any cycles of length greater than three? If so, pick a vertex on the cycle, and add an edge to another vertex on the cycle. By the Jordan curve theorem (!) this new edge seperates a face. Thus, we add a face and an edge. Note that ${V - E + F}$ is preserved. Continue adding edges until there are no cycles of length greater than three.

We now simplify our graph. In the next two steps we preserve the fact that the boundary of our graph is a simple cycle. At each stage, one of two things may happen:

1. There is face which has only one edge of the boundary of ${G}$. If so, remove the boundary edge. This will remove an edge and a face. Note that this preserves ${V - E + F}$.
2. There is a face with two edges on the boundary of ${G}$. Remove the two boundary edges. This will: remove a vertex, two edges, and a face. Note that ${-1 - (-2) - 1 = 0}$, so again ${V - E + F}$ is preserved.

The final case is when there is a face with three edges on the boundary. Since we’ve ensured that the boundary remains simple, this implies that ${G}$ has been reduce to a triangle. For a triangle, we compute ${V - E + F = 3 - 3 + 2 = 2}$.

It is a subtle point that there is a sequence of moves as above which preserves the boundary.

We can now use Euler’s theorem to prove an upper bound on the number of convex regular polyhedra.

Theorem 17 There are at most five convex regular polyhedra.

Take any convex polyhedra, inscribe it in a sphere, radially project to the sphere, then stereographically project on to the plane from some point which is not a vertex. This will give the edge graph of the polyhedra in the plane.

We now show that regularity strongly constrains the kinds of graphs that can appear.

Regularity implies that each face is bounded by ${k}$ edges, and each vertex is incident to ${\ell}$ edges. We get, by double counting: ${kF = 2E}$ and ${\ell V = 2E}$. Euler’s theorem then gives:

$\displaystyle \frac{ 2E }{ \ell } - E + \frac{2E }{ k } = 2 \Rightarrow \frac{ 1 }{ \ell } - \frac{ 1 }{ 2 } + \frac{ 1 }{ k } = \frac{ 1 }{ E }$

Thus, we want to classify all ${E, k, \ell}$ satisfying:

$\displaystyle \frac{ 1 }{ k } + \frac{ 1 }{ \ell } = \frac{ 1 }{ 2 } + \frac{ 1 }{ E }$

Note that we need ${1/k + 1/\ell \geq 1/2}$ for this to be possible.

$\displaystyle \begin{array}{c|cccccc} & k=3 & 4 & 5 & 6 & 7\\ \hline \ell = 3 & 2/3 & 7/12 & 8/15 & 1/2 &\\ 4 & 7/12 & 1/2 & & & \\ 5 & 8/15 & & & & \\ 6 & 1/2 & & & & \\ 7 & & & & & \\ \end{array}$

We have noted all the solutions ${(k, \ell)}$ such that ${1/k + 1/\ell \geq 1/2}$. Moving down a column, or across a row, can only decrease the value of ${1/k + 1/\ell}$ so this table is complete. There are no solutions of ${1/k + 1/\ell = 1/2 + 1/E}$ where ${1/k+1/\ell = 1/2}$, so we are left with five candidate entries in our table.

Question 18 Write down all solutions of ${1/k + 1/\ell = 1/2 + 1/E}$ and write down which Platonic solids they correspond to.

Question 19 Draw the edge graphs of all the Platonic solids.

Question 20 Suppose Euler’s formula said: ${V - E + F = 0}$. Can you get platonic solid like objects by following the proof above?

Session 6

6.1. Szilassi

For a closed orientable genus ${g}$ surface ${\Sigma_G}$ we have, and any edge intersection free graph whose faces bound topological discs drawn on it, we have:

$\displaystyle \chi(\Sigma_g) = V - E + F = 2 - 2g$

If we take a genus one surface, a torus, we get: ${V - E + F = 0}$. This causes us to ask: are there toroidal equivalents of the Platonic solids? (Note: I’m using terminology very loosely here, what I mean to ask is: are there toroidial solids with a heigh degree of regularity in their face adjacency structures?) Following the proof from the previous session: suppose all the faces are bounded by ${k}$ edges, and each vertex meets ${\ell}$ edges. We get:

$\displaystyle 2E = kF \quad \textrm{and} \quad 2E = \ell V$

We then arrange:

$\displaystyle \frac{ 2E }{ \ell } - E + \frac{ 2E }{ k } = 0 \iff \frac{ 1 }{ \ell } + \frac{ 1 }{ k } = \frac{ 1 }{ 2 }$

We have previously classified all solutions ${1/k + 1/\ell = 1/2}$. They are as follows: ${(k,\ell) = (4,4), (3,6), (6,3)}$. This computation doesn’t tell us about ${(V,E,F)}$. It is interesting to note that all three of these solutions correspond directly to tilings of the plane by regular squares, triangles, and hexagons respectively.

Going back to our question about genus one polyhedra. It turns out that there is a miraculous polyhedron with irregular faces corresponding to the ${(k,\ell) = (6,3)}$ solution. It has a dual which is the ${(k,\ell) = (3,6)}$ solution. These are the Szilassi and Császár polyhedra. They have: ${(V,E,F) = (14,21,7)}$ and ${(V,E,F)=(7,21,14)}$.

One surprising property of the tetrahedron is that all of its faces are adjacent. Its edge graph, when embedded in the plane, shows us that we need at least four colours to the regions of any planar graph. The Szilassi polyhedron provides a higher genus analogue of this phenomena. Suppose a polyhedron has all of its faces adjacent. We have that ${F(F-1) = 2E}$. Also, in such a polyhedron, we’d have three edges at each vertex. Thus, ${3V = 2E}$. We’d then have:

$\displaystyle \begin{array}{lcc} V - E + F & = & 2 - 2g\\ \Rightarrow\frac{ F(F-1) }{ 3 } - \frac{ F(F-1) }{ 2 } + F & = & 2 - 2g\\ \end{array}$

Some further algebraic manipulation gives:

$\displaystyle \frac{ (F-3)(F-4) }{ 12 } = g$

For ${g = 1}$, we have the solution ${F = 7}$, which gives the Szilassi polyhedra.

Question 21 (Open) Is there a polyhedra of genus six with ${F=12}$ such that every face is adjacent to every other face?

It is curious that the lower bound for the chromatic number of of graphs on genus zero and one surfaces both come from polyhedra.

6.2. The Dodecahedron Exists

We give a concrete construction of a dodecahedron.

Consider a regular pentagon of side length one and vertices ${ABCDE}$. Consider the trapezoid ${ABCD}$. We have that ${AB=BC=CD=1}$ and ${AD = 1 + 2\sin\left( \frac{ \pi }{ 10 } \right)}$. Note that the length of ${AD}$ is the fabled “golden ratio”. Construct a square ${PQRS}$ with ${PQ=AD}$. Attach ${ABCD}$ to ${PQ}$ along ${AD}$ and another copy of ${ABCD=UVWX}$ by gluing ${UX}$ to ${RS}$. Now fold the trapezoids in until ${BD=VW}$.

Question 22 Calculate the sum of the dihedral angles at ${PQ}$ and ${QR}$.

One may piece together several of the solids constructe above to form a dodecahedron as details on CutOutFoldUp.com.

Session 7 — Groups and symmetries.

Groups are a vital component of modern mathematics. We’ll use them to describe the symmetries of the Platonic solids rigorously. They were initially invented to describe such symmetries, but they are interesting mathematical objects in themselves.

Definition 26 A group ${G}$ is a set with a map ${\cdot : G \times G \rightarrow G}$ and a distinguished element ${e \in G}$ such that:

• (Identity) For all ${g \in G}$ we have ${e \cdot g = g \cdot e = g}$.
• (Inverses) For all ${g \in G}$ there is ${g' \in G}$ such that ${g \cdot g' = g' \cdot g = e}$.
• (Associativity) For all ${g, h, k \in G}$ we have: ${g \cdot (h \cdot k) = (g \cdot h) \cdot k}$.
• Question 27 Show that there is a unique identity element in any group. Show that for any ${g \in G}$ there is a unique ${g' \in G}$ such that ${g \cdot g' = e}$. In every group ${(g \cdot h)' = h' \cdot g'}$. Give an example of a non-group.

Definition 28 We say that a map ${f : X \rightarrow Y}$ is bijective if:

• Injectvity If ${f(x) = f(y)}$ then ${x = y}$.
• Surjectivity For all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$.
• We usually work with ${S_{\{1, 2, \dots, n\}}}$, which we will abbreviate to ${S_n}$.

Cycle notation It is cumbersome to write down permutations, so we write down the following cycle notation’ instead. We write ${(x_1\ x_2\ \dots\ x_k)}$ for the map ${f(x_1) = x_2,\ f(x_2) = x_3,\ \dots ,\ f(x_k) = x_1}$. A cyclic permutation of two elements is a transposition. Try the following computations:

$\displaystyle (1 2)(3 2 4)(1 2) = (3 1 4) , \quad (1 2 3)(2 3) = (1 2), \quad (12)(23)(34) = (1234)$

Question 29 Show that every permutation can be written as a product of cycles. Show that every permuation can be written as a product of transpositions.

Also, try these computations:

$\displaystyle (12)(23) = (123) ,\ (23)(12) = (132)$

Definition 30 We say that a group ${G}$ is abelian (commutative) if: ${g \cdot h = h \cdot g}$ for all ${g, h \in G}$.

Groups arise as the symmetries of some object. A symmetry is broadly interpreted to mean `a transformation of an object which preserves its structure.’

Question 31 Describe the group of rotational symmetries of a tetrahedron. Describe the group of rotational symmetries which fix a cube.

Definition 32 Given a graph ${\Gamma = (V,E)}$ we say that a map ${f : V \rightarrow V}$ is an isomorphism of the graph if ${f}$ is bijective and ${\{u,v\} \in E}$ implies ${\{f(u), f(v)\} \in E}$.

Question 33 Describe the group of isomorphisms of a cyclic graph.