Classifying Regular Polyhedra and Polytopes using Wythoff's Construction
CClassifying Regular Polyhedra and Polytopes using Wythoff ’sConstruction
Spencer Whitehead ∗ orcid.org/0000-0001-8747-8186.Department of Pure Mathematics, University of Waterloo [email protected] January 29, 2020
Abstract
A polytope is the generalization of a polyhedron to any number of dimensions. The regularpolyhedra are the Platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodec-ahedron. The hypercubes, hyperoctahedra, simplices, and regular polygons form four infinitefamilies of regular polytopes. Ludwig Schl¨afli proved that with the addition of five exceptionalsolids (the icosahedron and dodecahedron in 3 dimensions, and the 24-cell, 120-cell, and 600-cell in 4 dimensions) this list is complete. This paper provides an alternate proof to Schl¨afli’sresult, using Wythoff’s construction and the theory of decorated Coxeter diagrams.
A regular polyhedron is a polyhedron whose faces and vertex figures are all regular polygons.Theaetetus proved around 400 BC that the only regular polyhedra are the tetrahedron, cube, octa-hedron, dodecahedron, and icosahedron. An axiomatic proof of this fact occurred one hundred yearslater, in the twelfth book to Euclid’s
Elements . Descartes’s proved in 1623 a theorem on the totalspherical excess of a convex polyhedron, from which the classification of regular polyhedra follows.Perhaps the simplest proof is due to Euler, who derived the classification from the observation thatfor a convex polyhedron with V vertices, E edges, and F faces, the equation V − E + F = 2.In 1852, Schl¨afli introduced in [1] the concept of a polytope , a generalization of a polyhedronto higher dimensions. Schl¨afli defined regular polytopes , and proved that they occur in four infinitefamilies (the regular polygons, hypercubes, hyperoctahedra, and simplices), along with five excep-tional structures (the dodecahedron, icosahedron, 24-cell, 120-cell, and 600-cell). The purpose ofthis paper is to re-derive Schl¨afli’s result using Coxeter’s theory of reflection groups. As a specialcase of this result, we find another classification of the regular polyhedra.Geometers of the early 20th century interested themselves with uniform polytopes: generaliza-tions of regular polytopes requiring a weaker symmetry condition. Wythoff described a way tounderstand these polytopes using kaleidoscopes in [2], where he investigated a similarity betweentwo 4-dimensional uniform polytopes of H symmetry. This result was generalized by Coxeter in ∗ The author was partially supported by a NSERC USRA grant. The author thanks Benoit Charbonneau for hisassistance in the preparation of this paper. a r X i v : . [ m a t h . M G ] J a n igure 1: The left is a typical example of a flag vertex ⊂ edge ⊂ face. Unrolling it, one gets theimage on the right, motivating the name.[3] to a method of generating uniform polytopes in any dimension, called Wythoff ’s construction .Together with the results of Coxeter’s papers [4], [5], which classified all the finite reflection groups,it is easy to enumerate all polytopes arising from this construction, called the
Wythoffian polytopes.
To classify the regular polytopes using Wythoff’s construction, we need to know that regularpolytopes are Wythoffian, and we need a way to decide which polytopes arising from Wythoff’sconstruction are regular.That regular polytopes are Wythoffian was proved by Schulte and McMullen in [6]. This pa-per proves this fact again, with a more intuitive geometric construction, by demonstrating thatthe kaleidoscope with mirrors between boundaries of top dimensional faces of a regular polytopegenerates its dual polytope.The paper [7] of Champagne, Kjiri, Patera, and Sharp provides a rich set of combinatorial datadescribing the faces of a polytope generated by Wythoff’s construction. Using these data, thepossible regular polytopes can be restricted to those corresponding to a particular list of decoratedCoxeter diagrams. The diagrams on this list are known to all correspond to regular polytopes,which completes the classification.Section 2 defines polytopes and regular polytopes. Section 3 describes Wythoff’s constructionand Coxeter’s classification of finite groups generated by reflections. Section 4 proves that all regularpolytopes arise from Wythoff’s construction. Section 5 exploits the combinatorics of decoratedCoxeter diagrams to classify the regular polytopes. A polytope is a bounded set of solutions to an inequality of the form Ax ≤ b where A ∈ R m × n , b ∈ R m , or equivalently the convex hull of a finite set of points in R n . A k -face of a polytope is abounding element contained in an affine subspace of dimension k . A 0-face is called a vertex , a1-face is called an edge , and a 2-face is simply called a face , when there is no chance of confusion.A ( n − facet , and a ( n − ridge .A flag in a polytope is a chain (vertex ⊆ edge ⊆ · · · ⊆ facet); see Fig. 1 for an illustration ofthe terminology. Two flags are said to be adjacent if they differ in only one element.A regular polytope is defined recursively: a regular polygon is a polygon with equal edges andedge lengths, while a regular polytope is a polytope whose group of symmetries acts transitively on2igure 2: The five Platonic solids, from Kepler’s Harmonices Mundi [8]its flags, and whose facets are all themselves regular polytopes. In three dimensions, the regularpolytopes are the familiar platonic solids of Fig. 2.Another definition of regularity uses vertex figures . When the midpoints of the edges about avertex in a polytope all lie on a hyperplane, the intersection of the hyperplane and the polytope isa polytope of one lower dimension, called the vertex figure of the polytope about that vertex. Forexample, the vertex figure of a cube about any vertex is an equilateral triangle. Roughly speaking,the vertex figure at v is a “localization at v ”: edges containing v become vertices of the vertexfigure, faces containing v become edges of the vertex figure, and so forth, in a way maintainingthe incidence structure. Another definition of regularity is then that a polytope is regular whenall its vertex figures exist, they are all regular, and the facets are all regular. These definitions areequivalent, and there are many more equivalent definitions; see [9].All polytopes satisfy the following two properties (see, for instance, [6, Chap. 1]): Property 1 (Diamond property) . Given a ( k − F k − and a ( k + 1)-face F k +1 of a polytope,there exist exactly two k -faces F k , F (cid:48) k so that F k − ⊆ F k , F (cid:48) k ⊆ F k +1 . Property 2 (Flag-connectedness) . Given two flags
F, F (cid:48) of a polytope, there exists a sequence offlags F = F , F , . . . , F n − , F n = F (cid:48) having F i adjacent to F i +1 for all i .The diamond property remains valid even for the extremal values k ∈ { , n − } , so long as weinterpret a − n -face as the entire polytope. It says, for example, thateach edge contains two vertices, and that each vertex is incident to exactly two edges of every faceit is contained in. A consequence of the diamond property is that each flag is adjacent to exactly n other flags: if n − P we may associate its dual polytope , whose vertices are the centers of the facets of P . A k -face of P becomes a ( n − k )-face of its dual, and this map reverses inclusions. A consequenceof this fact is that the dual polytope is regular. Moreover, up to scaling, the dual of the dual of P is again P . Among the Platonic solids, the cube is dual to the octahedron, the icosahedron is dualto the dodecahedron, and the tetrahedron is self-dual.3igure 3: The left image demonstrates the diamond property. With k = 0 (red), each edge has twovertices. With k = 1 (green), each vertex is incident to two edges in a face. With k = 2 (blue),each edge borders two faces. The right shows two flags in a cube: to get from one to the other, firstchange the blue vertex to the red vertex, then the blue face to the red face, then the blue edge tothe red edge. A n ··· B n ··· I ( k ) k D n ··· H H F E E E Figure 4: A list of the irreducible Coxeter diagrams. The subscript indicates the number of vertices. A uniform polytope is defined in a similar way to a regular polytope. In two dimensions, the uniformpolygons are defined to be exactly the regular polygons. In higher dimensions, a polytope is saidto be uniform if all its facets are uniform, and its group of symmetries acts transitively on itsvertices. Given a finite subgroup of O ( n ) generated by reflections and a point in R n , we considerthe convex polytope whose vertices are the orbit of the point under the group. Coxeter showed in[3] that a judicious choice of point results in a uniform polytope. The uniform polytopes arisingfrom this construction are called Wythoffian . Almost all known uniform polytopes arise from thekaleidoscopic construction of Wythoff and Coxeter. Conway and Guy show in [10] that of the 63four-dimensional uniform polytopes not occurring in infinite families, only three are not Wythoffian.An finite group generated by reflections through hyperplanes through the origin in R n is calleda (spherical) Coxeter group . Coxeter showed in [4] that these groups have presentations of the form (cid:104) r , . . . , r n | ( r i r j ) m ij = 1 , i, j = 1 , . . . n (cid:105) for some symmetric positive integer matrix ( m ij ) with m ii = 1 for all i . Such a group can bemodelled in O ( n ) by associating to each r i a reflection through a hyperplane in R n containing theorigin. To ensure the condition ( r i r j ) m ij = 1, the hyperplanes associated to r i and r j should meetat dihedral angle πm ij . This information is encoded into a Coxeter diagram consisting of a vertex4or every mirror, and a line between mirrors i, j with label m ij . Conventionally, if m ij ≤ m ij = 3 the label is omitted. Coxeter’s classification showed that the Coxeterdiagram of any spherical Coxeter group is a disjoint union of the diagrams in Fig. 4To pick a point for Wythoff’s construction, select some subset S of these mirrors to be astabilizer. Then take any point of norm 1 that lies on every mirror of S , and is equidistant fromevery mirror not in S . The resulting polytope is uniform, and moreover, all points resulting inuniform polytopes arise from this procedure. This polytope is represented by drawing the Coxeterdiagram of the group and putting a cross through each box corresponding to a mirror in S ; thisis called the Wythoff-decorated Coxeter diagram , or a
Wythoff-decorated diagram for short. Twoexamples of Wythoff-decorated diagrams and the polytopes they represent are given in Fig. 5. If S contains a full connected component of the Coxeter diagram, any mirror in that component fixesthe whole polytope, and the polytope is contained within a hyperplane.The rest of the paper is dedicated to showing the completeness of Table 1, which shows howregular polytopes correspond to certain Wythoff-decorated diagrams. r r r r Figure 5: Two examples of Coxeter diagrams and the polytopes they represent—the left is a square,and the right is an equilateral triangle. The thick lines are the reflecting mirrors, and the whitepoints are images of the black point in the group generated by reflection through the mirrors.
Theorem 3.
Let P ⊆ R n be a regular polytope. Then P is Wythoffian.Proof. To the polytope P we may associate a center , obtained by averaging its vertices. A symmetryof the polytope is an affine isometry that permutes the vertices, and hence fixes the center. Assumewithout loss of generality that P is centered at the origin. Then symmetries of P are orthogonaltransformations—that is, rotations and reflections.Assume without loss of generality that P is not contained in a hyperplane. If it is, it can beprojected through hyperplanes until this condition holds in lower dimension, and then the followingproof applies. If some facet contains the origin, then P is contained in a half-space of the form { x ∈ R n : a T x ≥ } . Since P is not contained in the hyperplane a T x = 0, for each vertex v wehave a T v ≥
0, and for at least one vertex, a T v >
0. Then if c is the center of P , we find that a T c >
0. On the other hand, c is the origin, so a T c = 0. A contradiction is reached, and so nofacet contains the origin. Since each lower dimensional face is contained in a facet, no face of anydimension contains the origin.Let R be a ridge of P . Then R is contained in an affine subspace of dimension n −
2, anddoes not contain the origin, so it has linear span of dimension n −
1. Denote by H the unique5roup Diagram Polytope I ( k ) k regular k -gon I ( k ) k regular 2 k -gon A octahedron H icosahedron H dodecahedron H H F B D A n ··· n -simplex B n ··· n -hypercube B n ··· n -hyperoctahedron D n ··· n -hyperoctahedronTable 1: A list of Wythoff-decorated diagrams whose polytopes are regular.hyperplane spanned by R . By Property 1, R is contained in exactly two facets, say F and F (cid:48) . Byflag transitivity, there exists a symmetry s R of P fixing R and sending F to F (cid:48) . Since s R fixes R ,it fixes H by linearity. Because s R is an isometry and is not the identity, it must be reflection in H , and in particular, it is unique.Let W be the subgroup of the symmetries of P generated by all the s R . Since P has finitesymmetry group, W is finite also. By Property 2, given any two facets F, F (cid:48) there is a sequenceof adjacent flags Φ , . . . , Φ n such that Φ has F as a facet, and Φ n has F (cid:48) as a facet. Write F i forthe facet of Φ i , and R i for the ridge of Φ i . When F i (cid:54) = F i +1 , adjacency of Φ i and Φ i +1 guaranteesthat R i = R i +1 , so that F i +1 = s R i F i . If F i = F i +1 , let w i = 1. Otherwise, let w i = s R i . Then w n · · · w F = F (cid:48) , so W acts transitively on the facets of P .Pick a facet F of P . Since F is also a polytope of one lower dimension, we may pick the center x of F , and consider the set Q = conv( W x ). As W is finite, Q is a polytope. In fact, Q is theconvex hull of the centers of the facets of P , and is therefore dual to P . Since Q is the polytopegenerated by taking the orbit of a point under a finite reflection group, it is Wythoffian. What wehave shown is that when P is regular, the dual of P is Wythoffian. Consequently, since the dual ofa regular polytope is itself regular, P is Wythoffian. A decoration of a Coxeter diagram attaches some additional data to a typical diagram. Section 3described how Wythoff-decorated Coxeter diagrams could be used to construct uniform polytopes.To distinguish Coxeter diagrams of regular polytopes, we use the decorations described in [7], whichaugment the decorations for Wythoff’s construction. Starting from a Wythoff-decorated Coxeter6
Figure 6: Two CKPS-decorations based on giving 2-faces are shown, correspondingto an octagon and a triangle. The triangle decoration is obtained by first circling the middle vertex.The leftmost vertex is then uncrossed by the transformation rule, so it may be circled.diagram, pick an uncrossed box and replace it with a circle. Then uncross all boxes adjacentto the circle. This rule is called the (CKPS) transformation rule. Decorated Coxeter diagramsobtained from repeatedly applying this rule to a Wythoff-decorated Coxeter diagram are called
CKPS-decorated Coxeter diagrams , or
CKPS-decorated diagram for short. Such a CKPS-decorateddiagram is said to be based on the Wythoff-decorated diagram from which it was created.
Theorem 4 ([7]) . The orbits of k -faces of a Wythoffian polytope are in correspondence with theCKPS-decorated diagrams having k circles in them, obtained from a Wythoff-decorated diagram byapplying the CKPS transformation rule repeatedly. The correspondence sends a CKPS-decorateddiagram X based on Y to a Wythoff-decorated diagram by taking the sub-diagram of Y whose verticesin X are decorated with circles. For an example of the theorem, see Fig. 6. This example motivates the use of CKPS-decorateddiagrams to classify regular polytopes: the fact that two distinct diagrams with 2 circles exist provesthat the polytope cannot be regular. One can also think of the transformation rule “backwards”;by deleting k vertices from a Wythoff-decorated diagram in such a way that there is no connectedcomponent with no uncrossed square, we obtain a ( n − k )-face of the polytope it represents, andmoreover, all ( n − k )-faces are obtained in this way.As an application of these rules, we can restrict the number of uncrossed squares in connectedWythoff-decorated diagrams representing regular polytopes. Precisely, if such a diagram has sizeat least 3, it has exactly one uncrossed square. As before, and for the rest of this section, polytopesare assumed to not be contained in a hyperplane, so that each connected component has at leastone uncrossed square. Lemma 5.
If a connected Wythoff-decorated diagram with at least three nodes represents a regularpolytope, then it has exactly one uncrossed square.Proof.
Suppose that i, j are two uncrossed squares in the diagram, and pick a third square k adjacentto one of them. Assume without loss of generality that k is adjacent to i . Then since the Coxeterdiagrams are all acyclic, k is not adjacent to j .If k is uncrossed, then circling i and k gives a 2 m ik -gon face, while circling k and j gives a2 m jk -gon face. Since j, k are not adjacent, m jk = 2. On the other hand, m ik ≥
3. So 2 m ik (cid:54) = 2 m jk and the diagram does not represent a regular polytope.7f k is crossed, then circling i and j gives a 2 m ij -gon face, while circling i and k gives a m ik -gonface. The only way the polytope can be regular is to have 2 m ij = m ik . Since the labels on theCoxeter diagrams of size at least 3 are between 2 and 5, to have 2 m ij = m ik only happens when m ik = 4 , m ij = 2. Additionally, any connected diagram with a label of 4 has exactly one 4, withthe rest of the labels being 2 or 3. Since the diagram is connected, there exists a node (cid:96) adjacentto j and we have m j(cid:96) = 3. Circling j then (cid:96) gives a triangular face if (cid:96) is uncrossed, or a hexagonalface if (cid:96) is crossed. In either case, we conclude that the polytope has a non-square face, and ishence not regular.Given a disconnected Coxeter diagram with components X and Y of size n and m respectively,choose n hyperplanes with normals v , . . . , v n ∈ R n so that the group generated by reflection throughthese hyperplanes is isomorphic to the Coxeter group of X . Similarly, choose w , . . . , w m ∈ R m so the group generated by reflection through the w i is isomorphic to the Coxeter group of X . Let v (cid:48) i = ( v i , , , . . . , ∈ R n + m , and w (cid:48) i = (0 , , . . . , w i ) ∈ R n + m . Then the angle between v (cid:48) i and v (cid:48) j isequal to the angle between v i and v j , and v (cid:48) i is orthogonal to each w (cid:48) j . Similarly, the angle between w (cid:48) i and w (cid:48) j is equal to the angle between w i and w j , and so the group generated by reflections throughall the v (cid:48) i , w (cid:48) j is the Coxeter group of the diagram X ∪ Y . Considering how v (cid:48) i and w (cid:48) j act on elementsof R n + m shows that the polytope represented by a disjoint union of connected Wythoff-decorateddiagrams is the Cartesian product of the polytopes represented by the connected components. Forexample, adding a single uncrossed box to a diagram for a polytope P gives us a “ P -prism”, whichlooks like [ − δ, δ ] × P for the appropriate δ . We have Lemma 6.
The only regular polytope represented by a disconnected Wythoff-decorated diagram isa hypercube.Proof.
Suppose our polytope is P . Take X to be a connected component of the diagram, say ofsize k , and denote by P X the polytope represented by X .If k = n −
1, then by circling a vertex of X and the vertex not in X we see that P X has a squareface. By regularity, every face of P , and hence of P X , is square. Let v be an uncrossed vertex in X . Circling v and then any adjacent vertex must produce a square face; so every neighbour of v iscrossed, and the edge between them has label 4. The only Coxeter diagrams having label 4 are F and B n for various n . But X cannot be a Wythoff-decoration of F , because the vertices incidentto an edge of label 4 also are incident to edges with label 3. So X = · · · , and P X is a hypercube.If k + 1 < n , by deleting n − k − X we obtain a diagram Y that is the disjointunion of X , with an additional isolated uncrossed vertex u . Denote by P Y the polytope representedby Y . Since P Y is a ( k + 1)-face of the regular polytope P , it is regular, and since Y is disconnectedand has size k + 1 < n , we see inductively that P Y is a hypercube. Delete u from Y to see that P X is a facet of P Y . Since the facets of hypercubes are lower dimensional hypercubes, P X is also ahypercube.In this fashion, we find that every connected component of the diagram represents a hypercubeof dimension equal to its size. Then P is the product of the respective hypercubes, which is againa hypercube. 8 heorem 7. The regular polytopes are comprised of the infinite families of simplices, hypercubes,hyperoctahedra, and regular polygons, as well as five exceptional structures: the icosahedron, dodec-ahedron, 120-cell, 600-cell, and 24-cell.Proof.
By Theorem 3, it suffices to classify the Wythoff constructions that are regular. Moreover,by Lemma 6 we can without loss of generality take our Coxeter diagram to be connected. Thus itmust be one of the diagrams from Fig. 4.All non-empty decorations of I ( k ) give a regular polytope for any k ∈ Z + ; k gives theregular k -gon, and k gives the regular 2 k -gon. These are all the 2-dimensional Wythoff-decorated diagrams. Henceforth, we turn our attention to connected Wythoff-decorated diagramsof size at least 3. By Lemma 5, it suffices to consider the diagrams having a single uncrossed square. · · ·· · ·· · · n -hyperoctahedron · · ·· · · n -simplexFigure 7: Two distinct faces of a putative regular polytope of E k symmetry for some k Suppose that the diagram has a branch point, as in the case of D n ( n ≥ , E , E , E . Takethe uncrossed square in the diagram, and repeatedly circle vertices towards the direction of thebranch point. When the branch point is reached, there are two cases. If one branch has lengthlonger than two, then circling the first vertex of each branch and circling the first two vertices ofthe longer branch gives two different decorations; the first one is a hyperoctahedron, and the secondis a simplex. In particular, no decoration of E , E , E can be regular—as each branch point hastwo branches of length at least two. For an illustration of this process, see Fig. 7.In the case of D n , n ≥
5, since both small branches of D n have length 1, the uncrossed squaremust always lie on the long branch. In fact, the uncrossed vertex must be on the end of the longbranch; otherwise, once the branch point is reached, circling one of the two branches and thena vertex behind our initial one gives a different decoration than circling two branches; see, forexample, Fig. 8. Finally, one can check that this decoration of D n gives the n -hyperoctahedron.In the case where n = 4, one additional decoration giving a regular polytope is possible, where thecenter vertex is uncrossed. This diagram represents the 24-cell.When the diagram does not have a branch point, it must be a path. In the case of A n , when n > a b b Figure 8: In an decoration of D having only the a vertex uncrossed, one sequence of transformationsleads us to this step. In the next two steps, choosing to circle b and b gives a different diagramthan choosing to circle b and b square. The decoration of A n with uncrossed vertex at either end corresponds to the n -simplex.The only remaining case is when n = 3 with the decoration ; we already saw thispolytope was an octahedron.If the diagram is not a Wythoff-decoration A n and is a path, it must be a decoration of oneof B n , H , H , or F . Numbering the vertices in this path 1 , , . . . , n , we have some j so that m j − ,j >
3. Let i be the square that is uncrossed, and suppose that i is not at one of the ends ofthe path. Up to possibly relabeling the vertices of the path, we may assume that 1 < i < j . Sinceeach connected Coxeter diagram has at most one label not a 2 or 3, we obtain two diagrams: oneby circling vertices i, . . . , j , and another by circling i − , . . . , j −
1. If the polytope is to be regular,these ( j − i + 1)-faces must be the same and both themselves regular. The case of A n handled aboveshows that the only way this condition can occur is if j − i +1 = 3, and the cells mj − ,j andare the same. Since is an octahedron, we must have m j − ,j = 4.The only possible diagram satisfying this information is , which representsthe 24-cell.Otherwise, the only decorations of H , H , F , B n representing regular polytopes must have theuncrossed square at one of the ends of the path. All such decorations do give regular polytopes: is the icosahedron, is the dodecahedron, isthe 600-cell, is the 120-cell, and is the 24-cell. For adecoration of B n , when the uncrossed square is on the edge with a 4, the result is the n -hypercube.If it is instead on the edge marked with a 3, it is the n -hyperoctahedron. References [1] L. Schl¨afli,
Theorie der vielfachen Kontinuit¨at . Cornell University Library historical mathmonographs, Z¨urcher & Furrer, 1901.[2] W. A. Wijthoff, “A relation between the polytopes of the C600-family,”
Koninklijke Neder-landse Akademie van Wetenschappen Proceedings Series B Physical Sciences , vol. 20, pp. 966–970, 1918.[3] H. S. M. Coxeter, “Wythoff’s Construction for Uniform Polytopes,”
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Annals of Mathematics , vol. 35,no. 3, pp. 588–621, 1934.[5] H. S. M. Coxeter, “The complete enumeration of finite groups of the form r i = ( r i r j ) k ij = 1,” J. London Math. Soc., (1) , vol. 10, pp. 21–25, 1935.106] P. McMullen and E. Schulte,
Abstract Regular Polytopes . Cambridge University Press, 2002.[7] B. Champagne, M. Kjiri, J. Patera, and R. T. Sharp, “Description of reflection-generatedpolytopes using decorated Coxeter diagrams,”
Canadian J. Physics , vol. 73, pp. 566–584,1995.[8] J. Kepler,
Harmonices Mundi . Linz, 1619.[9] P. McMullen, “On the combinatorial structure of convex polytopes,” 1968.[10] J. H. Conway and M. J. T. Guy, “Four-dimensional Archimedean polytopes,” in