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Wythoff Construction and -Embedding Wythoff construction and -embedding Michel Deza Serguei Shpectorov ENS/CNRS, Paris and ISM, Tokyo Bowling Green State University Mathieu Dutour-Sikiric Hebrew University, Jerusalem – p.1/47 I. Wythoff kaleidoscope construction W.A. Wythoff (1918) and H.S.M. Coxeter (1935) – p.2/47 Polytopes and their faces A polytope of dimension is defined as the convex hull ¢ of a finite set of points in ¡ . A valid inequality on a polytope £ is an inequality of the ¥§¦ ¨ ¤ ¤ £ £ form © on with linear. A face of is the set of ¥ ¨ ¤ £ ¦ points satisfying to on . A face of dimension , , , is called, respectively, vertex, edge, ridge and facet. – p.3/47 Face-lattice There is a natural inclusion relation between faces, which define a structure of partially ordered set on the set of faces. This define a lattice structure, i.e. every face is uniquely defined by the set of vertices, contained in it, or by the set of facets, in which it is contained. ¢¡¤£ ¢¡ ¨ ¦ ¥ ¥ § Given two faces of dimension © and ¨ ¨ , there are exactly two faces of dimension , such ¦ ¦ ¡¤£ ¥ ¡ ¥ that § . This is a particular case of the Eulerian property satisfied by the lattice: Nr. faces of even dimension=Nr. faces of odd dimension – p.4/47 Skeleton of polytope The skeleton is defined as the graph formed by vertices, with two vertices adjacent if they form an edge. The dual skeleton is defined as the graph formed by facets with two facets adjacent if their intersection is a ridge. In the case of -dimensional polytopes, the skeleton is a planar graph and the dual skeleton is its dual, as a plane graph. Steinitz’s theorem: a graph is the skeleton of a -polytope if and only if it is planar and -connected. – p.5/47 Complexes We will consider mainly polytopes, but the Wythoff construction depends only on combinatorial information. Also not all properties of face-lattice of polytopes are necessary. The construction will apply to complexes: which are partially ordered sets, which have a dimension function associated to its elements. This concerns, in particular, the tilings of Euclidean -space. – p.6/47 Wythoff construction ¨ ¥ Take a © -dimensional complex . ¥ ¨ ¤ A flag is a sequence ¡ of faces with ¤¢¡ ¤ ¤¦¥ ¦ ¦¤£ ¦ ¥ £ £ § ¥ ¨ ¤ ¨ ¨ The type of a flag is the sequence ¡ . © © Given a non-empty subset of § § § , the ¥ ¨ © Wythoff construction is a complex £ , whose vertex-set is the set of flags with fixed type © . ¥ ¨ The other faces of © are expressed in terms of flags of the original complex . – p.7/47 Formalism of faces of Withoffian ¡ ¢ © £ ¦ ¤ Set § § § and fix an . For two ¦ ¦ © ¥ ¥ ¥ ¥ ¤ subsets , we say that blocks (from ) if, ¦ ¦ © ¥ ¥ ¤ ¤ ¤ for all § and ¨ , there is an § with ¦ ¦ © © © © § § § § ¨ ¨ or . This defines a binary relation ¦ ¥ ¥ © on (i.e. on subsets of § § § ), denoted by . ¦ ¦ ¦ ¦ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ © © Write , if and , and write if ¦ ¦ ¢ ¥ ¥ ¥ ¥ © © and . Clearly, is reflexive and transitive, i.e. an equivalence. ¥ ¥ is equivalence class containing . Minimal elements of equivalence classes are types of ¥ ¨ © faces of © ; vertices correspond to type , edges to ¦ ¦ © © © "next closest" type with , etc. – p.8/47 Example: the case ¡ , vertices 0,1 ¥ ¨ ¢ £¥¤ § One type of vertices for : (i.e. type © ). – p.9/47 .10/47 p – edges , and ¡ : ¨ ¥ ¤ £ § ¢ 1 or f case 0,2 edges the of types o w T Example: Example: the case ¡ , faces 2 0 ¥ ¨ ¢ £¥¤ § Two types of faces for : and – p.11/47 .12/47 p – with ¨ ¨ ¨ © © ¥ ¨ ¥ ¤ ¨ identified ¥ £ ¨ ¢¡ ¥ are £ ¤ ¥ £ ¢¡ es aph map r x ¥ ¢¡ g £ complexes uncated ¢¡ uncated tr tr iginal uncated plane comple tr or © ian aph r g set Euler . plane aphs a r -dimensional g is -dimensional If plane .13/47 p – ¨ ¡§ ¤ ¦ £¥¤ ¢ £¡ § cube ¢ ¨ the £¥¤ § ¢ on ¥ ¤ )= ythoff W Cube( .13/47 p – ¨ ¡§ ¤ ¦ cube £¥¤ ¢ the ¤ £¥¤ § ¢ on )= ythoff Cube( W .13/47 p – £¥¤ § ¢ cube ¤ £ ¢ ¤ ¨ the § § £ on )= ythoff Cube( W .13/47 p – ¨ ¡§ ¤ ¦ £ ¤ ¢ cube ¤ £ the ¢ ¤ ¨ § on § £ )= ythoff W Cube( .13/47 p – ¨ ¡§ ¤ ¦ £¥¤ ¢ £ § ¢ ¨ £ ¨ ¦ ¡ cube ¨ ¨ the ¡§ ¤ ¦ on £ ¤ ¢ £ § ¢ ¥ ¤ ythoff W )= Cube( .13/47 p – ¨ ¡§ ¤ ¦ £¥¤ ¢ £¡ § cube ¢ ¤ £ the ¢ ¤ ¨ § on § £ )= ythoff W Cube( Properties of Wythoff construction ¨ ¥ If is a © -dimensional complex, then: ¥ ¨ . ¨ ¥ ¤ © (dual complex). ¥ ¨ is median complex. ¨ ¥ ¨ ¥ ¤ £ £ £ £ ¤ ¨ ¨ © , where © © . ¢ admits at most different © Wythoff constructions. if is self-dual, then it admits at most different ¢ ¤ ¥ ¡ ¢ ¥ £ £ ¤ © Wythoff constructions. ¨ ¥ © § § § is called order complex. Its skeleton is bipartite and the vertices are full flags. Edges are full flags minus some face. ¨ ¨ Flags with faces correspond to faces of dim. © . – p.14/47 II. -embedding – p.15/47 .16/47 p – ¡ ), . ¡ £¥§ ¨ ¥ © ¡ points ¡ . ¦ . and o ¡ x-set x-set ¢ on : ¨ ¨ ¤ £ te te tw ¡ en ¨ ¨ © er er ¦ v v v e of ¨ ¢ ¨ on een ¦ ¦ ¥ ¥ © is © ¡ with with ¨ if if ¦ betw Half-cube ¨ © ¡ erence aph aph path-distance r r g ¦ ¥ g diff ¢ ¡ ic the and the the adjacent adjacent ¨ denotes path-distance £¥¤ is is ¡ ¡ ¤ ¨ the twice ¦ ¥ ¨ tices tices ¥ © is is distance cube symmetr ¦ er er is v v ¡ of o o (where e tw tw £¥§ siz ¦ ypercube ¤ Hyper Hamming h half-cube distance distance with with £¥¤ the . The = The and The and The The i.e ¦ .17/47 p – is is ¡ ¦ ¡ ¨ . aph r g cubes and a ¦ ypercube that ¨ ¨ § h into ¡ een yper ¥§¦ ¦ ¥ into such ¡ h ¥¤ embedding, ¥¤ , ¡ betw embedding. aph r g ¨ that aph ¨ ¨ a r ¨ into ¥ g £ ¢ ¥ of ¢ a ypercube ¨ such ¡ h ¨ of half-cube , ¦ ¦ ¥ ¡ ¢ ¦ ¥ is is ¢ ¢ ¢ ¥ £ ¥ path-distance ¥¤ ¦ ¡ the embedding ¢ ¢ ic embedding mapping embedding embedding being embedding x ¥¤ te isometr er scale mapping v A with An Scale scale a a Scale Examples of half-cube embeddings 34 14 45 3458 1458 1267 2367 4590 12 23 1267 2378 3478 1456 34 15 1256 2378 12 23 3489 1560 1456 3478 2378 14 34 1256 15 34 45 1458 3458 1560 3489 2367 1267 2378 1267 4590 23 12 23 12 Rhombicuboctahedron ¥ ¨ ¥ ¡ Dodecahedron © ¥ embeds into ¨ ¨ ¥ ¡ ¥ ¡ © embeds into (moreover, into : add ¢ to vertex-addresses) – p.18/47 Johnson and ¡ -embedding ¥ ¨ ¨ ¢ the Johnson graph is the graph formed by all © £ ¢ ¨ subsets of size of § § § with two subsets and ¦ © £ adjacent if . ¥ ¥ ¨ ¨ ¨ ¨ ¨ © ¡ ¡ © embeds in , which embeds in . ¤ A metric is ¥ -embeddable if it embeds isometrically ¥ ¦ ¤ into the metric space ¥ for some dimension . ¤ A graph is ¥ -embeddable if and only if it is scale embeddable (Assouad-Deza). The scale is or even. – p.19/47 Further examples 145 4 146 2379 27 2579 467 67 0 25 1259 3679 14 1245 7 46 367 3467 1468 1458 279 259 2 125 snub Cube embeds into ¥ twisted ¨¡ © , but not in any Rhombicuboctahedron is Johnson graph not ¡ -gonal – p.20/47 .21/47 p – the ic inequality the metr called § (Deza). then © is ¨ § ¨ £ ¨ yper iangular ¥ § h ¨ ¨ tr ¥ ¨ ¥ is ¨ the satisfied ¦ £ £ ¥ ¨ ¨ ¡ £ ¨ ys then ¡ a inequality , ¨ embedding, then ¤ ¦ ¥ , ¦ alw ¡ ¥ © ¤ then § is , § ¡ ¨ £ ¡ ¨ § scale ¡ ¡ a © ¡£¢ . ¦ ¥ § ¨ £ § ¥ § ¨ © inequality © and admits ic is ¥ ic inequality ¡ § Hypermetric ¥ ¥ metr ¤ metr £ £ £ a yper -gonal ¡ If If If If inequality h Embedding of graphs The problem of testing scale embedding for general metric spaces is NP-hard (Karzanov). ¡ ¨ Theorem(Jukovic-Avis): a graph embeds into ¡ if and only if: ¡ is bipartite and ¥¤ satisfies the ¡ -gonal inequality. ¡ ¨ In particular, testing embedding of a graph into ¡ is polynomial. The problem of testing scale embedding of graphs into ¥ ¨ ¡ © is also polynomial problem (Deza-Shpectorov). – p.22/47 III. -embedding of Wythoff construction – p.23/47 Regular (convex) polytopes A regular polytope is a polytope, whose symmetry group acts transitively on its set of flags. The list consists of: regular polytope group ¥ ¨ £ ¨ © regular polygon ¡ Icosahedron and Dodecahedron ¨¢¡ ¨¥¤ -cell and £ -cell ¦ -cell ¤ ¨ © § ¡ ¡ ¡ (hypercube) and (cross-polytope) ¥ ¨ © ¨ ¨ ¡ ¡ (simplex) = ¦ ¦ £ There are regular tilings of Euclidean plane: © , and £ , and an infinity of regular tilings of hyperbolic plane. ¥ ¨ Here is shortened notation for . – p.24/47 2-dim. regular tilings and honeycombs Columns and rows indicate vertex figures and facets, resp. ¡ Blue are elliptic (spheric), red are parabolic (Euclidean). Å ¤ 2 3 4 5 6 7 m 2 22 23 24 25 26 27 2m ¨ £ ¡ 3 32 ¡ Ico 37 3m ¦ § © 4 42 ¡ 45 46 47 4m ¡ 5 52 Do 54 55 56 57 5m £ 6 62 63 64 65 66 67 6m ¡ 7 72 73 74 75 76 77 7m m m2 m3 m4 m5 m6 m7 mm ¨ ¡ ¦ £ ¡ ¨ – p.25/47 .26/47 p – ¨ , ¤ ¨§¦ ) ¨¥¤ ) ¥ © ¥ © © £ embed . into ¥ . into or or ely (f ¥ . ; ¨ . en and v ely embed embeds ¦ e i.e ¥ , ¡ ¨ is ) embeds ¥ © respectiv ) , ¥ © ic if ¡ ¡ © ¢ ¥ ¥ © respectiv ¡ (elliptic ¥ , ¨ holds: ic spher it ¥ ¦ , ¨ odd, and ¥ ¥ into . ¡ ¡ . y i.e Spher and since , Dodecahedron . and an (i.e ¡ ¨ . en into or v f , and resp ¥ ¨ ¡ e odd ely ¡ ¨ ¨ , abolic ¥ ¡ is embed, tiling (par or embed ¡ § f if £ ¡ ¢ into and ¨ , ¡ © ¥ ¡ ¨ respectiv tilings ¥ © ¥ © , e ¡ ¥ ¨ v and © £ ¥ © into Hyperbolic Euclidean embeds and into Icosahedron abo All 3-dim. regular tilings and honeycombs ¥§¦ ¤ ¢¡ £ ¡ ¡ Do Ico 63 36 ¤ ¨¡ © © 600- 336 ¤ ¡ 24- 344 ¥ £ £ © ¡ ¡ 435* 436* Ico 353 Do 120- 534 535 536 ¥ ¦ 443* 444* 36 363 63 633* 634* 635* 636* ¨ © ¢ ¢ ¥ ¥ § All emb. ones with are, besides and § : all £ § § ¢ ¢ ¢ ¢ ¥ ¥ § bipartite ones (i.e. with cell , £ or ): , and ¢ ¡ ¦ ¡ £ hyperbolic tilings with . Last 11 embed into . – p.27/47 .28/47 p – © ¢ ¥ ¡ © ¢ -space ¡ ¢£ ¢£ ¢ ¢ ¢ © 600- 5335 yperbolic h honeycombs 120- of ¨ ¤ © ¢ © § and ¢ 24- 4343 © ¤¦¥ ¤ © ¤ ¥ © 5334 tilings ¨ ¤ © £ © § ¦ (non-compact) . ¢ ¡ and © ¡ £ egular 5333 r into 4335 © ¤ £ © ¥ © ¡ 24- 600- 120- Tilings embed 4-dim. .29/47 p – . ¢ , ¢ ¤ ¨ planes . and e ¢ 34334 , ¨ ely
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