Regular polytopes with Hd symmetry Johannes Kepler’s mathematical stars Gabriel Pallier 1 / 34 Outline Background on convex and star polyhedra Star polytopes and platonic Riemann surfaces Schläfli-Hess polytopes 2 / 34 Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. 3 / 34 Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 3-gon. 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 4-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 5-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 5-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if 4 / 34 I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and 4 / 34 Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. 4 / 34 There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. 4 / 34 Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. 4 / 34 2 The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric sphere S2. 5 / 34 Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. 5 / 34 As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q 5 / 34 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 5 / 34 where the last equality accounts for χ(S2) = 2, after possibly triangulating the 2-cells. Cell numbers Using the expression for Area(4) and Euler’s formula, one can recover the full combinatorial data of fp; qg: denoting by fp; qg(k) the set of k-cells 4π 4q fp; qg(2) = = ; (Conv 2) 2p · Area(4) 2p + 2q − pq 4p fp; qg(0) = fq; pg(2) = ; (Conv 0) 2p + 2q − pq 2p + 2q + pq fp; qg(1) = fp; qg(2) + fp; qg(0) − 2 = ; (Conv 1) 2p + 2q − pq 6 / 34 Cell numbers Using the expression for Area(4) and Euler’s formula, one can recover the full combinatorial data of fp; qg: denoting by fp; qg(k) the set of k-cells 4π 4q fp; qg(2) = = ; (Conv 2) 2p · Area(4) 2p + 2q − pq 4p fp; qg(0) = fq; pg(2) = ; (Conv 0) 2p + 2q − pq 2p + 2q + pq fp; qg(1) = fp; qg(2) + fp; qg(0) − 2 = ; (Conv 1) 2p + 2q − pq where the last equality accounts for χ(S2) = 2, after possibly triangulating the 2-cells. 6 / 34 This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively. The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) 7 / 34 However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively. The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. 7 / 34 The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively.