Abstract
It is well known that the unique 7-vertex triangulation of the 2-dimensional torus S1×S1 is a consequence of the relationship between two hexagonal lattices in the euclidean plane: it is just the quotient of the triangular tessellation of the plane by a translation group. Each vertex star is a regular hexagon and the symmetry group of this triangulation is the affine group A(1,ℤ7) in one dimension over ℤ7. In this paper we describe a particular 15-vertex triangulation of the 3-dimensional torus S1×S1×S1 whose symmetry group is the affine group A(1,ℤ15) and which is similarly related to two lattices in euclidean 3-space: it is just the quotient of a particular tessellation of 3-space by a translation group. Each vertex star happens to be a rhombidodecahedron, the dual of a (semiregular) cuboctahedron.
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References
ALTSHULER,A.:Combinatorial 3-manifolds with few vertices, J. Comb. Th. (A)16, 165–173 (1974)
ALTSHULER,A.:Neighborly 4-polytopes and neighborly combinatorial 3-manifolds with ten vertices, Can. J. Math.29, 400–420 (1977)
ALTSHULER,A., STEINBERG,L.:Neighborly combinatorial 3-manifolds with 9 vertices, Discr. Math.8, 113–137 (1974)
COXETER,H.S.M., MOSER,W.O.J.:Generators and Relations for Discrete Groups (4th ed.), Berlin-Heidelberg-New York: Springer 1980 (Ergebnisse der Mathematik und ihrer Grenzgebiete 14)
CROWE,D.W.:Steiner triple systems, Heawood's torus coloring, Császár's polyhedron, Room designs, and bridge tournaments, Delta3, 27–32 (1972)
CSÁSZÁR,A.:A polyhedron without diagonals, Acta Sci. Math. (Szeged)13, 140–142 (1949)
EDMONDS,A.L., EWING,J.H., KULKARNI,R.S.:Regular tessellations of surfaces and (p,q,2)-triangle groups, Ann. Math.116, 113–132 (1982)
EDMONDS,A.L., EWING,J.H., KULKARNI,R.S.;Torsion free subgroups of Fuchsian groups and tessellations of surfaces, Invent. math.69, 331–346 (1982)
GRÜNBAUM,B., SHEPHARD,G.C.:Tilings with congruent tiles, Bull. Amer. Math. Soc.3, 951–973 (1980)
HEAWOOD,P.J.:Map-colour theorem, Quart. J. Math.24, 332–338 (1890)
KÜHNEL,W., LASSMANN,G.:The unique 3-neighborly 4-manifold with few vertices, J. Comb. Th. (A)35, 173–184 (1983)
KÜHNEL,W., LASSMANN,G.:Neighborly combinatorial 3-manifolds with dihedral symmetry group, (in prep.)
MÖBIUS,A.:Gesammelte Werke, Vol.2, Leipzig 1886
RINGEL,G.:Map Color Theorem, Berlin-Heidelberg-New York: Springer 1974 (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 209)
STREET,A.P.:Some designs with block size three, Combinatorial Mathematics VII, 224–237, Berlin-Heidelberg-New York: Springer 1980 (Lecture Notes in Mathematics 829)
THURSTON,W.P.:Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc.6, 357–381 (1982)
TSUZUKU,T.:Finite Groups and Finite Geometries, Cambridge University Press 1982 (Cambridge Tracts in Mathematics 78)
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Kühnel, W., Lassmann, G. The rhombidodecahedral tessellation of 3-space and a particular 15-vertex triangulation of the 3-dimensional torus. Manuscripta Math 49, 61–77 (1984). https://doi.org/10.1007/BF01174871
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DOI: https://doi.org/10.1007/BF01174871