Existence of A-avoiding paths in abstract polytopes

1974 ◽  
pp. 41-42 ◽  
Author(s):  
Ilan Adler ◽  
George Dantzig ◽  
Katta Murty
Keyword(s):  
Abstract Polytopes

Free Extensions of Chiral Polytopes

1995 ◽  
Vol 47 (3) ◽  
pp. 641-654 ◽  
Author(s):  
Egon Schulte ◽  
Asia Ivić Weiss
Keyword(s):  
Classical Theory ◽  
Convex Polytope ◽  
Regular Polytopes ◽  
Geometric Structures ◽  
Maximal Symmetry ◽  
Abstract Polytopes ◽  
Chiral Polytopes ◽  
Classical Notion ◽  
Extension Result ◽  
Amalgamated Product

AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

Eulerian abstract polytopes

2011 ◽  
Vol 82 (1-2) ◽  
pp. 1-23
Author(s):  
Michael I.  Hartley
Keyword(s):  
Abstract Polytopes

On the flag graphs of regular abstract polytopes: Hamiltonicity and Cayley index

2020 ◽  
Vol 343 (1) ◽  
pp. 111599
Author(s):  
Leah Wrenn Berman ◽  
István Kovács ◽  
Gordon I. Williams
Keyword(s):  
Abstract Polytopes

scholarly journals A new Petrie-like construction for abstract polytopes

2008 ◽  
Vol 115 (6) ◽  
pp. 997-1007 ◽  
Author(s):  
Michael I. Hartley ◽  
Dimitri Leemans
Keyword(s):  
Abstract Polytopes

scholarly journals Internal and external invariance of abstract polytopes

2012 ◽  
Author(s):  
Gabe Cunningham
Keyword(s):  
Abstract Polytopes

scholarly journals Constructions of k-orbit abstract polytopes

2013 ◽  
Author(s):  
Ilanit Helfand
Keyword(s):  
Abstract Polytopes

Symmetry Groups Associated with Tilings of a Flat Torus

2014 ◽  
Vol 70 (a1) ◽  
pp. C1428-C1428
Author(s):  
Mark Loyola ◽  
Ma. Louise Antonette De Las Peñas ◽  
Grace Estrada ◽  
Eko Santoso
Keyword(s):  
Symmetry Group ◽  
Full Rank ◽  
Flat Torus ◽  
Space Forms ◽  
Color Symmetry ◽  
Abstract Polytopes ◽  
Symmetry Properties ◽  
Transformation Geometry ◽  
Groups Of Isometries ◽  
Injective Maps

A flat torus E^2/Λ is the quotient of the Euclidean plane E^2 with a full rank lattice Λ generated by two linearly independent vectors v_1 and v_2. A motif-transitive tiling T of the plane whose symmetry group G contains translations with vectors v_1 and v_2 induces a tiling T^* of the flat torus. Using a sequence of injective maps, we realize T^* as a tiling T-of a round torus (the surface of a doughnut) in the Euclidean space E^3. This realization is obtained by embedding T^* into the Clifford torus S^1 × S^1 ⊆ E^4 and then stereographically projecting its image to E^3. We then associate two groups of isometries with the tiling T^* – the symmetry group G^* of T^* itself and the symmetry group G-of its Euclidean realization T-. This work provides a method to compute for G^* and G-using results from the theory of space forms, abstract polytopes, and transformation geometry. Furthermore, we present results on the color symmetry properties of the toroidal tiling T^* in relation with the color symmetry properties of the planar tiling T. As an application, we construct toroidal polyhedra from T-and use these geometric structures to model carbon nanotori and their structural analogs.

Maximum diameter of abstract polytopes

1974 ◽  
pp. 20-40 ◽  
Author(s):  
Ilan Adler ◽  
George B. Dantzig
Keyword(s):  
Maximum Diameter ◽  
Abstract Polytopes

scholarly journals Representing the sporadic Archimedean polyhedra as abstract polytopes

2010 ◽  
Vol 310 (12) ◽  
pp. 1835-1844 ◽  
Author(s):  
Michael I. Hartley ◽  
Gordon I. Williams
Keyword(s):  
Abstract Polytopes

scholarly journals Contractions of polygons in abstract polytopes

2015 ◽  
Author(s):  
Ilya Scheidwasser
Keyword(s):  
Abstract Polytopes
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