Restrictedly-Regular Map Representation of n-Dimensional Abstract Polytopes

Regularity has often been present in the form of regular
polyhedra or tessellations; classical examples are the nine regular
polyhedra consisting of the five Platonic solids (regular convex
polyhedra) and the four Kleper-Poinsot polyhedra. These polytopes
can be seen as regular maps. Maps are cellular embeddings of
graphs (with possibly multiple edges, loops or dangling edges) on
compact connected (closed) surfaces with or without boundary. The
n-dimensional abstract polytopes, particularly the regular ones, have
gained popularity over recent years. The main focus of research
has been their symmetries and regularity. Planification of polyhedra
helps its spatial construction, yet it destroys its symmetries. To our
knowledge there is no “planification” for n-dimensional polytopes.
However we show that it is possible to make a “surfacification”
of the n-dimensional polytope, that is, it is possible to construct a
restrictedly-marked map representation of the abstract polytope on
some surface that describes its combinatorial structures as well as
all of its symmetries. We also show that there are infinitely many
ways to do this; yet there is one that is more natural that describes
reflections on the sides ((n−1)-faces) of n-simplices with reflections
on the sides of n-polygons. We illustrate this construction with the
4-tetrahedron (a regular 4-polytope with automorphism group of size
120) and the 4-cube (a regular 4-polytope with automorphism group
of size 384).

Authors:

• Antonio Breda d’Azevedo

Keywords:

• Maps
• representation
• polytopes.

References:

[1] A. Breda d’Azevedo, A theory of restricted regularity of hypermaps, J. Korean
Math. Soc. 43 (2006), No.5, 991–1018.
[2] A. Breda d’Azevedo, Riemann surfaces and restrictively-marked hypermaps, Ars
Mathematica Contemporanea 3 (2010), 87-98.
[3] A. Breda d’Azevedo, Planification of n-dimensional abstract polytopes and
hypertopes, submitted.
[4] A. Breda d’Azevedo, D. Catalano, Classification of the thin regular
map-representation of hypermaps, Submitted.
[5] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin,
1977.
[6] Lynne D. James Representation of Maps, Dissertacciones Del Seminario de
Matem´aticas Fundamentales, Universidad Nacinal de Educaci´on a Distancia,
Lecture notes 27 March 1990.
[7] P. McMullen, E. Schulte, Abstract Regular Polytopes, Encyclopedia of
Mathematics and Its Applications, Vol. 92. Cambridge Univ. Press,
Cambridge-New York, 2002.