Two Classes Of Regular Polytopes

This article mainly studies an open problem proposed by Schulte and Weiss in[Problem on polytopes,their groups,and realizations,Periodica Math.Hungarica2006,53:231-255][1]:Characterize the groups of order 2~n or 2~np,with n a positive integer and p an odd prime,which are automorphism groups of regular or chiral polytopes.This article considers the case of p=3.Two types of special 3·2~n regular polytopes are constructed.The paper is organized as follows.Chapter 1,Introduction:A brief introduction to the historical process of regular polytopes research and the current research results,is given in this area at home and abroad.Chapter 2,Preliminaries:Introduces the background knowledge of regular polytopes and their automorphism groups needed in this article,as well as some group theory concepts and results.Chapter 3,The main results:First,for any positive integer m,regular 3-polytopes of type{3,8m}of order 192m~2and 384m~2are constructed.Taking m to the power of 2,this is an answer to a special case of the question in[1].Then for any n≥7,we construct a regular polytope of order 3·2~n of type{6,4}.In particular,the automorphism groups of the polytopes we constructed above are all solvable.