Rotation of a polytope in another one

TL;DR

This paper investigates the conditions under which a polytope can be rotated within another of the same dimension, providing a criterion and exploring specific cases like simplices, with findings influenced by the dimension's parity.

Contribution

It introduces a criterion for rotating a polytope within another and analyzes special cases such as simplices, highlighting the impact of dimension and parity on feasibility.

Findings

Rotation of even-dimensional simplices is generally possible under certain conditions.

In three dimensions, the rotation possibility is limited and not as general.

Parity of the dimension affects the feasibility of rotation in elementary cases.

Abstract

We are interested in the naive problem whether we can move a solid object in a solid box or not. We restrict move to rotation. In the case we can, the centre and the ``direction'' of rotation may be restricted. Simplifying, we consider possibility of rotation of a polytope within another one of the same dimension and give a criterion for the possibility. Consider the particular case of simplices of the same dimension assuming that the vertices of the inner simplex are contained in different facets of the outer one. Premising further that simplices are even dimensional, rotation is possible in a very general situation. However, in dimension 3, the possible case is not not general. Even in these elementary phenomena, the parity of the dimension seems to yield difference.