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Rotation of a polytope in another one

Shuzo Izumi

TL;DR

The paper develops a framework for rotating a polytope inside another of the same dimension using infinitesimal rotations from $\mathfrak{so}(n)$ and the exponential map to $SO(n)$. A central admissibility criterion is derived via boundary-vertex analysis, involving skew-symmetric $S$, normals to facets, and sign-based regions that determine feasible rotation centres. The key results show that in even dimensions, a generic rotatability holds for simplices with vertices on facet interiors, while in 3D (and more generally odd $n$) rotatability is not generic and can fail for open sets of directions, highlighting a parity-dependent distinction. These findings provide both a practical criterion for rotatability and a deeper geometric understanding of rigid-body motion within polytopal constraints.

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.

Rotation of a polytope in another one

TL;DR

The paper develops a framework for rotating a polytope inside another of the same dimension using infinitesimal rotations from and the exponential map to . A central admissibility criterion is derived via boundary-vertex analysis, involving skew-symmetric , normals to facets, and sign-based regions that determine feasible rotation centres. The key results show that in even dimensions, a generic rotatability holds for simplices with vertices on facet interiors, while in 3D (and more generally odd ) rotatability is not generic and can fail for open sets of directions, highlighting a parity-dependent distinction. These findings provide both a practical criterion for rotatability and a deeper geometric understanding of rigid-body motion within polytopal constraints.

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.
Paper Structure (5 sections, 4 theorems, 17 equations, 3 figures)

This paper contains 5 sections, 4 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Infinitesimal Rotation.
  3. Rotation of a polytope in another one.
  4. Rotation of an even dimensional simplex in another one.
  5. Rotation of an odd dimensional simplex in another one.

Key Result

Lemma 2

Let $\alpha$ be an affine hyperplane of $\mathbb{R}^n$ and take a point X$=({\bf x})\in\alpha$ and Q$=(\bf q)\in\mathbb{R}^n$. Let ${\bf n}$ be a unit normal vector of $\alpha$. Let $S$ be a real skew symmetric matrix of size $n\times n$. Then we have the following.

Figures (3)

  • Figure 1: wall and a infinitesimal rotation
  • Figure 2: seldom non-rotatable case in dimension 2
  • Figure 3: non-rotatable case in dimension 3

Theorems & Definitions (11)

  • Remark 1
  • Lemma 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Example 8
  • ...and 1 more