Conway's spiral and a discrete G\"omb\"oc with 21 point masses

TL;DR

This paper presents the first explicit 3D construction of a convex mono-monostatic polyhedron with 21 vertices and faces, providing an upper bound for such shapes and contrasting with homogeneous mass distributions.

Contribution

It introduces the first known mono-monostatic polyhedral solid with equal vertex masses and establishes bounds on minimal face and vertex counts.

Findings

Constructed a 21-vertex mono-monostatic polyhedron in 3D

Serves as an upper bound for minimal face and vertex counts

Homogeneous mass distribution cannot produce mono-monostatic solids

Abstract

We show an explicit construction in 3 dimensions for a convex, mono-monostatic polyhedron (i.e., having exactly one stable and one unstable equilibrium) with 21 vertices and 21 faces. This polyhedron is a 0-skeleton, with equal masses located at each vertex. The above construction serves as an upper bound for the minimal number of faces and vertices of mono-monostatic 0-skeletons and complements the recently provided lower bound of 8 vertices. This is the first known construction of a mono-monostatic polyhedral solid. We also show that a similar construction for homogeneous distribution of mass cannot result in a mono-monostatic solid.