DeepAI

# Any Regular Polyhedron Can Transform to Another by O(1) Refoldings

We show that several classes of polyhedra are joined by a sequence of O(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain at most 6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.

08/12/2019

### Regular Separability and Intersection Emptiness are Independent Problems

The problem of regular separability asks, given two languages K and L, w...
12/23/2020

### A triangle process on regular graphs

Switches are operations which make local changes to the edges of a graph...
03/31/2021

### Revisiting regular sequences in light of rational base numeration systems

Regular sequences generalize the extensively studied automatic sequences...
04/15/2018

### Regular Polygon Surfaces

A regular polygon surface M is a surface graph (Σ, Γ) together with a co...
12/03/2021

### A Web of Confocal Parabolas in a Grid of Hexagons

If one erects regular hexagons upon the sides of a triangle T, several s...
02/06/2020

### Which convex polyhedra can be made by gluing regular hexagons?

Which convex 3D polyhedra can be obtained by gluing several regular hexa...
12/03/2019

### Geodesic Folding of Tetrahedron

In this work, we show the geometric properties of a family of polyhedra ...