# Research on Rubik's Snake and Quasi-Surfaces

5-2023

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Computational Analysis and Modeling

Songming Hou

Jinku Kanno

## Abstract

In this dissertation, we have two topics in applied math. For the first one, we studied the Rubik’s snake by mathematical methods.

The Rubik’s snake is a toy invented by Professor Erno Rubik in 1981. It consists of right isosceles triangular prisms. Since the shape of a snake depends only on how we twist it, we can get a unique sequence from every snake. Those sequences record the way we twist snakes. In this dissertation, we will study the relations between some structures of the snake and the sequences.

For a snake with a specific structure, we will find out what type of sequence may be possible from such a structure. Also, we can predict the properties of a snake if we know its sequences.

The second chapter of the dissertation delves into Wagner’s theorem, first introduced in 1936, and extend the same questions to a new type of topological space, referred to as a quasi-surface. The diagonal flip is an operation on the edges of a graph. Wagner proved that two triangulations of the sphere could be transformed into each other by a sequence of the diagonal flips. The resulting graph after each transformation is still a triangulation. Many mathematicians have proved that this theorem works not only for the triangulations of the sphere but also for other surfaces, such as the torus.

In the second part of this dissertation, we will study multitriangulations of Tk, a quasi-surface first introduced by Turner. Similar to Wagner’s theorem, we aim to show that every two multitriangulations of Tk can be transformed into each other by a sequence of graph operations. In order to achieve the goal, we determine two more operations which appear to help solve those problems; and we introduce a number of results which put us within close reach of a Wagner-type theorem for Tk.

COinS