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Convex Polytopes: A group theoretic and a graph theoretic perspective

Boi, Carlotta (2025) In Master's theses in Mathematical Sciences MATM03 20251
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
We present classical results for convex polytopes. After reviewing quaternions, we apply these algebraic tools to polyhedral graphs. In 1891, Eberhard wondered if there exists a trivalent polyhedron with an odd number of multitrivalent faces. In 1964, Motzkin was able to find a solution to the problem. He used the group formed by the 24 units in the ring of Hurwitz quaternion integers to obtain the result. After that, different proofs were discovered. Among others, Gru ̈nbaum gave a graph-theoretical proof (1964), which we present here. Finally, we will focus on connectedness, in particular on a famous result which states that the graph of a d-polytope is d-connected (1995).
Popular Abstract
This thesis explores discoveries about convex polytopes, using algebra and tools from graph theory. In particular, we will see how Motzkin, in 1964, solved a famous problem posed by Eberhard in the nineteenth century, using fascinating numbers, called quaternions. Eberhard posed a question on the existence of a polyhedron in which every vertex has degree three, every face has length divisible by three, and the total number of faces is odd. Motzkin’s result, which we present here, ascertains that no such polyhedron exists. We will also see how it is possible to prove the same theorem using an alternative approach, which consists of using a particular technique in graph theory. Finally, we will see another result. Consider two geometric... (More)
This thesis explores discoveries about convex polytopes, using algebra and tools from graph theory. In particular, we will see how Motzkin, in 1964, solved a famous problem posed by Eberhard in the nineteenth century, using fascinating numbers, called quaternions. Eberhard posed a question on the existence of a polyhedron in which every vertex has degree three, every face has length divisible by three, and the total number of faces is odd. Motzkin’s result, which we present here, ascertains that no such polyhedron exists. We will also see how it is possible to prove the same theorem using an alternative approach, which consists of using a particular technique in graph theory. Finally, we will see another result. Consider two geometric objects, a pentagon and a cube. The pentagon is 2-dimensional, while the cube is 3-dimensional. If we remove a vertex from the pentagon, then its structure stays together, as we can still walk among the lines, and if we remove two vertices from the cube, its structure also stays together. Then, we can continue in higher dimensions. The result that we will see is that the structure of a d-dimensional polytope stays connected if we remove
d − 1 vertices. (Less)
Please use this url to cite or link to this publication:
author
Boi, Carlotta
supervisor
organization
course
MATM03 20251
year
type
H2 - Master's Degree (Two Years)
subject
keywords
convex polytopes, Hurwitz quaternions, d-connectedness
publication/series
Master's theses in Mathematical Sciences
report number
LUNFMA-3160-2025
ISSN
1404-6342
other publication id
2025:E97
language
English
id
9211686
date added to LUP
2025-09-05 09:38:11
date last changed
2025-09-05 09:38:11
@misc{9211686,
  abstract     = {{We present classical results for convex polytopes. After reviewing quaternions, we apply these algebraic tools to polyhedral graphs. In 1891, Eberhard wondered if there exists a trivalent polyhedron with an odd number of multitrivalent faces. In 1964, Motzkin was able to find a solution to the problem. He used the group formed by the 24 units in the ring of Hurwitz quaternion integers to obtain the result. After that, different proofs were discovered. Among others, Gru ̈nbaum gave a graph-theoretical proof (1964), which we present here. Finally, we will focus on connectedness, in particular on a famous result which states that the graph of a d-polytope is d-connected (1995).}},
  author       = {{Boi, Carlotta}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's theses in Mathematical Sciences}},
  title        = {{Convex Polytopes: A group theoretic and a graph theoretic perspective}},
  year         = {{2025}},
}