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Friezes, Triangulations, and Trees

Zeng, Yijia LU (2023) In Bachelor's Theses in Mathematical Sciences MATK11 20222
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
In this thesis, we focus on these three classes of objects: frieze patterns, polygon triangulations, and planar binary rooted trees. After proving that these objects are in pairwise bijective correspondence with each other, we introduce Catalan numbers through Dyck paths and prove that all these objects are Catalan objects. By studying the properties of triangulations and binary trees, we establish some properties of frieze patterns. In the final section, we provide a python code to generate frieze patterns together with their corresponding polygon triangulation. Additionally, we prove that the Pl¨ucker relations are satisfied in an arbitrary pattern.
Popular Abstract
Frieze Patterns have been used in architecture since ancient times. People put them high up on the wall either inside or outside to decorate buildings. Frieze patterns are usually repeating and form a band around a room. Among the architectural styles in human history, the ancient Greeks were famous for using frieze patterns.

In mathematics, Coxeter first used the word 'frieze' to describe a mathematical object in 1971 in order to give an instance to relations in Gauss' study of the pentagramma mirificum, a spherical pentagram formed by five successively orthogonal great-circle arcs. \cite{bau21} Objects of this type have rows full of numbers that are infinitely long and periodic. These properties made the word 'frieze' from... (More)
Frieze Patterns have been used in architecture since ancient times. People put them high up on the wall either inside or outside to decorate buildings. Frieze patterns are usually repeating and form a band around a room. Among the architectural styles in human history, the ancient Greeks were famous for using frieze patterns.

In mathematics, Coxeter first used the word 'frieze' to describe a mathematical object in 1971 in order to give an instance to relations in Gauss' study of the pentagramma mirificum, a spherical pentagram formed by five successively orthogonal great-circle arcs. \cite{bau21} Objects of this type have rows full of numbers that are infinitely long and periodic. These properties made the word 'frieze' from architecture very suitable to be their name.

Polygon triangulations and binary trees have been studied since the time of the ancient Egyptian and the ancient Greeks. Just like their names suggest, polygon triangulation is a way to divide a polygon into triangles; binary trees are graphical structures with the property that every node has two branches and one parent where the whole graphs look like trees. In this thesis, we will discover the relationship between each pair of these three mathematical objects and elucidate the properties of one object from the properties of the others. (Less)
Please use this url to cite or link to this publication:
author
Zeng, Yijia LU
supervisor
organization
course
MATK11 20222
year
type
M2 - Bachelor Degree
subject
keywords
Frieze pattern, polygon triangulation, binary tree, Catalan numbers
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4142-2023
ISSN
1654-6229
other publication id
2023:K2
language
English
id
9111047
date added to LUP
2023-12-19 16:07:20
date last changed
2023-12-19 16:07:20
@misc{9111047,
  abstract     = {{In this thesis, we focus on these three classes of objects: frieze patterns, polygon triangulations, and planar binary rooted trees. After proving that these objects are in pairwise bijective correspondence with each other, we introduce Catalan numbers through Dyck paths and prove that all these objects are Catalan objects. By studying the properties of triangulations and binary trees, we establish some properties of frieze patterns. In the final section, we provide a python code to generate frieze patterns together with their corresponding polygon triangulation. Additionally, we prove that the Pl¨ucker relations are satisfied in an arbitrary pattern.}},
  author       = {{Zeng, Yijia}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Friezes, Triangulations, and Trees}},
  year         = {{2023}},
}