Theory of Regular Polygon Tessellations

Introduction | Regular Tessellations | Typical Vertices

Semi-regular | Nonuniform periodic | Nonuniform non-periodic

Introduction

 

Tessellations (or tilings) are a vast area of study in mathematics. Escher's famous tessellations made the field accessible, but the mathematics behind this field is a rich area to begin the exploration of how a theory can be developed building from one idea to another.

Four different types of tessellations using regular polygons will be presented: regular, semi-regular, nonuniform periodic, and nonuniform non-periodic. The problem with non-regular polygons leads to Penrose tilings.

Tessellations (or tilings) are defined to be a covering of a surface by congruent (or non-congruent) shapes where the shapes all meet around a common vertex.

A Tessellation!
NOT a Tessellation!

Regular Tessellations

 

A regular tessellation is periodic and uniform and consists of congruent regular polygons. Periodic means that the tessellation is formed by repeated translations of polygon groupings. Uniform means that the same type of polygon(s) are at each vertex. Only three regular tessellations exist, because there are only three polygons whose interior angles divide evenly into 360 degrees: the triangle (60 degrees) the square (90 degrees) and the hexagon (120 degrees).

The regular tessellation generated with hexagons is perhaps the most famous one, as its use as a storage receptacle is prevalent everywhere. In nature, bees use it for storing honey. I visited the Oak Ridge's nuclear museum, where they use it for storage of nuclear cylinders.

Typical Vertices

Of the regular polygons, only triangles, squares, hexagons, octagons, and dodecagons can be used for tiling around a common vertex - again because of the angle value, and 14 such combinations exist!

This means that each common vertex of any type of regular polygon tessellation must use one of these combinations!

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Combination of Polygons
Combination of Angles

6 triangles

6X 60 = 360

4 squares

4 X 90 = 360

3 hexagons

3 X 120 = 360

3 triangles and 2 squares

(3 X 60) + (2 X 90) = 360

3 triangles and 2 squares - another formation

(3 X 60) + (2 X 90) = 360

2 hexagons and 2 triangles

(2 X 120) + (2 X 60) = 360

2 hexagons and 2 triangles - another formation

(2 X 120) + (2 X 60) = 360

1 hexagon, 2 squares, and 1 triangle

(1 X 120) + (2 X 90) + (1 X 60) = 360

1 hexagon, 2 squares, and 1 triangle - another formation

(1 X 120) + (2 X 90) + (1 X 60) = 360

1 hexagon and 4 triangles

(1 x 120) + (4 X 60) = 360

2 octagons and 1 square

(2 X 135) + (1 X 90) = 360

1 dodecagon, 1 square, and 1 hexagon

(1 X 150) + (1 X 90) + (1 X 120) = 360

2 dodecagons and 1 triangle

(2 X 150) + (1 X 60) = 360

1 dodecagon, 2 triangles, and 1 square

(1 X 150) + (2 X 60) + (1 x 90) = 360

Semi-regular Tessellations

A semi-regular tessellation is periodic and uniform. The most common one is the one below of the octagon and the square. However, there are 7 other semi-regular tessellations!

October 30 - Transformational Geometry

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