2013

Jessie & Sheena

Math 1900 – FINAL PROJECT ESSAY

4/3/2013

Tessellations

Introduction

Tessellation comes from the Latin word “tessella” which is a small stone or piece of glass used to make mosaics. Tessella means small square and is usually referred to as a tile. A tesselation is a 2 dimensional tiled plane with no overlaps or gaps. Tessellations can be found in the form of art, nature and best known for tiling floors. Throughout the essay we will be discussing the mathematics behind tessellations, the creator, Escher and how he manipulated them into works of art as well as Penrose Tilings introduced by Roger Penrose and how he brought a new twist to tessellations. Math Behind Tessellations

In order to create a regular tessellation the first step is to choose one single regular polygon, whether it be an equilateral triangle, a square, or a hexagon, these three regular polygons are the only prototiles that will tile the field with no overlaps or gaps when completed. The rule for regular polygon is that all sides are the same length and all angles are of the same degree. A regular polygon has rotational and reflexive symmetry. Reflectional symmetry is when a shape can be cut directly in half and be identical on both sides. Rotational symmetry is when an object can be rotated on any degree and remain its original shape. In numerous tessellations but not in all transitional symmetry occurs; translational symmetry is when you can identify a shape or area of a tessellation and it will be tiled throughout and remain the same.

Figure 1. An equilateral triangle, square, and hexagon showing that all sides and angles are the same in each and how they have consistent rotational symmetry

A semi-regular tessellation is the use of two or more regular polygons that plane the field. In order to have a semi-regular tessellation you need all the vertices to be the same. A vertices is the point of a polygon.

Figure 2. Different systems for Semi-Regular Tessellations

There is an endless amounts of irregular tessellations. These are made by first tiling a plane with a regular polygon like a square, then transforming it. The steps for this process goes as follows: 1. Draw a regular polygon

2. Change one side of the polygon, for example make it a curvy line (section AD) 3. Do the exact opposite to the other side of the polygon (section BC) Once you get a basic understanding of how to manipulate a plane you can repeat this with the other sides. Another way you can manipulate a plane is to cut it into sections and flip it as a mirror image: different image

Figure 3. Irregular Tessellation made using the 3 steps

Tessellations are not just basic shapes and math. They can be seen in the world and be shown as drawings of animals like birds, fish and lizards.

Escher’s Contribution

Tessellations were created by Maurits Cornelis Escher in the early 20th century. Escher was a failed architect with a degree in graphic art. In 1922 he started travelling around Europe and painting landscapes and portraits. While touring through the Mediterranean Escher became interested with symmetry and the concept of infinity. He started incorporating symmetry and polymeric shapes into his drawings. He is known for some of his symmetrical drawings like `drawing hands`` or the optical illusion relativity. In order to create a tessellation he used the translation technique. This is when you take a shape like a square then redraw one the sides. After that you translate a copy of the opposite side of the square. So whatever you do to one side of the shape you do the reverse to the opposite side. When the new side is copied a new tessellation occurs. Escher created hundreds of these tessellations with drawings of animals, plants and humans.

Figure 4 one of escher’s tessellations

Penrose Tiling’s

In the 1970’s a man by the name of Roger Penrose created the idea of a non-periodic tiling system built by...