Tessellations
Tessellations
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
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