Mathematics of Tessellations

The basic mathematics of tessellations are based on geometry. Every tessellation starts out as a geometric shape that has been manipulated and then repeated many times. One thing to note is that not all geometric shapes will tessellate. For a tessellation to be produced, the geometric shape has to be able to fit together like a puzzle where there are no holes or overlap. In a sense they have to fit together perfectly. To better understand this, we should look at tiling. Regular tiling are tilings that use regular polygons. For a polygon to tessellate the total of all the angles around a point must be 360 degrees. Shapes that can produce tessellations by themselves are triangles, squares, rectangles, and hexagons. If we think about the interior angle measures of theses polygons, it makes sense why they can tessellate. Their angle measures can add up to 360 degrees. There are also polygons that can be put together with other polygons to produce tessellations as well, such as squares and triangles. (Campbell, 1996)

This link shows the regular polygons that tessellate and also shows some examples.

Shapes don't have to be regular polygons to tessellate. Irregular triangles, quadrilaterals, and hexagons can be tiled and tessellated. Triangles and quadrilaterals are awesome because they can all be tessellated. Hexagons are a little more complex. K. Reinhardt came up with three cases for when irregular hexagons will tessellate. (Campbell, 1996)



Campbell, 1996

The different shapes can produce so many different tessellations on their own, but there are also many different ways to modify polygons to produce an endless number of different tessellations. The most basic and well known tessellation is a translation tessellation. Some of the others I want to point out are vertex rotation, midpoint rotation/ reflection, and glide-reflection/ half turn. These are all tessellations that M. C. Escher used in his works.

Translation Tessellations

"Each tile is a translation of each other one, since we can move one to coincide with another without doing any rotation or reflection."(Campbell, 1996) Not all the shapes that can be tessellated can be a translated to form a tessellation. Polygons "which displace every point by a specified vector" produce a tessellation by translation (Kaplan, 2009).
"A tile can tile the plane by translations if either
1. There are four consecutive points A, B, C, and D on the boundary such that
(a) the boundary part from A to B is congruent by translation to the boundary part from D to C, and
(b) the boundary part from B to C is congruent by translation to the boundary part from A to D (see Figure 20.12a) or
2. There are six consecutive points A, B, C, D, E, and F on the boundary such that the boundary parts AB, BC, and CD are congruent by translation, respectively, to the boundary parts ED, FE, and AF (see Figure 20.12b)."(Campbell, 1996)

Translation Tessellation This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Megan Urbanik, 4 December 2013, Created with GeoGebra

One Side Transformation Tessellation Move the blue points to change the shape of the translated object. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Megan Urbanik, 29 November 2013, Created with GeoGebra

Midpoint Rotation Tessellations

Midpoint rotation tessellations are just what the title says. A polygon side is modified from one vertex to the midpoint and then rotated 180 degrees. The polygon can then be rotated 180 degrees to form the tessellation. (Britton, 2000)

Escher's tessellation that uses midpoint rotations and translations.

(Britton, 2000)

Vertex Rotation Tessellations

Vertex rotation tessellations must have equal and adjacent sides so that one side can be modified and rotated about a vertex. Regular polygons can do this type of tessellation, but not all irregular polygons can. (Britton, 2000)

Escher's tessellation that uses vertex rotations.

(Britton, 2000)

Glide Reflection

To create a glide reflection tessellation modify one side of the polygon and then "flip and slide" it with an equal side (Britton, 2009). Because the sides of the polygon have to have equal sides, not all irregular polygons can tessellate this way. (Britton, 2000)

Escher's tessellation that uses glide reflections and translations.

(Britton, 2000)