“For me it remains an open question whether [this work]
pertains to the realm of mathematics or to that of art.”

- M.C. Escher

Contents

Page 1
Introduction
The Art of Alhambra
Our Area of Focus
Our Aim

Page 2
The Principals behind Tessellations
- Translation
- Rotation
- Reflection
- Glide Reflection

Page 3
Mathematics in Escher's Art
- Translation
- Rotation
- Glide Reflection
- Combination

Page 4
Our Original Tessellations
- The Catch (Translation)
- Under the Sea (Rotation)
- The Herd (Glide Reflection)
- Dumbo & Butterfly (Combination)


Page 5
Possible links with Architecture
Conclusion
References


Mathematics in Escher's Art

In this page we attempt to try to use simple mathematical terms to explain a few of Escher's pieces.

One of the basic principals behind his tessellations is the use of what we call an "addition and subtraction" method within the grid. The diagrams below help illustrate this method.

Starting with a simple square.
We "subtract" a portion of it from one side,
and "add" it to the corresponding opposite side.

Thus the resultant shape still preserves the ability to tile.

The translation animation below will also help to illustrate this.


Translation

One of Escher's first explorations into the tessllations is that of the use of translation. Below is Liberal Christian Lyceum at The Hague, 1960, which uses one of Escher's pieces as a cladding.

The following animation will now explain how this tessellation was formed:


Rotation

Next, we look at the following image and try to understand another method of forming tessellations the way Escher does it, this method is rotation.

We will attempt to show the mathematical properties of this image through the following animation and through it understand how the tessellation was formed:



Glide Reflection

In glide reflection, reflection and translation are used concurrently much like the following piece by Escher, Horseman.

In this piece we can see that there is no reflectional symmetry, nor is there rotational symmetry, rather it is a combination of translational and reflectional symmetry to form this tessellation. The following animation will illustrate this point:


Combination

The following tessellation called Angels & Devils, 1941, combines reflection, rotation and glide reflection all into the same tessellation!

The following animation will show where the symmetries lie in this mind boggling tessellation:


After we disected these tessellations and placed the mathematics into it, it was time to use this knowledge and come up with our own tessellations. However, it proved to be more difficult than it had originally seemed as we will explain on the following page.