Glide Reflection and other Combined Transformations

Abstract

Glide Reflection is one of the four (translation, rotation, reflection and glide reflection) symmetry transformations we use to classify the regular divisions of the plane. In a way glide reflection is somewhat different from the other three, because it’s not a simple tranformation. In fact it is a combination of two transformations: translation and reflection. A combination of two but used as a single combined transformation. This article is about my work with other combinations of transformations that could be used as a combined transformation and how these transformations could be used in the design of patterns and objects. I define three other tranformations which will be explained in the article. The transformations are not restricted to the 2D plane, but are used in the 3D space. Most of the examples are 3D objects or designs.


1. What is Glide Reflection

Figure 1: Horseman - M.C. Escher.

First let's have a closer look at Glide Reflection. Maybe one of the most well known examples is M.C. Escher's Horseman (Figure 1) [1]. The shapes of the light and dark horseman are the same apart from the operation of reflection. If you want to place a tile of a light horseman on a tile of a dark horseman, you can't do that with just reflection, rotation or translation. You have to use both translation and reflection. These two transformations have to be used together (Figure 2, 3, 4).

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Figure 2,3,4: Line of Reflection, Translation, Reflection.