Tessellations by Recognizable Figures

Horseman. M.C. Escher, 1946.

Relevant examples from Escher's work:

• Regelmatige vlakverdeling, Plate I
• Regular Division of the Plane Drawings, particularly those which also show the underlying geometric tessellation:
  • Sketch #67 (Man on horse) and related work Visions of Symmetry pg. 111.
  • Sketch #96 (Birds)
  • Sketch #127 (Birds)
  • Sketch #128 (Birds)

When you go looking for something specific, your chances of finding it are very bad. Because of all the things in the world, you're only looking for one of them. When you go looking for anything at all, your chances of finding it are very good. Because of all the things in the world, you're sure to find some of them. - Daryl Zero, The Zero Effect

Introduction

Escher created his tessellations by using fairly simple polygonal tessellations, which he then modified using isometries.

Escher's Polygon Systems

Escher's classified his tessellations based on the type of polygon used combined with the symmetries present in the tessellation. Escher used the following polygons:

• A - Parallelogram
• B - Rhombus
• C - Rectangle
• D - Square
• E - Isoceles Right Triangle

Escher used ten different systems of symmetry:

System	Translations	Rotations	Glide-Reflections
I	Translation in both transversal and diagonal directions	none	none
II	Translations in one transversal direction	2-fold rotations on the vertices; 2-fold rotations on the centers of the parallel sides	none
III	Translations in both diagonal directions	2-fold rotations on the centers of all sides	none
IV	Translations in both diagonal directions	none	Glide-reflections in both transversal directions
V	Translations in one transversal direction	none	Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions
VI	Translations in one diagonal direction	2-fold rotations on the centers of two adjacent sides	Glide-reflections in one diagonal direction. Glide-reflections in both transversal directions, but only in the direction of the sides without rotation point
VII	none	2-fold rotations on the centers of the parallel sides	Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions.
VIII	none	2-fold rotations on the four vertices	Glide-reflections in both transversal directions
IX	none	4-fold rotations on diagonal vertices; 2-fold rotations on diagonal vertices	none
X	none	4-fold rotations on three vertices; 2-fold rotations on the center of the hypothenuse	none

If you look at any of the prints in Visions of Symmetry, for instance in print 96 (above), you will notice the system denoted below the print. Print 96 belongs to system IV-D-. This means that the underlying geometric tessellation is based on a square and that there must be translations in both diagonal directions, no rotations and glide-reflections in both transversal directions. The latter is not entirely clear. There seem to be two different glide-reflections along a vertical axis, but no glide-reflections along a horizontal axis.

Tessellating With Translations

Escher's Type I.

Tessellating With Glide Reflections

Escher's Type IV

Tessellating With Rotations

Escher's Type II

Other Interesting Methods

Cutting into two tiles. Pentagon tessellation. Escher's favorite Alhambra pattern.

Related Sites

• Tessellation Database by Snels-Design.
• Escher in the Classroom by Jill Britton.