Tessellations by Recognizable Figures

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Horseman. M.C. Escher, 1946.

Relevant examples from Escher's work:

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

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


Regular-division-96.jpg

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.

Heesch Types

Related Sites