Difference between revisions of "Tessellations by Recognizable Figures"

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==Tessellating With Glide Reflections==
 
==Tessellating With Glide Reflections==
Escher's Type IV
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Escher's Type IV: Rhombus and square tessellations can be used to create tessellations that have translations in both diagonal directions, no rotations and glide-reflections in both transversal directions.
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[[Image:TypeIV-Escher.png|thumb|center|500px| Type IV tessellations based on the rhombus (left) and on the square (right) ]]
  
 
==Tessellating With Rotations==
 
==Tessellating With Rotations==

Revision as of 11:42, 15 May 2007

Horseman. M.C. Escher, 1946.

Relevant examples from Escher's work:

Escher's Polygon Systems

Escher created his tessellations by using fairly simple polygonal tessellations, which he then modified using isometries. This chapter gives a brief overview of Escher's own classification system for tessellations and contains instructions for creating tessellations by recognizable figures using some of Escher's simpler techniques.

Escher organizes his tessellations into two classes, systems based on quadrilaterals, and triangle systems built on the regular tessellation by equilateral triangles. The bulk of Escher's tessellations are based on quadrilaterals, which the novice will find much easier to work with. The less common triangle systems are easily identified because three or six motifs will meet at a point, and the entire tessellation will have order 3 or order 6 rotation symmetry.

For a complete discussion of Escher's classification system, read Visions of Symmetry (Chapter 2), which also reprints each page of Escher's notebook "Regular Division of the plane into asymmetric congruent polygons". Here, we give only a brief summary of his quadrilateral systems.

Within the quadrilateral systems, Escher's classification has two factors: the type of polygon and the symmetries present in the tessellation. Specifically, he assigns a capital letter to each type of polygon and a Roman numeral to each type of symmetry. The polygons are:

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

Note that each is a special type of quadrilateral except for E, the isoceles right triangle. These 45°-45°-90° triangles fit together to form a square grid with order 4 rotation symmetry.

Escher's ten different systems of symmetry do not correspond exactly to the wallpaper groups. The wallpaper group for a figure describes the isometries of the figure. Escher's systems also describe the isometries, but are additionally concerned with the relative positions of the different motifs. There are five wallpaper groups that have no reflections and no order 3 rotations, and all of Escher's symmetry systems correspond to one of these five wallpaper groups.

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: These tessellations can be based on a paralellogram, rhombus, rectangle or square. The Tessellations have translations in both ttransversal directions and in both diagonal directions. There are no rotations or glide-reflections.


Escher's Type I tessellations.

Tessellating With Glide Reflections

Escher's Type IV: Rhombus and square tessellations can be used to create tessellations that have translations in both diagonal directions, no rotations and glide-reflections in both transversal directions.

Type IV tessellations based on the rhombus (left) and on the square (right)

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