Difference between revisions of "Tessellations by Recognizable Figures"

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Escher used ten different systems of symmetry:
 
Escher used ten different systems of symmetry:
  
* I - Translation in both transversal and diagonal directions; (no rotations and no glide-reflections)
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{| border="1"
* II - Translations in one transversal direction; 2-fold rotations on the vertices; 2-fold rotations on the centers of the parallel sides. (no glide-reflections)
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!width="150"|System
* III - Translations in both diagonal directions2-fold rotations on the centers of all sides.(no glide-reflections)
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!width="150"|Translations
* IV - Translations in both diagonal directions; (no rotations); Glide-reflections in both transversal directions.
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!width="150"|Rotations
* V - Translations in one transversal direction; (no rotations); Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions.
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!width="150"|Glide-Reflections
* 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.
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|-
* VII - No translations; 2-fold rotations on the centers of the parallel sides; Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions.
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| <div style="height:15px"></div> I || Translation in both transversal and diagonal directions || none || none
* VIII - No translations; 2-fold rotations on the four vertices; Glide-reflections in both transversal directions.
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|-  
* IX - No translations; 4-fold rotations on diagonal vertices; 2-fold rotations on diagonal vertices; No glide-reflections
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| <div style="height:15px"></div> II || Translations in one transversal direction || 2-fold rotations on the vertices; 2-fold rotations on the centers of the parallel sides || none
* X - 4-fold rotations on three vertices; 2-fold rotations on the center of the hypothenuse; No glide-reflections
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|-  
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| <div style="height:15px"></div> III || Translations in both diagonal directions || 2-fold rotations on the centers of all sides || none
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|-  
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| <div style="height:15px"></div> IV || Translations in both diagonal directions || none || Glide-reflections in both transversal directions
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|-
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| <div style="height:15px"></div> V || Translations in one transversal direction || none || Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions
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|-
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| <div style="height:15px"></div> 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
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|-
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| <div style="height:15px"></div> VII || none || 2-fold rotations on the centers of the parallel sides || Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions.
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|-
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| <div style="height:15px"></div> VIII || none || 2-fold rotations on the four vertices || Glide-reflections in both transversal directions
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|-
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| <div style="height:15px"></div> IX || none ||4-fold rotations on diagonal vertices; 2-fold rotations on diagonal vertices || none
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|-  
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| <div style="height:15px"></div> X || none || 4-fold rotations on three vertices; 2-fold rotations on the center of the hypothenuse || none
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|}
  
 
==Tessellating With Translations==
 
==Tessellating With Translations==

Revision as of 14:48, 1 May 2007

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

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