Visualization of Polyhedra tessellation with Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Escher's works
The figures or works that are chosen as tesserae cover or pave the plane or a surface. The result is called tessellation, tiling, paving or mosaic.
Tessellations have been used since ancient times in floors, walls, carpets and other objects.
This work shows the tessellations made on all or some faces of polyhedrons, taking advantage of triangular, square, pentagonal or hexagonal symmetries of the faces of these solids.
On this page, we have some tessellation works by Maurits Cornelis Escher (1898-1972) inserted into the faces of common polyhedrons, such as the Platonic tetrahedron, cube or icosahedron, as well as rarer polyhedrons, such as the Johnson solids.
3D models
1. Alhambra tessellation
description: Escher painting 1922 of islamic mosaic tile at the Alhambra
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
2. Alhambra tessellation
description: vector image based on Escher’s work of Alhambra
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
3. Alhambra tessellation
description: vector image based on Escher’s work of Alhambra
tessellation type: quadrilateral
polyhedra: medial rhombic triacontahedron, small dodecacronic hexecontahedron, great ditrigonal dodecacronic hexecontahedron, medial deltoidal hexecontahedron, small icosacronic hexecontahedron, great rhombic triacontahedron
4. Alhambra tessellation
description: vector image based on Escher’s work of Alhambra
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, stella octangula, Escher solid, hexakis tetrahedron, octahedron, Möbius octakis hexahedron, pentagonal dipyramid
5. Alhambra tessellation
description: vector image based on Escher’s work of Alhambra
tessellation type: triangular
polyhedra: tetrahedron, Möbius 10-akis dodecahedron, icosahedron, Möbius hexakis icosahedron, Möbius deltahedron, tetrakis hexahedron, snub disphenoid
6. Tessellation with 4 transformations
description: features butterflies, birds, and moths; work from 1950
tessellation type: quadrilateral and triangular
polyhedra: icosahedron, pentagonal dipyramid, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, cube, deltoidal hexecontahedron, deltoidal icositetrahedron, rhombic dodecahedron, rhombic hexecontahedron, pentakis dodecahedron
7. Tessellation with 4 transformations
description: features butterflies, birds, and moths; work from 1950
tessellation type: quadrilateral
polyhedra: cube, deltoidal hexecontahedron, deltoidal icositetrahedron, rhombic dodecahedron, rhombic triacontahedron, rhombic hexecontahedron
8. Tessellation with 4 transformations
description: features butterflies, birds, and moths; work from 1950
tessellation type: quadrilateral and triangular
polyhedra: Möbius octakis hexahedron, deltoidal hexecontahedron, deltoidal icositetrahedron, rhombic dodecahedron, rhombic triacontahedron, Möbius hexakis octahedron, Möbius 10-akis dodecahedron, octahedron, tetrakis hexahedron
9. Tessellation with 4 transformations
description: features butterflies, birds, and moths; work from 1950
tessellation type: quadrilateral and triangular
polyhedra: Möbius deltahedron, stella octangula, tetrahedron, rhombic dodecahedron, rhombic triacontahedron, cube, rhombic hexecontahedron, triakis tetrahedron, triakis octahedron, triakis icosahedron
10. Circle limit I
description: based on the Poincaré disc model of hyperbolic geometry; work from 1958
tessellation type: triangular
polyhedra: icosahedron, great icosahedron, Escher solid, hexakis tetrahedron, snub disphenoid, Möbius deltahedron, octahedron, Möbius hexakis octahedron, pentakis dodecahedron
11. Circle limit I
description: based on the Poincaré disc model of hyperbolic geometry; work from 1958
tessellation type: triangular
polyhedra: Möbius hexakis icosahedron, pentagonal dipyramid, Möbius octakis hexahedron, Möbius 10-akis dodecahedron, stella octangula, tetrahedron, tetrakis hexahedron
12. Development III
description: abstract art with a spiral design; work from 1939
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
13. Earth
description: new year’s greeting card (ants); work from 1952
tessellation type: quadrilateral and triangular
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron, tetrakis hexahedron
14. Symmetry watercolor: fish
description: work from 1942
tessellation type: quadrilateral and triangular
polyhedra: stella octangula, tetrahedron
15. Fish, duck, turtle
description: work from 1948
tessellation type: quadrilateral and triangular
polyhedra: Möbius hexakis octahedron, deltoidal hexecontahedron, Escher solid, hexakis tetrahedron, octahedron, Möbius octakis hexahedron
16. Fish, duck, turtle
description: work from 1948
tessellation type: quadrilateral and triangular
polyhedra: rhombic dodecahedron, Möbius 10-akis dodecahedron, rhombic hexecontahedron, Möbius hexakis icosahedron, Möbius deltahedron, tetrakis hexahedron
17. Fish, duck, turtle
description: Pol de Miguel’s adaptation of Escher’s work; work from 2019
tessellation type: quadrilateral and triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
18. Heaven and hell
description: work from 1962
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
19. Lizard, fish, bat
description: work from 1952
tessellation type: quadrilateral and triangular
polyhedra: Möbius hexakis octahedron, deltoidal hexecontahedron, Escher solid, hexakis tetrahedron, octahedron, Möbius octakis hexahedron
20. Lizard, fish, bat
description: work from 1952
tessellation type: quadrilateral and triangular
polyhedra: rhombic dodecahedron, Möbius 10-akis dodecahedron, rhombic hexecontahedron, Möbius hexakis icosahedron, Möbius deltahedron, tetrakis hexahedron
21. Path of life II
description: work from 1958
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
22. Shells and starfish
description: work from 1963
tessellation type: triangular and quadrilateral
polyhedra: Möbius hexakis octahedron, stella octangula, Escher solid, hexakis tetrahedron, octahedron, Möbius octakis hexahedron, pentagonal dipyramid, pentakis dodecahedron
23. Shells and starfish
description: work from 1963
tessellation type: triangular and quadrilateral
polyhedra: tetrahedron, Möbius 10-akis dodecahedron, icosahedron, Möbius hexakis icosahedron, Möbius deltahedron, tetrakis hexahedron, snub disphenoid
24. Shells and starfish
description: work from 1963
tessellation type: triangular and pentagonal
polyhedra: pentagonal icositetrahedron, tetartoid, pentagonal hexecontahedron, dodecahedron, augmented dodecahedron, metabiaugmented dodecahedron, parabiaugmented dodecahedron, triaugmented dodecahedron
25. Symmetry (lizards)
description: work from 1965
tessellation type: triangular and quadrilateral
polyhedra: rhombic hexecontahedron, Escher solid, deltoidal hexecontahedron, hexakis tetrahedron, rhombic dodecahedron, Möbius deltahedron
26. Symmetry (lizards)
description: work from 1965
tessellation type: triangular and quadrilateral
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Möbius hexakis icosahedron, octahedron, Möbius octakis hexahedron, tetrakis hexahedron
27. Snakes
description: Escher’s last impression; work from 1969
tessellation type: hexagonal
polyhedra: cubitruncated cuboctahedron, great dodecahemicosahedron, great dodecicosahedron, great truncated cuboctahedron, great truncated icosidodecahedron, icositruncated dodecadodecahedron, octahemioctahedron
28. Snakes
description: Escher’s last impression; work from 1969
tessellation type: hexagonal
polyhedra: rhombicosahedron, small dodecicosahedron, small hexagonal hexecontahedron, small triambic icosahedron, great truncated icosahedron, truncated icosahedron, truncated tetrahedron, truncated octahedron
29. Square Limit
description: work from 1964
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
30. Sun and moon
description: work from 1948
tessellation type: quadrilateral
polyhedra: deltoidal hexecontahedron, rhombic triacontahedron, cube, rhombic dodecahedron, rhombic hexecontahedron, deltoidal icositetrahedron
31. Symmetry and tessellation: fish
description: work from 1955
tessellation type: triangular and quadrilateral
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, deltoidal hexecontahedron, hexakis tetrahedron, rhombic dodecahedron, Möbius deltahedron, Möbius octakis hexahedron
32. Symmetry and tessellation: fish
description: work from 1955
tessellation type: triangular and quadrilateral
polyhedra: rhombic hexecontahedron, rhombic triacontahedron, Escher solid, Möbius hexakis icosahedron, octahedron, deltoidal icositetrahedron, tetrakis hexahedron
33. Tessellation
description: work from 1959
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
34. Symmetry Drawing
description: work from 1955
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
35. Butterfly
description: work from 1932
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron, tetrahedron
36. Circle limit II
description: work from 1959, adapted to square by Chamberlain (2016)
tessellation type: triangular and quadrilateral
polyhedra: Möbius hexakis octahedron, Escher solid, hexakis tetrahedron, Möbius deltahedron, Möbius octakis hexahedron, deltoidal hexecontahedron, deltoidal icositetrahedron, cube, great ditrigonal dodecacronic hexecontahedron, truncated dodecadodecahedron, great truncated icosidodecahedron
37. Circle limit II
description: work from 1959, adapted to square by Chamberlain (2016)
tessellation type: triangular and quadrilateral
polyhedra: rhombic dodecahedron, Möbius 10-akis dodecahedron, rhombic hexecontahedron, Möbius hexakis icosahedron, octahedron, tetrakis hexahedron, rhombic triacontahedron, pentagonal dipyramid, medial deltoidal hexecontahedron
38. Circle limit II
description: work from 1959, adapted to square by Chamberlain (2016)
tessellation type: triangular and quadrilateral
polyhedra: great dirhombicosidodecahedron, medial rhombic triacontahedron, small dodecacronic hexecontahedron, great hexacronic icositetrahedron, medial icosacronic hexecontahedron, rhombidodecadodecahedron, stella octangula, small icosacronic hexecontahedron, great rhombic triacontahedron, great rhombihexahedron, great rhombidodecahedron
39. Path of life I
description: work from 1958
tessellation type: triangular and quadrilateral
polyhedra: Möbius hexakis octahedron, Escher solid, hexakis tetrahedron, Möbius deltahedron, Möbius octakis hexahedron, deltoidal hexecontahedron, deltoidal icositetrahedron, cube, great ditrigonal dodecacronic hexecontahedron, truncated dodecadodecahedron, great truncated icosidodecahedron
40. Path of life I
description: work from 1958
tessellation type: triangular and quadrilateral
polyhedra: rhombic dodecahedron, Möbius 10-akis dodecahedron, rhombic hexecontahedron, Möbius hexakis icosahedron, octahedron, tetrakis hexahedron, rhombic triacontahedron, pentagonal dipyramid, medial deltoidal hexecontahedron
41. Path of life I
description: work from 1958
tessellation type: triangular and quadrilateral
polyhedra: great dirhombicosidodecahedron, medial rhombic triacontahedron, small dodecacronic hexecontahedron, great hexacronic icositetrahedron, medial icosacronic hexecontahedron, rhombidodecadodecahedron, stella octangula, small icosacronic hexecontahedron, great rhombic triacontahedron, great rhombihexahedron, great rhombidodecahedron
42. Path of life III
description: work from 1966
tessellation type: hexagonal
polyhedra: cubitruncated cuboctahedron, great dodecahemicosahedron, great dodecicosahedron, great truncated cuboctahedron, great truncated icosidodecahedron, icositruncated dodecadodecahedron, octahemioctahedron
43. Path of life III
description: work from 1966
tessellation type: hexagonal
polyhedra: rhombicosahedron, small dodecicosahedron, small hexagonal hexecontahedron, small triambic icosahedron, great truncated icosahedron, truncated icosahedron, truncated tetrahedron, truncated octahedron
44. Flower tessellation
description: coloring book published in 2021, based on Escher tessellations
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
45. Leaves tessellation
description: Regolo Bizzi work, based on Escher tessellations
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
46. Leaves tessellation
description: Regolo Bizzi work, based on Escher tessellations
tessellation type: triangular
polyhedra: Möbius hexakis octahedron, Möbius 10-akis dodecahedron, Escher solid, hexakis tetrahedron, Möbius hexakis icosahedron, octahedron, Möbius deltahedron, Möbius octakis hexahedron, tetrakis hexahedron
Escher’s works: polyhedra tessellation and visualization with Virtual Reality by Paulo Henrique Siqueira is licensed with a license Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International.
How to cite this work:
Siqueira, P.H., "Escher's works: polyhedra tessellation and visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/tessellation/escher/>, February 2025.
References:
Weisstein, Eric W. “Tessellation.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Tessellation.html
Mohr, R. “Tiled Art” https://tiled.art/en/home
McCooey, D. I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/