Visualization of Polyhedra with Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Escher's transformations in polyhedra
The technical work Gravitation was completed in 1952 by the Dutch artist Maurits Cornelis Escher. This work shows the non-convex small stellated dodecahedron of Kepler-Poinsot with geometric transformations on each face. Using the midpoints of parts of the faces, trapezoidal indentations are used by Escher to insert the heads and legs of shell-less turtles.
This work shows the transformations and some compositions made with polyhedra, based on geometric transformations with trapezoidal cutouts on the faces or parts of the faces of these solids, modeled for viewing in Virtual Reality.
3D models
1. Concave dodecahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
2. Cube and octahedron compound
Composition of a cube with the dual octahedron, with the faces of these polyhedra transformed by removing squares and triangles.
3. Disdyakis dodecahedron
The faces of this polyhedron were transformed by removing trapezoids.
4. Disdyakis triacontahedron
The faces of this polyhedron were transformed by removing trapezoids.
5. Dodecahedron and icosahedron compound
A composition of a dodecahedron with its dual icosahedron, with the faces of these polyhedra transformed by removing pentagons and triangles.
6. Escher solid
The faces of this polyhedron were transformed by removing trapezoids.
7. Escher solid v2
The faces of this polyhedron were transformed by removing trapezoids.
8. Great disdyakis dodecahedron
The faces of this polyhedron were transformed by removing trapezoids.
9. Great dodecahedron
A composition of a great dodecahedron with its dual small stellated dodecahedron, with parts of the faces of great dodecahedron transformed by removing trapezoids.
10. Great dodecahedron v2
A composition of a great dodecahedron with the great icosahedron, with parts of the faces of these polyhedra transformed by removing trapezoids.
11. Great dodecahedron v3
A composition of a great dodecahedron with the great icosahedron, with parts of the faces of these polyhedra transformed by removing trapezoids.
12. Great dodecahedron v4
A composition of a great dodecahedron with the great icosahedron, with parts of the faces of these polyhedra transformed by removing trapezoids.
13. Great dodecicosacron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
14. Great dodecicosacron v2
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
15. Great icosahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
16. Great icosahedron v2
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
17. Great icosahedron v3
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
18. Great icosahedron v4
A composition of a great icosahedron with a great dodecahedron, with parts of the faces of these polyhedra transformed by removing trapezoids.
19. Great icosahedron v5
A composition of a great icosahedron with a great dodecahedron, with parts of the faces of these polyhedra transformed by removing trapezoids.
20. Great rhombihexacron
The faces of this polyhedron were transformed by removing trapezoids.
21. Great stellated dodecahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
22. Great triambic icosahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
23. Heptagrammic dipyramid
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
24. Hexakis tetrahedron
The faces of this polyhedron were transformed by removing trapezoids.
25. Medial deltoidal hexecontahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
26. Medial deltoidal hexecontahedron v2
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
27. Medial deltoidal hexecontahedron v3
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
28. Medial icosacronic hexecontahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
29. Medial icosacronic hexecontahedron v2
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
30. Medial rhombic triacontahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
31. Medial rhombic triacontahedron v2
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
32. Medial triambic icosahedron
Parts of the faces of this polyhedron were transformed by removing trapezoids and triangles.
33. Merkaba
A composition of two tetrahedra, with the faces of these polyhedra transformed by removing triangles.
34. Möbius deltahedron
The faces of this polyhedron were transformed by removing trapezoids.
35. Möbius octakis hexahedron
The faces of this polyhedron were transformed by removing trapezoids.
36. Möbius hexakis octahedron
The faces of this polyhedron were transformed by removing trapezoids.
37. Möbius hexakis icosahedron
The faces of this polyhedron were transformed by removing trapezoids.
38. Möbius 10-akis dodecahedron
The faces of this polyhedron were transformed by removing trapezoids.
Escher’s transformations in polyhedra: 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 transformations in polyhedra: visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/polycompound/escher1/>, June 2026.
References:
Weisstein, Eric W. “Polyhedron Compound” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCompound.html
Weisstein, Eric W. “Uniform Polyhedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/UniformPolyhedron.html
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/