Visualization of polyhedra with Augmented Reality (AR) and Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Self-intersecting truncated polyhedra
A polyhedron is truncated regular if it is vertex-transitive with isosceles triangular vertex figures. Vertex transitivity means that for any two vertices of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one vertex to the other. A vertex figure is the polygon produced by connecting the midpoints of the edges meeting at the vertex in the same order that the edges appear around the vertex.
Augmented Reality | 3D Models | Home
Immersive rooms
🔗 room 1 | 🔗 room 2 | 🔗 room 3
Augmented Reality
To view self-intersecting truncated polyhedra in AR, simply visit:
https://paulohscwb.github.io/polyhedra/selfintersecttruncated/ra.html
with any browser with a webcam device (smartphone, tablet or notebook).
Access to the VR sites is done by clicking on the blue circle that appears on top of the marker.
3D models
1. Stellated Truncated Hexahedron
U19 The stellated truncated hexahedron is the uniform polyhedron also called the quasitruncated hexahedron, whose dual polyhedron is the great triakis octahedron. The convex hull of the stellated truncated hexahedron is the Archimedean small rhombicuboctahedron.
Faces: 14 equilateral triangles and 6 regular octagrams | Edges: 36 | Vertices: 24 | Dihedral angles: 54.74° and 90°. More…
2. Great Triakis Octahedron
The great triakis octahedron is the dual of the uniform stellated truncated hexahedron. It has 24 intersecting isosceles triangles faces and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 24 isosceles triangles | Edges: 36 | Vertices: 14 | Dihedral angle: 60.72°. More…
3. Truncated Great Dodecahedron
U37 The truncated great dodecahedron is the uniform polyhedron whose dual is the small stellapentakis dodecahedron. It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron.
Faces: 12 regular pentagrams and 12 regular decagons | Edges: 90 | Vertices: 60 | Dihedral angles: 63.43° and 116.57°. More…
4. Small Stellapentakis Dodecahedron
The small stellapentakis dodecahedron is the polyhedron whose dual is the uniform truncated great dodecahedron. It has 60 intersecting triangular faces and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 60 isosceles triangles | Edges: 90 | Vertices: 24 | Dihedral angle: 149.1°. More…
5. Small Stellated Truncated Dodecahedron
U58 The small stellated truncated dodecahedron is the uniform polyhedron also called the quasitruncated small stellated dodecahedron, whose dual polyhedron is the great pentakis dodecahedron. Part of each face lies within the solid, hence is invisible in solid models.
Faces: 12 regular pentagons and 12 regular decagrams | Edges: 90 | Vertices: 60 | Dihedral angles: 63.43° and 116.57°. More…
6. Great Pentakis Dodecahedron
The great pentakis dodecahedron is the polyhedron whose dual is the uniform small stellated truncated dodecahedron. The pentagonal faces pass close to the center in the uniform polyhedron, causing this dual to be very spikey and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 60 isosceles triangles | Edges: 90 | Vertices: 24 | Dihedral angle: 108.22°. More…
7. Great Stellated Truncated Dodecahedron
U66 The great stellated truncated dodecahedron is the uniform polyhedron also called the quasitruncated great stellated dodecahedron, whose dual is the great triakis icosahedron. Part of each face lies within the solid, hence is invisible in solid models.
Faces: 20 equilateral triangles and 12 regular decagrams | Edges: 90 | Vertices: 60 | Dihedral angles: 63.43° and 79.19°. More…
8. Great Triakis Icosahedron
The great triakis icosahedron is the polyhedron whose dual is the uniform great stellated truncated dodecahedron. Its faces are isosceles triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 60 isosceles triangles | Edges: 90 | Vertices: 32 | Dihedral angle: 81°. More…
9. Great Truncated Icosahedron
U55 The great truncated icosahedron is the uniform polyhedron, also called the truncated great icosahedron, whose dual is the great stellapentakis dodecahedron. Part of each face lies within the solid, hence is invisible in solid models.
Faces: 20 regular hexagons and 12 regular pentagrams | Edges: 90 | Vertices: 60 | Dihedral angles: 41.81° and 100.81°. More…
10. Great Stellapentakis Dodecahedron
The great stellapentakis dodecahedron (or great astropentakis dodecahedron) is the polyhedron whose dual is the uniform great truncated icosahedron. Its faces are isosceles triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 60 isosceles triangles | Edges: 90 | Vertices: 32 | Dihedral angle: 123.32°. More…
11. Cubitruncated Cuboctahedron
U16 The cubitruncated cuboctahedron (or cuboctatruncated cuboctahedron) is the uniform polyhedron whose dual is the tetradyakis hexahedron and is a faceted octahedron. Its convex hull is a nonuniform truncated cuboctahedron.
Faces: 8 regular hexagons, 6 regular octagons and 6 regular octagrams | Edges: 72 | Vertices: 48 | Dihedral angles: 54.74°, 90° and 125.26°. More…
12. Tetradyakis Hexahedron
The tetradyakis hexahedron (or great disdyakis dodecahedron) is the polyhedron whose dual is the uniform cubitruncated cuboctahedron. Its faces are scalene triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 48 obtuse triangles | Edges: 72 | Vertices: 20 | Dihedral angle: 135.58°. More…
13. Great Truncated Cuboctahedron
U20 The great truncated cuboctahedron (or quasitruncated cuboctahedron or stellatruncated cuboctahedron) is the uniform polyhedron whose dual is the great disdyakis dodecahedron. Its convex hull is a nonuniform truncated cuboctahedron.
Faces: 8 regular hexagons, 12 squares and 6 regular octagrams | Edges: 72 | Vertices: 48 | Dihedral angles: 35.26°, 54.73° and 135°. More…
14. Great Disdyakis Dodecahedron
The great disdyakis dodecahedron is the polyhedron whose dual is the uniform great truncated cuboctahedron. The great disdyakis dodecahedron is topologically identical to the convex Catalan solid, disdyakis dodecahedron, which is dual to the truncated cuboctahedron.
Faces: 48 obtuse triangles | Edges: 72 | Vertices: 26 | Dihedral angles: 123.85° and 236.15°. More…
15. Icositruncated Dodecadodecahedron
U45 The icositruncated dodecadodecahedron (or icosidodecatruncated icosidodecahedron) is the uniform polyhedron whose dual is the tridyakis icosahedron. Its convex hull is a nonuniform truncated icosidodecahedron.
Faces: 20 regular hexagons, 12 regular decagons and 12 regular decagrams | Edges: 180 | Vertices: 120 | Dihedral angles: 100.81°, 116.57° and 142.62°. More…
16. Tridyakis icosahedron
The tridyakis icosahedron is the polyhedron whose dual is the uniform icositruncated dodecadodecahedron. Its faces are scalene triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 120 obtuse triangles | Edges: 180 | Vertices: 44 | Dihedral angle: 151.04°. More…
17. Truncated Dodecadodecahedron
U59 The truncated dodecadodecahedron (or quasitruncated dodecahedron or stellatruncated dodecadodecahedron) is the uniform polyhedron whose dual is the medial disdyakis triacontahedron. The central region of the polyhedron is connected to the exterior via 20 small triangular holes.
Faces: 30 squares, 12 regular decagons and 12 regular decagrams | Edges: 180 | Vertices: 120 | Dihedral angles: 58.28°, 63.43° and 148.28°. More…
18. Medial Disdyakis Triacontahedron
The medial disdyakis triacontahedron is the polyhedron whose dual is the uniform truncated dodecadodecahedron. Its faces are scalene triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 120 obtuse triangles | Edges: 180 | Vertices: 54 | Dihedral angles: 144.9° and 215.09°. More…
19. Great Truncated Icosidodecahedron
U68 The great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is the uniform polyhedron whose dual is the great disdyakis triacontahedron. It can be alternated into the great inverted snub icosidodecahedron after equalizing edge lengths.
Faces: 30 squares, 20 regular hexagons and 12 regular decagrams | Edges: 180 | Vertices: 120 | Dihedral angles: 69.09°, 79.19° and 121.72°. More…
20. Great Disdyakis Triacontahedron
The great disdyakis triacontahedron (or trisdyakis icosahedron) is the polyhedron whose dual is the uniform great truncated icosidodecahedron. Its faces are scalene triangles and part of each triangle lies within the solid, hence is invisible in solid models.
Faces: 120 obtuse triangles | Edges: 180 | Vertices: 62 | Dihedral angle: 121.34°. More…
21. Self-intersecting truncated polyhedra and their duals
Representation with each Self-intersecting truncated polyhedron and its respective dual. In this project, we have polyhedra simulating a DNA ribbon with the respective connections between the dual of Self-intersecting truncated polyhedra.
Self-intersecting truncated polyhedra - Visualization of polyhedra with Augmented Reality and 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., "Self-intersecting truncated polyhedra - Visualization of polyhedra with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra/selfintersecttruncated/>, January 2023.
References:
Weisstein, Eric W. “Miscellaneous Polyhedra” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/topics/MiscellaneousPolyhedra.html
Weisstein, Eric W. “Uniform Polyhedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/UniformPolyhedron.html
Wikipedia https://en.wikipedia.org/wiki/List_of_uniform_polyhedra
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/