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
Non convex polyhedra fractals
Using the same principle as the construction of the Sierpinski triangle or the Koch curve, we can construct fractals from other regular polygons. When these polygons form a polyhedron, we have the construction of a fractal polyhedron.
Augmented Reality | 3D Models | Home
Immersive rooms
🔗 room 1 | 🔗 room 2 | 🔗 room 3 | 🔗 room 4
🔗 room 5 | 🔗 room 6 | 🔗 room 7 | 🔗 room 8
Augmented Reality
To view fractal polyhedra in AR, simply visit the pages indicated in the 3D solid models using 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. Escher solid fractal
Applying the construction principle of the Sierpinski triangle to the 48 faces of the Escher solid, we obtain an Escher solid fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 48 | 72 | 26 |
| 1 | 12 | 576 | 864 | 312 |
| 2 | 144 | 6912 | 10368 | 3744 |
| 3 | 1728 | 82944 | 124416 | 44928 |
2. Small stellated dodecahedron fractal
Applying the construction principle of the Koch curve to the 12 faces of the small stellated dodecahedron, we obtain a small stellated dodecahedron fractal. In the first order of fractal construction, we construct a new solid at each vertex of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 12 | 30 | 12 |
| 1 | 12 | 144 | 360 | 144 |
| 2 | 144 | 1728 | 4320 | 1728 |
| 3 | 1728 | 20736 | 51840 | 20736 |
3. Great stellated dodecahedron fractal
Applying the construction principle of the Koch curve to the 12 faces of the great stellated dodecahedron, we obtain a great stellated dodecahedron fractal. In the first order of fractal construction, we construct a new solid at each vertex of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 12 | 30 | 20 |
| 1 | 21 | 252 | 630 | 420 |
| 2 | 441 | 5292 | 13230 | 8820 |
| 3 | 9261 | 111132 | 277830 | 185220 |
4. Great icosahedron fractal
Applying the construction principle of the Koch curve to the 20 faces of the great icosahedron, we obtain a great icosahedron fractal. In the first order of fractal construction, we construct a new solid at each vertex of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 20 | 30 | 12 |
| 1 | 12 | 240 | 360 | 144 |
| 2 | 144 | 2880 | 4320 | 1728 |
| 3 | 1728 | 34560 | 51840 | 20736 |
5. Great dodecahedron fractal
Applying the construction principle of the Koch curve to the 12 faces of the great dodecahedron, we obtain a great dodecahedron fractal. In the first order of fractal construction, we construct a new solid at each vertex of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 12 | 30 | 12 |
| 1 | 12 | 144 | 360 | 144 |
| 2 | 144 | 1728 | 4320 | 1728 |
| 3 | 1728 | 20736 | 51840 | 20736 |
6. Great stellapentakis dodecahedron fractal
Applying the construction principle of the Koch curve to 20 faces of the great stellapentakis dodecahedron, we obtain a great stellapentakis dodecahedron fractal. In the first order of fractal construction, we construct a new solid at 20 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 90 | 32 |
| 1 | 21 | 1260 | 1890 | 672 |
| 2 | 441 | 26460 | 39690 | 14112 |
| 3 | 9261 | 555660 | 833490 | 296352 |
7. Pentagramic dipyramid fractal
Applying the construction principle of the Koch curve to the edges that form the pentagram of the pentagramic dipyramid, we obtain a pentagramic dipyramid fractal. In the first order of fractal construction, we construct a new solid at 5 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 10 | 15 | 7 |
| 1 | 6 | 60 | 90 | 42 |
| 2 | 36 | 360 | 540 | 252 |
| 3 | 216 | 2160 | 3240 | 1512 |
| 4 | 1296 | 12960 | 19440 | 9072 |
8. Medial triambic icosahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the medial triambic icosahedron, we obtain a medial triambic icosahedron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 20 | 60 | 24 |
| 1 | 13 | 260 | 780 | 312 |
| 2 | 169 | 3380 | 10140 | 4056 |
| 3 | 2197 | 43940 | 131820 | 52728 |
9. Great rhombic triacontahedron fractal
Applying the construction principle of the Koch curve to 20 faces of the great rhombic triacontahedron, we obtain a great rhombic triacontahedron fractal. In the first order of fractal construction, we construct a new solid at 20 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 30 | 60 | 32 |
| 1 | 21 | 630 | 1260 | 672 |
| 2 | 441 | 13230 | 26460 | 14112 |
| 3 | 9261 | 277830 | 555660 | 296352 |
10. Medial rhombic triacontahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the medial rhombic triacontahedron, we obtain a medial rhombic triacontahedron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 30 | 60 | 24 |
| 1 | 13 | 390 | 780 | 312 |
| 2 | 169 | 5070 | 10140 | 4056 |
| 3 | 2197 | 65910 | 131820 | 52728 |
11. Small ditrigonal dodecacronic hexecontahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the small ditrigonal dodecacronic hexecontahedron, we obtain a small ditrigonal dodecacronic hexecontahedron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 120 | 44 |
| 1 | 13 | 780 | 1560 | 572 |
| 2 | 169 | 10140 | 20280 | 7436 |
| 3 | 2197 | 131820 | 263640 | 96668 |
12. Rhombicosacron fractal
Applying the construction principle of the Koch curve to 20 faces of the rhombicosacron, we obtain a rhombicosacron fractal. In the first order of fractal construction, we construct a new solid at 20 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 120 | 50 |
| 1 | 21 | 1260 | 2520 | 1050 |
| 2 | 441 | 26460 | 52920 | 22050 |
| 3 | 9261 | 555660 | 1111320 | 463050 |
13. Small hexacronic icositetrahedron fractal
Applying the construction principle of the Koch curve to 6 faces of the small hexacronic icositetrahedron, we obtain a small hexacronic icositetrahedron fractal. In the first order of fractal construction, we construct a new solid at 6 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 48 | 20 |
| 1 | 7 | 168 | 336 | 140 |
| 2 | 49 | 1176 | 2352 | 980 |
| 3 | 343 | 8232 | 16464 | 6860 |
14. Great triakis octahedron fractal
Applying the construction principle of the Koch curve to 8 faces of the great triakis octahedron, we obtain a great triakis octahedron fractal. In the first order of fractal construction, we construct a new solid at 8 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 36 | 14 |
| 1 | 9 | 216 | 324 | 126 |
| 2 | 81 | 1944 | 2916 | 1134 |
| 3 | 729 | 17496 | 26244 | 10216 |
15. Great disdyakis dodecahedron fractal
Applying the construction principle of the Koch curve to 8 faces of the great disdyakis dodecahedron, we obtain a great disdyakis dodecahedron fractal. In the first order of fractal construction, we construct a new solid at 8 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 48 | 72 | 20 |
| 1 | 9 | 432 | 648 | 180 |
| 2 | 81 | 3888 | 5832 | 1620 |
| 3 | 729 | 34992 | 52488 | 14580 |
16. Small rhombidodecacron fractal
Applying the construction principle of the Koch curve to 12 faces of the small rhombidodecacron, we obtain a small rhombidodecacron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 120 | 42 |
| 1 | 13 | 780 | 1560 | 546 |
| 2 | 169 | 10140 | 20280 | 7098 |
| 3 | 2197 | 131820 | 263640 | 92274 |
17. Great triakis icosahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the great triakis icosahedron, we obtain a great triakis icosahedron fractal. In the first order of fractal construction, we construct a new solid at 12 faces of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 90 | 32 |
| 1 | 13 | 780 | 1170 | 416 |
| 2 | 169 | 10140 | 15210 | 5408 |
| 3 | 2197 | 131820 | 197730 | 70304 |
18. Great truncated icosahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the great truncated icosahedron, we obtain a great truncated icosahedron fractal. In the first order of fractal construction, we construct a new solid at 12 faces of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 32 | 90 | 60 |
| 1 | 13 | 416 | 1170 | 780 |
| 2 | 169 | 5408 | 15210 | 10140 |
| 3 | 2197 | 70304 | 197730 | 131820 |
19. Pentagrammic dipyramid fractal
Applying the principle of repetition of solids at vertices of the pentagrammic dipyramid, we obtain a pentagrammic dipyramid fractal. In the first order of fractal construction, we construct a new solid at each vertice of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2, 3, 4 and 5.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 10 | 15 | 7 |
| 1 | 6 | 60 | 90 | 42 |
| 2 | 11 | 110 | 165 | 77 |
| 3 | 21 | 210 | 315 | 147 |
| 4 | 41 | 410 | 615 | 287 |
| 5 | 81 | 810 | 1215 | 567 |
20. Heptagrammic dipyramid fractal
Applying the principle of repetition of solids at vertices of the heptagrammic dipyramid, we obtain a heptagrammic dipyramid fractal. In the first order of fractal construction, we construct a new solid at each vertice of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2, 3, 4 and 5.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 14 | 21 | 9 |
| 1 | 8 | 112 | 168 | 72 |
| 2 | 15 | 210 | 315 | 135 |
| 3 | 29 | 406 | 609 | 261 |
| 4 | 57 | 798 | 1197 | 513 |
| 5 | 113 | 1582 | 2373 | 1017 |
21. Great pentakis dodecahedron fractal
Applying the construction principle of the Koch curve to 12 vertices of the great pentakis dodecahedron, we obtain a great pentakis dodecahedron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 90 | 24 |
| 1 | 13 | 780 | 1170 | 312 |
| 2 | 169 | 10140 | 15210 | 4056 |
| 3 | 2197 | 131820 | 197730 | 52728 |
22. Icosidodecadodecahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the icosidodecadodecahedron, we obtain an icosidodecadodecahedron fractal. In the first order of fractal construction, we construct a new solid at 12 faces of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 44 | 120 | 60 |
| 1 | 13 | 572 | 1560 | 780 |
| 2 | 169 | 7436 | 20280 | 10140 |
| 3 | 2197 | 96668 | 263640 | 131820 |
23. Rhombicosahedron fractal
Applying the construction principle of the Koch curve to 12 faces of the rhombicosahedron, we obtain an rhombicosahedron fractal. In the first order of fractal construction, we construct a new solid at 12 faces of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 50 | 120 | 60 |
| 1 | 13 | 650 | 1560 | 780 |
| 2 | 169 | 8450 | 20280 | 10140 |
| 3 | 2197 | 109850 | 263640 | 131820 |
24. Medial inverted pentagonal hexecontahedron fractal
Applying the construction principle of the Koch curve to 12 vertices of the medial inverted pentagonal hexecontahedron, we obtain a medial inverted pentagonal hexecontahedron fractal. In the first order of fractal construction, we construct a new solid at 12 vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 60 | 150 | 84 |
| 1 | 13 | 780 | 1950 | 1092 |
| 2 | 169 | 10140 | 25350 | 14196 |
| 3 | 2197 | 131820 | 329550 | 184548 |
25. Heptagrammic dipyramid fractal
Applying the construction principle of the Koch curve to vertices of the heptagrammic dipyramid, we obtain a heptagrammic dipyramid fractal. In the first order of fractal construction, we construct a new solid at vertices of the original polyhedron. In this example, we have representations of the solid in orders 0, 1, 2, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 14 | 21 | 9 |
| 1 | 8 | 112 | 168 | 72 |
| 2 | 64 | 896 | 1344 | 576 |
| 3 | 512 | 7168 | 10752 | 4608 |
| 4 | 4096 | 57344 | 86016 | 36864 |
26. Hexagonal trapezohedron-antiprism toroid fractal
Applying the Antoine necklace construction principle to a hexagonal trapezohedron-antiprism toroid, we obtain a fractal of the hexagonal trapezohedron-antiprism toroid. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 48 | 24 |
| 1 | 16 | 384 | 768 | 384 |
| 2 | 256 | 6144 | 12288 | 6144 |
| 3 | 4096 | 98304 | 196608 | 98304 |
27. Tetragonal toroid fractal
Applying the Antoine necklace construction principle to a tetragonal toroid, we obtain a fractal of the tetragonal toroid. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 18 | 36 | 18 |
| 1 | 16 | 288 | 576 | 288 |
| 2 | 256 | 4608 | 9216 | 4608 |
| 3 | 4096 | 73728 | 147456 | 73728 |
28. Hexagonal trapezohedron toroid fractal
Applying the Antoine necklace construction principle to a hexagonal trapezohedron toroid, we obtain a fractal of the hexagonal trapezohedron toroid. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 60 | 36 |
| 1 | 16 | 384 | 960 | 576 |
| 2 | 256 | 6144 | 15360 | 9216 |
| 3 | 4096 | 98304 | 245760 | 147456 |
29. Hexagonal antiprism-trapezohedron toroid fractal
Applying the Antoine necklace construction principle to a hexagonal antiprism-trapezohedron toroid, we obtain a fractal of the hexagonal antiprism-trapezohedron toroid. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 48 | 24 |
| 1 | 16 | 384 | 768 | 384 |
| 2 | 256 | 6144 | 12288 | 6144 |
| 3 | 4096 | 98304 | 196608 | 98304 |
30. Anti-octagonal iris toroid fractal
Applying the Antoine necklace construction principle to a anti-octagonal iris toroid, we obtain a fractal of the anti-octagonal iris toroid. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 32 | 48 | 16 |
| 1 | 16 | 512 | 768 | 256 |
| 2 | 256 | 8192 | 12288 | 4096 |
| 3 | 4096 | 131072 | 196608 | 65536 |
31. Geometric Christmas tree v1
Construction of a Christmas tree using the following solids: right circular cone frustum, pyramid frustums, a pentagrammic dipyramid and a conical helix with dodecahedrons.
32. Geometric Christmas tree v2
Construction of a Christmas tree using the following solids: right circular cone frustum, pyramid frustums, a pentagrammic dipyramid and a conical helix with small stellated dodecahedrons.
33. Geometric Christmas tree v3
Construction of a Christmas tree using the following solids: right circular cone frustum, pyramid frustums, a heptagrammic dipyramid and a conical helix with great stellapentakis dodecahedrons.
34. Geometric Christmas tree v4
Construction of a Christmas tree using the following solids: right circular cone frustum, stellated pyramid frustums, a heptagrammic dipyramid and a conical helix with icosahedrons.
35. Geometric Christmas tree v5
Construction of a Christmas tree using the following solids: right circular cone frustum, stellated pyramid frustums, a heptagrammic dipyramid and a conical helix with icosidodecadodecahedrons.
36. Geometric Christmas tree v6
Construction of a Christmas tree using the following solids: right circular cone frustum, stellated pyramid frustums, a heptagrammic dipyramid and a conical helix with truncated great icosahedrons.
37. Geometric Christmas tree v7
Construction of a Christmas tree using the following solids: right circular cone frustum, stellated pyramid frustums, a pentagrammic dipyramid and a conical helix with small rhombidodecacrons.
Non convex polyhedra fractals - 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., "Non convex polyhedra fractals - Visualization of polyhedra with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra2/fractalnonconvex/>, October 2023.
References:
Weisstein, Eric W. “Fractal” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Fractal.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/