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
Catalan 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.
This work shows Catalan and some Archimedean polyhedra forming fractals, modeled for visualization in Virtual Reality.
3D models
1. Deltoidal hexecontahedron
Applying the construction principle of the Koch curve to faces of the deltoidal hexecontahedron, we obtain a deltoidal hexecontahedron 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 | 62 |
| 1 | 21 | 1260 | 2520 | 1302 |
| 2 | 441 | 26460 | 52920 | 27342 |
| 3 | 9261 | 555660 | 1111320 | 574182 |
2. Deltoidal icositetrahedron
Applying the construction principle of the Sierpinski curve to faces of the deltoidal icositetrahedron, we obtain a deltoidal 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 48 | 26 |
| 1 | 6 | 144 | 288 | 156 |
| 2 | 36 | 864 | 1728 | 936 |
| 3 | 216 | 5184 | 10368 | 5616 |
| 4 | 1296 | 31104 | 62208 | 33696 |
3. Disdyakis dodecahedron
Applying the construction principle of the Sierpinski curve to faces of the disdyakis dodecahedron, we obtain a disdyakis dodecahedron 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 48 | 72 | 26 |
| 1 | 6 | 288 | 432 | 156 |
| 2 | 36 | 1728 | 2592 | 936 |
| 3 | 216 | 10368 | 15552 | 5616 |
| 4 | 1296 | 62208 | 93312 | 33696 |
4. Disdyakis triacontahedron
Applying the construction principle of the Koch curve to faces of the disdyakis triacontahedron, we obtain a disdyakis 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 | 120 | 180 | 62 |
| 1 | 21 | 2520 | 3780 | 1302 |
| 2 | 441 | 52920 | 79380 | 27342 |
| 3 | 9261 | 1111320 | 1666980 | 574182 |
5. Pentagonal hexecontahedron
Applying the construction principle of the Koch curve to faces of the pentagonal hexecontahedron, we obtain a pentagonal hexecontahedron 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 | 150 | 92 |
| 1 | 21 | 1260 | 3150 | 1932 |
| 2 | 441 | 26460 | 66150 | 40572 |
| 3 | 9261 | 555660 | 1389150 | 852012 |
6. Pentagonal icositetrahedron
Applying the construction principle of the Sierpinski curve to faces of the pentagonal icositetrahedron, we obtain a pentagonal icositetrahedron 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 | 60 | 38 |
| 1 | 8 | 192 | 480 | 304 |
| 2 | 64 | 1536 | 3840 | 2432 |
| 3 | 512 | 12288 | 30720 | 19456 |
7. Pentakis dodecahedron
Applying the construction principle of the Koch curve to faces of the pentakis dodecahedron, we obtain a pentakis 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 |
8. Rhombic dodecahedron
Applying the construction principle of the Sierpinski curve to faces of the rhombic dodecahedron, we obtain a rhombic dodecahedron 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 12 | 24 | 14 |
| 1 | 6 | 72 | 144 | 84 |
| 2 | 36 | 432 | 864 | 504 |
| 3 | 216 | 2592 | 5184 | 3024 |
| 4 | 1296 | 15552 | 31104 | 18144 |
9. Archimedean rhombicosidodecahedron
Applying the construction principle of the Koch curve to faces of the rhombicosidodecahedron, we obtain a rhombicosidodecahedron 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 | 62 | 120 | 60 |
| 1 | 21 | 1302 | 2520 | 1260 |
| 2 | 441 | 27342 | 52920 | 26460 |
| 3 | 9261 | 574182 | 1111320 | 555660 |
10. Rhombic triacontahedron
Applying the construction principle of the Koch curve to faces of the rhombic triacontahedron, we obtain a 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 |
11. Archimedean snub dodecahedron
Applying the construction principle of the Koch curve to faces of the snub dodecahedron, we obtain a snub 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 | 92 | 150 | 60 |
| 1 | 21 | 1932 | 3150 | 1260 |
| 2 | 441 | 40572 | 66150 | 26460 |
| 3 | 9261 | 852012 | 1389150 | 555660 |
12. Tetrakis hexahedron - Menger sponge
Applying the construction principle of the Sierpinski carpet to tetrakis hexahedron faces, we obtain a tetrakis hexahedron fractal. In the first order of construction of the fractal, we construct 8 new solids on each face of the original polyhedron, all with ⅓ the measurement of the tetrakis hexahedron’s edge. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
13. Tetrakis hexahedron - Menger's Cross Jerusalem
Consider a tetrakis hexahedron. We can increase the edge sizes of the corner tetrakis hexahedrons and decrease the edge sizes of the intermediate tetrakis hexahedrons to reveal a cross. In this version, we have 8 homothetic tetrakis hexahedrons with an aspect ratio of ⅖ and 12 homothetic tetrakis hexahedrons with a proportion of ⅕.
14. Tetrakis hexahedron - Menger's Cross Jerusalem v2
Consider a tetrakis hexahedron. We can increase the edge sizes of the corner tetrakis hexahedrons and decrease the edge sizes of the intermediate tetrakis hexahedrons to reveal a cross. In this version, we have 8 homothetic tetrakis hexahedrons with an aspect ratio of √2 - 1 and 12 homothetic tetrakis hexahedrons with a proportion of (√2 - 1)².
15. Tetrakis hexahedron - Mosely snowflake
The Mosely snowflake is a type of Sierpinski-Menger fractal obtained in two variants by the operation used in creating the Sierpinski-Menger snowflake. In this case, we removed eight corner tetrakis hexahedrons and the center tetrakis hexahedron in each iteration.
16. Triangular fractal tree with pentakis dodecahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentakis dodecahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 6.
17. Pentagonal fractal tree with pentakis dodecahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentakis dodecahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 5.
18. Triangular fractal tree with rhombic triacontahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added rhombic triacontahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 6.
19. Pentagonal fractal tree with rhombic triacontahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added rhombic triacontahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 5.
20. Triangular fractal tree with triakis icosahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added triakis icosahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 6.
21. Pentagonal fractal tree with triakis icosahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added triakis icosahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 5.
22. Triangular fractal tree with pentagonal hexecontahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentagonal hexecontahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 6.
23. Pentagonal fractal tree with pentagonal hexecontahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentagonal hexecontahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 5.
24. Triangular fractal tree with pentagonal icositetrahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentagonal icositetrahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 6.
25. Pentagonal fractal tree with pentagonal icositetrahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added pentagonal icositetrahedrons as the “fruits” or “flowers” of the tree. In the first order of construction the fractal, we build three new cone frustums connected with a face of the original cone frustum. In this example, we have solid representations in orders from 0 to 5.
26. Triakis icosahedron
Applying the construction principle of the Koch curve to faces of the triakis icosahedron, we obtain a triakis icosahedron 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 |
27. Triakis octahedron
Applying the construction principle of the Sierpinski curve to faces of the triakis octahedron, we obtain a triakis octahedron 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 24 | 36 | 14 |
| 1 | 6 | 144 | 216 | 84 |
| 2 | 36 | 864 | 1296 | 504 |
| 3 | 216 | 5184 | 7776 | 3024 |
| 4 | 1296 | 31104 | 46656 | 18144 |
28. Triakis tetrahedron
Applying the construction principle of the Sierpinski curve to faces of the triakis tetrahedron, we obtain a triakis tetrahedron fractal. In the first order of fractal construction, we construct a new solid at 4 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 | 12 | 18 | 8 |
| 1 | 4 | 48 | 72 | 32 |
| 2 | 16 | 192 | 288 | 128 |
| 3 | 64 | 768 | 1152 | 512 |
| 4 | 256 | 3072 | 4608 | 2048 |
29. Archimedean truncated icosahedron
Applying the construction principle of the Koch curve to faces of the truncated icosahedron, we obtain a truncated icosahedron 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 | 32 | 90 | 60 |
| 1 | 21 | 672 | 1890 | 1260 |
| 2 | 441 | 14112 | 39690 | 26460 |
| 3 | 9261 | 296352 | 833490 | 555660 |
30. Archimedean truncated icosidodecahedron
Applying the construction principle of the Koch curve to faces of the truncated icosidodecahedron, we obtain a truncated icosidodecahedron 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 | 62 | 180 | 120 |
| 1 | 21 | 1302 | 3780 | 2520 |
| 2 | 441 | 27342 | 79380 | 52920 |
| 3 | 9261 | 574182 | 1666980 | 1111320 |
Catalan fractals: 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., "Catalan fractals: visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra3/fractal-catalan/>, February 2026.
References:
Weisstein, Eric W. “Archimedean Solid” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedeanSolid.html
Weisstein, Eric W. “Catalan Solid” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/CatalanSolid.html
McCooey, D. I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/