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
Platonic 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
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. Fractal tetrahedron
Applying the construction principle of the Sierpinski triangle to the 4 faces of the regular tetrahedron, we obtain a regular fractal tetrahedron. 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 4 | 6 | 4 |
| 1 | 4 | 16 | 24 | 16 |
| 2 | 16 | 64 | 96 | 64 |
| 3 | 64 | 256 | 384 | 256 |
| 4 | 256 | 1024 | 1536 | 1024 |
2. Fractal octahedron
Applying the construction principle of the Sierpinski triangle to the 8 faces of the regular octahedron, we obtain a regular fractal octahedron. 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, 3 and 4.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 8 | 12 | 6 |
| 1 | 6 | 48 | 72 | 36 |
| 2 | 36 | 288 | 432 | 216 |
| 3 | 216 | 1728 | 2592 | 1296 |
| 4 | 1296 | 10368 | 15552 | 7776 |
3. Fractal cube
Applying the construction principle of the Sierpinski carpet to the 6 faces of the cube, we obtain a fractal cube. 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 cube’s edge. In this example, we have representations of the solid in orders 0, 1, 2 and 3.
| order | polyhedra | faces | edges | vertices |
|---|---|---|---|---|
| 0 | 1 | 6 | 12 | 8 |
| 1 | 20 | 120 | 240 | 160 |
| 2 | 400 | 2400 | 4800 | 3200 |
| 3 | 8000 | 48000 | 96000 | 64000 |
4. Fractal icosahedron
Applying the construction principle of the Koch curve to the 20 faces of the regular icosahedron, we obtain a regular fractal icosahedron. 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. Fractal dodecahedron
Applying the construction principle of the Koch curve to the 12 faces of the regular dodecahedron, we obtain a regular fractal dodecahedron. 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 | 20 | 240 | 600 | 400 |
| 2 | 400 | 4800 | 12000 | 8000 |
| 3 | 8000 | 96000 | 240000 | 160000 |
6. Tetrahedron dragon fractal
Applying the construction principle of the Dragon curve with regular tetrahedron, we obtain a tetrahedron dragon fractal. In the first order of construction of the fractal, we construct two new tetrahedra corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
7. Fractal tree
Applying the principle of repetitions with cone frustum, we obtain a fractal 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 7.
8. Fractal tree with dodecahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added 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 7.
9. Menger's Cross - Jerusalem: Cube v1
Consider a fractal cube. We can increase the edge sizes of the corner cubes and decrease the edge sizes of the intermediate cubes to reveal a cross. In this version, we have 8 homothetic cubes with an aspect ratio of ⅖ and 12 homothetic cubes with a proportion of ⅕.
10. Menger's Cross - Jerusalem: Cube v2
Consider a fractal cube. We can increase the edge sizes of the corner cubes and decrease the edge sizes of the intermediate cubes to reveal a cross. In this version, we have 8 homothetic cubes with an aspect ratio of √2 - 1 and 12 homothetic cubes with a proportion of (√2 - 1)².
11. Mosely snowflake: Cube
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 cubes and the center cube from each previous iteration.
12. Fractal tree with icosahedrons
Applying the principle of repetitions with cone frustum, we obtain a fractal tree. In this example, we added 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 7.
13. Tetrahedron dragon fractal (3 rotations)
Applying the construction principle of the Dragon curve with regular tetrahedron and 3 rotations, we obtain a tetrahedron dragon fractal. In the first order of construction of the fractal, we construct three new tetrahedra corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
14. Cube dragon fractal
Applying the construction principle of the Dragon curve with a cube and 3 rotations, we obtain a cube dragon fractal. In the first order of construction of the fractal, we construct three new cube corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
15. Octahedron dragon fractal
Applying the construction principle of the Dragon curve with a regular octahedron and 3 rotations, we obtain a octahedron dragon fractal. In the first order of construction of the fractal, we construct three new octahedra corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
16. Icosahedron dragon fractal
Applying the construction principle of the Dragon curve with a regular icosahedron and 3 rotations, we obtain a icosahedron dragon fractal. In the first order of construction of the fractal, we construct three new icosahedra corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
17. Dodecahedron dragon fractal
Applying the construction principle of the Dragon curve with a regular dodecahedron and 3 rotations, we obtain a dodecahedron dragon fractal. In the first order of construction of the fractal, we construct three new dodecahedra corresponding to one original polyhedron. In this example, we have solid representations in orders from 0 to 10.
Platonic 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., "Platonic polyhedra fractals - Visualization of polyhedra with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra2/fractalplatonic/>, October 2023.
References:
Weisstein, Eric W. “Fractal” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Fractal.html
Weisstein, Eric W. “Platonic Solid” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html
Wikipedia https://en.wikipedia.org/wiki/Platonic_solid
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/