Polyhedra

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

An Platonic solid is a regular and convex polyhedron. It is constructed by congruent and regular polygonal faces with the same number of faces meeting at each vertex. They are named by the ancient Greek philosopher Plato who classified that the classical elements were made from these regular solids.

Augmented Reality | 3D Models | Home

Immersive room

🔗 room link

Augmented Reality

To view Platonic polyhedra in AR, simply visit

https://paulohscwb.github.io/polyhedra/platonic/ra.html

with any browser with a webcam device (smartphone, tablet or notebook).
Access to 3D models pages is done by clicking on the blue circle that appears on top of the marker.

3D models

1. Icosahedron

U₂₂ The icosahedron has five equilateral triangular faces meeting at each vertex. A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron.

Faces: 20 triangles | Edges: 30 | Vertices: 12 | Sphericity: 0.939 | Dihedral angle: 138.19°. More…

2. Dodecahedron

U₂₃ The dodecahedron has three regular pentagonal faces meeting at each vertex. The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges.

Faces: 12 pentagons | Edges: 30 | Vertices: 20 | Sphericity: 0.91 | Dihedral angle: 116.57°. More…

3. Octahedron

U₅ The octahedron has four equilateral triangular faces meeting at each vertex. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.

Faces: 8 triangles | Edges: 12 | Vertices: 6 | Sphericity: 0.846 | Dihedral angle: 109.47°. More…

4. Cube

U₆ The cube has three square faces meeting at each vertex. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained.

Faces: 6 squares | Edges: 12 | Vertices: 8 | Sphericity: 0.806 | Dihedral angle: 90°. More…

5. Tetrahedron

U₁ The tetrahedron has three equilateral triangular faces meeting at each vertex. The tetrahedron is also known as a triangular pyramid and it is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres.

Faces: 4 triangles | Edges: 6 | Vertices: 4 | Sphericity: 0.671 | Dihedral angle: 70.53°. More…

6. Platonic polyhedra compound

A polyhedron compound is an arrangement of several interpenetrating polyhedra, all the same or of several distinct types, usually with visually appealing symmetrical properties. In this example, we have a Platonic polyhedra compound using diagonals, vertices and edge perpendicular bisectors.
More…

7. Platonic polyhedra and their duals

Representation with each Platonic polyhedron and its respective dual. In this project, we have polyhedra simulating a DNA ribbon with the respective connections between the dual Platonic polyhedra.

Platonic 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., "Platonic polyhedra - Visualization of polyhedra with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra/platonic/>, September 2019.

References:
Weisstein, Eric W. “Platonic Solid” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html
Weisstein, Eric W. “Uniform Polyhedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/UniformPolyhedron.html
Wikipedia https://en.wikipedia.org/wiki/Platonic_solid
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/