The mathematics
Only five convex regular polyhedra exist: equal faces, equal angles, identical vertices. The proof is in Euclid's Elements, and Euler's formula checks each one: vertices minus edges plus faces always equals 2.
Platonic Solids in 3D
The five perfect forms of geometry, tied by Plato to the elements of the world. Spin each one in your hands, reveal the dual solid hiding inside it and watch them all rest within Metatron's Cube.
Choose the form
tetrahedron
Four triangular faces, the smallest and sharpest of the five forms. Plato tied it to fire for its rising point; it is the only solid that is its own dual.
- Element
- Fire
- Dual
- tetrahedron
- Vertices
- 4
- Edges
- 6
- Faces
- 4
Drag to rotate; on a keyboard, use the arrow keys.
Three layers
What you are seeing
Nature
Salt crystals grow as cubes, fluorite forms octahedra and entire viruses organize as icosahedra. The microscopic radiolarians drawn by Haeckel reproduce the five forms on the ocean floor.
The symbol
In the Timaeus, Plato made the five forms the alphabet of the elements: fire, earth, air, water and the cosmos. Kepler nested them between the orbits of the planets; the hermetic tradition keeps them inside Metatron's Cube.
Questions
Frequently asked questions
What are the platonic solids?
They are the only five convex regular polyhedra: tetrahedron, cube, octahedron, dodecahedron and icosahedron. In all of them, the faces are identical regular polygons and every vertex gathers the same number of faces.
Why are there only five?
Because at each vertex the face angles must add up to less than 360 degrees. Only five combinations of regular polygons respect that condition, as Euclid demonstrated in the Elements.
What is Metatron's Cube in 3D?
The name says the essence: it is a cube within a cube, contemplated along its own diagonal. From that direction, the 16 vertices appear as the 13 circles of the flat diagram, joined by 78 lines, because the 4 vertices on the axis overlap at the center. As you rotate the piece, the spheres separate and the connections among them all rise to 120: the count changes from C(13,2) to C(16,2).
What does the dual solid mean?
It is the solid born by placing a vertex at the center of each face of the original: the cube generates the octahedron, the dodecahedron generates the icosahedron and the tetrahedron generates another tetrahedron. In the tool, the dual appears drawn inside the form.
How do the solids relate to the elements?
The association comes from Plato's Timaeus: tetrahedron fire, cube earth, octahedron air, icosahedron water and dodecahedron the cosmos. It is a philosophical and symbolic reading, not a physical theory; the beauty lies in the marriage of form and quality.
Take the solids with you on your journey
The Initiatory Map of Sacred Geometry, our eBook, walks through these forms step by step. Leave your email to receive the Lab's next instruments and news from the portal.