The Platonic Solids


Platonic Solids

Five key sacred patterns that make up all matter in this universe. Known as the Platonic Solids, these shapes are Tetrahedron, Hexahedron, Octahedron, Dodecahedron, and Icosahedron. The Platonic Solids are called the perfect solids because, in each of them, every facet has equal side length, equal facet size, and equal angle. Moreover, all these structures fit perfectly within a sphere. The five Platonic Solids are regarded as the building blocks of the universe and equated with the five classical elements with which everything is made – fire, earth, air, water, and aether.

So how did the Platonic Solids come to be?

The ancient Greeks discovered the first three, the Pythagoreans had the knowledge of only the tetrahedron, the cube, and the dodecahedron. Later, Theaetetus a mathematician topped up the octahedron and the icosahedron. The shapes are called Platonic solids after an ancient Greek philosopher known as Plato, he stated that the shapes were fundamental components of the physical universe.

Platonic Solids

The five Platonic solids

Tetrahedron

The tetrahedron also known as a triangular pyramid, it has four triangular faces, four vertex corners, and six straight edges. It is the simplest of the solids and the only one with less than five faces. A tetrahedron is a 3-simplex. This is because all its vertices are equidistant from each other unlike the rest of the platonic shapes.
 
The tetrahedron is known as the triangular pyramid due to its triangular base which could be any of the four faces. It is a unique pyramid with the faces connecting the base to a common point. The tetrahedron is termed as self-dual meaning that its other dual is another tetrahedron, combined they form a stellated octahedron or stella octangula (a compound figure of two dual regular tetrahedron).
 

Cube

A cube also known as a hexahedron is a three-dimensional object made up of six square faces, twelve edges, and eight vertices. It is the only finite perfectly symmetrical whose faces are squared, at each vertex, three identical square faces meet.
 
The duplication of a cube is among the popular unsolved mathematical problems (Delian problem). The ancient Greek mathematicians used a compass and straightedge only, to construct a cube that should have twice the volume of the given cube. They found it to be impossible and later it was discovered that it is because the cube root of 2 is not a constructible number.

 

Octahedron

Octahedron is a three-dimensional shape with eight faces, twelve edges, and six vertices. The eight faces are a composition of equilateral triangles, four meeting at the same vertex creating a square-bottomed pyramid. When the two shapes are put together base to base, they create a complete octahedron. It is the dual polyhedron of a cube and a rectified tetrahedron.
 
The Octahedron is a simple well covered polyhedral meaning it requires complete removal of the four vertices to disconnect all the vertices. Note: there are irregular octahedra and the one being discussed is a regular octahedron (all faces are the same size and shape) and it is the one that qualifies as a platonic shape.

 

Dodecahedron

The dodecahedron is any polyhedron with a total of twelve flat faces (it is where the name comes from - Greek dodeca - meaning 12). However, the regular dodecahedron is the Platonic solid. It is made up of twelve pentagonal faces (5-sided), thirty edges and twenty vertices where at each, three edges meet with an internal angle of 3240.
 
The twelve regular pentagons each have internal angles of 1080 and each face is the same identical polygon. The dodecahedron has geometrical relations with the icosahedron as the two shapes are duals.

Icosahedron

The Icosahedron is a polyhedron with 20 faces, the name comes from the Greek word icosa- meaning 20. It has twenty equilateral triangular faces, thirty edges, and twelve vertices. At each vertex, five edges and five faces meet.
 
As mentioned earlier, the icosahedron is the dual of the dodecahedron having three regular pentagonal faces around each vertex.

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