Geometry

Platonic Solids
The regular polyhedra – solid figures with identical regular polygonal faces – were already known in classical Greece and were named after Plato, who used them in his theory of the structure of the world.

Regular polygons
REGULAR POLYGONS

The equilateral triangle is the simplest regular polygon. The simplest polyhedron, the tetrahedron, can be constructed with 4 of these as its faces. A tetrahedron can be inscribed in a sphere.

Regular polygons
TETRAHEDRON

The octahedron’s 8 faces are also equilateral triangles. As the number of faces in a regular polyhedron increases, the form of the polyhedron becomes more spherical, a sphere having the smallest possible surface area in relation to its volume.

Regular polygons
OCTAHEDRON

The spherical form is clearly visible in the icosahedron, which is comprised of 20 equilateral triangular faces. When balanced on a point, the icosahedron can be understood as a “ring” of 10 triangular faces with a rosette of 5 faces above and another rosette of 5 below.

Regular polygons
ICOSAHEDRON

Besides these 3 polyhedra with triangular faces, we are familiar with one with 6 square faces, namely the regular hexahedron, also called the cube. The cube is probably the most common polyhedron, as the box shape has become a part of our culture.

Regular polygons
HEXAHEDRON

The last of the 5 Platonic Solids is the regular dodecahedron, which is comprised of 12 regular pentagonal faces with three corners meeting at each apex. It may be understood as a top and a bottom face, with 2 rings of 5 faces each between the top and bottom.

Regular polygons
DODECAHEDRON

When 3 regular hexagons are placed together they form a flat surface, therefore a “sphere” cannot be formed out of regular hexagonal faces only. Regular polygons with more than 6 sides cannot form a polyhedron, as 3 corners cannot meet when the corner angle is more than 120°.

Regular polygons

Archimedean Solids
Archimedes described a series of “semi-regular” polyhedra, whose faces are a combination of different regular polygons. The semi-regular polyhedra can also be inscribed in a sphere, and can have hexagonal, octagonal, or decagonal faces combined with triangular, square or pentagonal faces.

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The geometry behind the IQlight Lighting System™

Rhombic Polyhedra
There are numerous polyhedra comprised of non-regular polygonal faces. The two that are dealt with here are both comprised of rhombic faces:

The rhombic dodecahedron has 12 rhombic faces and is related to the cube, as seen when drawing the short diagonals of the rhombic faces. It is also related to the regular octahedron, as seen when drawing the long diagonals of the rhombic faces.

The rhombic triacontahedron has 30 rhombic faces.

All 5 Platonic solids can be inscribed in this interesting form, which leads us to the subject of this site: the IQlight Lighting System™