Rhombicosidodecahedron

In geometry, the rhombicosidodecahedron is an Archimedean solid. It is one of thirteen special convex shapes that are perfectly symmetrical.

These solids are made by joining different types of regular polygon faces together. They look the same from every angle. This property is called isogonal.

The rhombicosidodecahedron has 62 flat surfaces. It has 20 triangular faces, 30 square faces, and 12 pentagonal faces. These shapes fit together perfectly at points called vertices. Lines called edges connect them. There are 60 vertices and 120 edges.

This solid is important in science and art because of its symmetry and balance. It shows how different regular shapes can fit together to make a beautiful, uniform structure. Studying shapes like the rhombicosidodecahedron helps mathematicians and scientists learn more about space and symmetry.

Names

Johannes Kepler named this shape the rhombicosidodecahedron in his book in 1618. He called it a shorter way to say "truncated icosidodecahedral rhombus." There are a few ways to change a rhombic triacontahedron to make a rhombicosidodecahedron.

Dimensions

For a rhombicosidodecahedron with edge length a, we can find its surface area and volume. The surface area is about 59.31 times a squared. The volume is about 41.62 times a cubed. These numbers help us know how big the shape is and how much space it fills.

Geometric relations

Expanding an icosidodecahedron by moving its faces away from the origin and rotating them makes a rhombicosidodecahedron. This new shape has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with squares in between.

Another way to picture it is by expanding five cubes and placing them around a center point. This also forms a rhombicosidodecahedron, keeping the same number of squares as the five cubes. Some building kits use this shape to build fun structures.

Cartesian coordinates

The positions of the points, or vertices, of a rhombicosidodecahedron can be described using special numbers called Cartesian coordinates. For a rhombicosidodecahedron with an edge length of 2 and centered at the origin, these coordinates are all the even permutations of:

(±1, ±1, ±φ³),

(±φ², ±φ, ±2_φ_),

(±(2+φ), 0, ±φ²),

where φ = ⁠1 + √5/2⁠ is the golden ratio. The distance from the center to any vertex, called the circumradius, is √φ⁶+2 = √8φ+7 for edge length 2. If the edge length is 1, this distance is about 2.233.

Orthogonal projections

The rhombicosidodecahedron has six special views when looked at straight on. These views focus on a corner, on two kinds of edges, and on three kinds of faces: triangles, squares, and pentagons. The last two views match the A₂ and H₂ Coxeter planes.

Spherical tiling

The rhombicosidodecahedron can be shown as a pattern on a sphere, called a spherical tiling. When we copy this pattern to flat paper using a special method called stereographic projection, the straight lines become curved arcs. This method keeps the angles the same but changes the sizes and lengths of the shapes.

Related polyhedra

The rhombicosidodecahedron is connected to several other interesting shapes in geometry. It is part of a group of shapes called cantellated polyhedra. These shapes have a special pattern where their edges meet. They can also appear as patterns on curved surfaces.

There are 12 related shapes called Johnson solids. These are made by changing the rhombicosidodecahedron in different ways. The rhombicosidodecahedron also shares the same points with three other complex shapes and with special prisms.

Rhombicosidodecahedral graph

In graph theory, a rhombicosidodecahedral graph shows the points and lines of the rhombicosidodecahedron. The rhombicosidodecahedron is one of the Archimedean solids. This graph has 60 points and 120 lines connecting them. It is also called a quartic graph and an Archimedean graph.