Rhombicosidodecahedron

In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen special convex shapes that are perfectly symmetrical. These solids are made by joining different types of regular polygon faces together in a way that looks the same from every angle, a property called isogonal.

The rhombicosidodecahedron is made up of 62 flat surfaces. It has 20 triangular faces, 30 square faces, and 12 pentagonal faces. All these shapes fit together perfectly at points called vertices, and lines called edges connect them. There are 60 vertices and 120 edges in this amazing shape.

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

Names

Johannes Kepler named this shape the rhombicosidodecahedron in his book Harmonices Mundi in 1618. He called it a shorter version of "truncated icosidodecahedral rhombus." There are a few ways to change a rhombic triacontahedron to make a rhombicosidodecahedron, including a process called rectification.

Dimensions

For a rhombicosidodecahedron with edge length a, we can calculate its surface area and volume. The surface area is given by a special formula that uses square roots, and it is about 59.31 times a squared. The volume is also found using another formula with square roots, and it is about 41.62 times a cubed. These formulas help us understand the size and space the shape takes up.

Geometric relations

Expanding an icosidodecahedron by moving its faces away from the origin and rotating them creates 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 filling in the gaps.

Another way to imagine it is by expanding five cubes and arranging them around a center point. This also results in a rhombicosidodecahedron, keeping the same number of squares as the five cubes. Some building kits use a version of this shape to create interesting 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 orthogonal projections. These projections are centered on a vertex, on two types of edges, and on three types of faces: triangles, squares, and pentagons. The last two projections correspond to 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 flatten this pattern onto a flat piece of paper using a special method called stereographic projection, the straight lines on the sphere turn into 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, which have a specific pattern around each point where their edges meet. These shapes also appear as patterns that can be drawn on curved surfaces.

There are also 12 related shapes called Johnson solids, which are made by changing the rhombicosidodecahedron in different ways. Additionally, the rhombicosidodecahedron shares the same arrangement of points with three other complex shapes and with groups of special prisms.

Rhombicosidodecahedral graph

In the field of graph theory, a rhombicosidodecahedral graph shows the graph of vertices and edges of the rhombicosidodecahedron, which is one of the Archimedean solids. This graph has 60 points where lines meet and 120 lines connecting these points. It is also called a quartic graph and an Archimedean graph.