Rhombicuboctahedron

The rhombicuboctahedron or small rhombicuboctahedron is a special kind of shape called a polyhedron. It has 26 faces in total, made up of 8 equilateral triangles and 18 squares. This interesting shape was named by the famous astronomer and mathematician Johannes Kepler in his book Harmonices Mundi from 1618. He called it the "truncated cuboctahedral rhombus," which is a long way of saying it looks like a mix between a cube and an octahedron.

As an Archimedean solid, the rhombicuboctahedron is one of the many beautiful, symmetrical shapes that have been studied for centuries. Each of its vertices, where the edges meet, is identical, making it very special in geometry. The shape that is formed when you flip the rhombicuboctahedron inside out is called its dual, and it is a Catalan solid known as the deltoidal icositetrahedron.

You can find the rhombicuboctahedron in many places, from old buildings and modern art to toys and designs. Its balanced and symmetrical look makes it a favorite in many cultures and fields. Another shape that looks a bit like the rhombicuboctahedron is the elongated square gyrobicupola, which is the 37th Johnson solid, but it is not an Archimedean solid because its vertices are not all the same.

Construction

The rhombicuboctahedron can be made from a cube by drawing a smaller cube in the middle of each face. We can then add squares next to the original ones and fill the corners with equilateral triangles. Another way is to join two square cupolas to the sides of an octagonal prism.

It can also be formed by moving the faces of a regular octahedron away from the center and filling the gaps with squares and triangles. This process is called expansion. Using these methods, the rhombicuboctahedron ends up with 8 triangles and 18 squares as its faces.

Properties

The rhombicuboctahedron is a special shape with 26 faces: 8 are equilateral triangles, and 18 are squares. It belongs to a group of shapes called Archimedean solids, which are known for their symmetry and the different regular polygons that meet at each vertex. In this case, each vertex has one triangle and three squares meeting together.

This shape has the same symmetry as a cube and an octahedron, making it very balanced and symmetrical. It also has a special property where another shape can pass through its center, known as the Rupert property. The rhombicuboctahedron’s opposite shape, called its dual, is a Catalan solid named the deltoidal icositetrahedron.

Graph

The skeleton of a rhombicuboctahedron can be described as a polyhedral graph, which means it is a graph that is planar and 3-vertex-connected. This means the edges of the graph do not cross when drawn, and removing any two of its vertices still leaves a connected subgraph.

The rhombicuboctahedral graph has 24 vertices and 48 edges. Each vertex is connected to four others, making it a quartic graph. This graph is known as an Archimedean graph because it is related to the graph of an Archimedean solid.

Appearances

The rhombicuboctahedron appears in many interesting places. You can see shapes like it in buildings, such as the National Library in Minsk. It is also used in toys, like some versions of the Rubik's Cube.

Artists have used this shape too. For example, a famous painting from 1495, called Portrait of Luca Pacioli, shows a glass rhombicuboctahedron. The shape is also linked to religious symbols, like the Moravian star, which represents the Star of Bethlehem in Christianity.