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.

As an Archimedean solid, the rhombicuboctahedron is one of the beautiful, symmetrical shapes that have been studied for a long time. 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.

Construction

The rhombicuboctahedron can be made from a cube. You start by drawing a smaller cube in the middle of each face. Then you 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 made by moving the faces of a regular octahedron away from the center. You then fill the gaps with squares and triangles. This process is called expansion. Using these methods, the rhombicuboctahedron has 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 is one of the Archimedean solids. These solids are known for their symmetry and the different regular polygons that meet at each vertex. In the rhombicuboctahedron, 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. It also has a special property where another shape can pass through its center, known as the Rupert property. The opposite shape of the rhombicuboctahedron is called its dual, and it is a Catalan solid named the deltoidal icositetrahedron.

Graph

The skeleton of a rhombicuboctahedron can be described as a polyhedral graph. This means it is a graph that can be drawn without any lines crossing.

The rhombicuboctahedral graph has 24 vertices and 48 edges. Each point connects to four others. This graph is called an Archimedean graph because it is related to the shape of an Archimedean solid.

Appearances

The rhombicuboctahedron shows up in many cool places. You can spot shapes like it on buildings, such as the National Library in Minsk. It’s also used in fun toys, like some types 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 tied to special symbols, like the Moravian star, which stands for the Star of Bethlehem in Christianity.