Snub dodecahedron

In geometry, the snub dodecahedron, also called the snub icosidodecahedron, is a special shape known as an Archimedean solid. Archimedean solids are special three-dimensional shapes that are not made of just one kind of face, but they are very symmetrical and evenly shaped.

The snub dodecahedron has 92 faces in total, which is more than any other Archimedean solid. Twelve of its faces are pentagons, and the other eighty are equilateral triangles. It also has 150 edges and 60 points where the edges meet, called vertices.

This shape has two different versions that are mirror images of each other, called "enantiomorphs". When you put both of these mirror-image shapes together, they form a compound of two snub dodecahedra. If you take the outer shape that covers both of these mirror images, you get a truncated icosidodecahedron.

Construction

The snub dodecahedron can be made from a regular dodecahedron by moving its pentagonal faces apart and filling the spaces with equilateral triangles. This idea comes from Kepler in 1619. H. S. M. Coxeter also showed it could come from a regular octahedron or icosahedron.

Another way to create the snub dodecahedron is by starting with a truncated icosidodecahedron and using a process called alternation. This creates two shapes that are mirror images of each other.

Cartesian coordinates

To find the exact positions of the points (vertices) of the snub dodecahedron, we use special numbers and rotations. These positions are found by repeatedly applying two rotation rules to a starting point. The rules come from the symmetries of a regular icosahedron. The distance between connected points (edge length) is about 0.45 times a certain value.

Properties

The snub dodecahedron is a special 3D shape with 92 faces: 12 pentagons and 80 equilateral triangles. It also has 150 edges and 60 points where the edges meet.

One interesting fact is that the snub dodecahedron has the highest sphericity among all Archimedean solids. Sphericity measures how close a shape is to a perfect sphere. For the snub dodecahedron, this value is about 0.947, meaning it is almost as round as a sphere.

Geometric relations

The snub dodecahedron can be paired with its mirror image to form another shape called a semiregular truncated icosidodecahedron. This shows how these special shapes relate to each other.

The positions of the points, or vertices, of this shape can be described using special numbers, including the golden ratio. These positions are chosen from sets of numbers that follow certain rules about plus and minus signs.

Related polyhedra and tilings

This special shape is part of a group of special shapes called "snubbed" polyhedra and tilings. They all share a certain pattern at their corners and have a special diagram that shows their structure. These shapes and their matching shapes have a certain kind of symmetry. In flat space, this happens when a number called "n" equals 6, and in curved space, it happens when "n" is any number larger than 6. The group starts with "n" equal to 2, where some faces become very thin two-sided shapes called digons.

Snub dodecahedral graph

In the area of math called graph theory, a snub dodecahedral graph shows the points and lines of the snub dodecahedron. The snub dodecahedron is one of the Archimedean solids. This graph has 60 points and 150 lines connecting them, and it is known as an Archimedean graph.