# What platonic solids have duality?

## What platonic solids have duality?

Interestingly enough, if you take a Platonic solid and connect the centers of adjacent faces with line segments, those segments will be the edges of another Platonic solid. This relationship is known as duality. It has many interesting properties. For instance: The dual of a tetrahedron is another tetrahedron.

## What are the 5 types of platonic solids?

Also known as the five regular polyhedra, they consist of the tetrahedron (or pyramid), cube, octahedron, dodecahedron, and icosahedron.

What are the duals of the different regular polyhedra?

Thus the five regular polyhedra fall into three groups: two dual pairs and one polyhedron that is dual to itself. The self-dual tetrahedron (left); the dodecahedron is dual to the icosahedron (center); the icosahedron is dual to the dodecahedron (right).

Are there only 5 Platonic solids?

There are only five! The Greeks recognized that there are only five platonic solids. But why is this so? The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees.

### What is the dual of a dodecahedron?

regular icosahedron
The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex.

### How many Dodecahedrons have been found?

In 1739, a strange, twelve-sided hollow object from Roman times was discovered in England. Since then, more than a hundred dodecahedrons have been unearthed, but their purpose remains unknown.

Why are there only 5 types of Platonic solids?

The tetrahedron,with 4 triangular faces:

• The cube,with 6 square faces:
• The octahedron,with 8 triangular faces:
• The dodecahedron,with 12 pentagonal faces.
• The icosahedron,with 20 triangular faces:
• What are the names of the five Platonic solids?

our understanding of the universe. There are only five solids that can be called platonic solids – the tetrahedron, the hexahedron or cube, the octahedron, the dodecahedron and the icosahedron. They are also called regular geometric solids or polyhedra and are 3D in shape.

#### What are real life uses of Platonic solids?

Platonic solids are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as d n where n is the number of faces (d8, d20, etc.); see dice notation for more details.

#### Who invented the five Platonic solids?

the tetrahedral group T,

• the octahedral group O (which is also the symmetry group of the cube),and
• the icosahedral group I (which is also the symmetry group of the dodecahedron).