Intrigued by regular polyhedra for years

In summary, a regular polyhedron is a polyhedron with all regular faces and congruent polyhedral angles. If it is also convex, it falls into the category of the 5 Platonic solids. However, if concave polyhedra are allowed, there are 4 additional regular polyhedra. The polyhedral angles refer to the angles between two adjacent polygons.
  • #1
DaveC426913
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I've been intrigued by regular polyhedra for years, and have oft tinkered with them.

Can someone provide the "textbook definition" of a regular polyhedron?
 
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  • #2
A "regular polyhedron" is a polyhedron, all of whose faces are regular polygons AND all of whose polyhedral angles are congruent.

If you add the requirement that they be convex (and was classically the case) then only the 5 "Platonic solids" are regular polyhedra. If you allow concave polyhedra, then there are 4 additional regular polyhedra.
 
  • #3
Fantastic. I was specifically looking to find out whether convexity was a requirement.


Can you point me at some links that explore the additional 4 polyhedra?

Also, the "polyhedral angles" - are they merely the angle joining two adjacent polygons?
 

1. What are regular polyhedra?

Regular polyhedra are three-dimensional shapes that have identical regular polygons as their faces. They are also known as Platonic solids and include shapes like the cube, tetrahedron, and dodecahedron.

2. What makes regular polyhedra intriguing?

Regular polyhedra have been studied for centuries and continue to fascinate scientists and mathematicians due to their unique geometric properties and symmetry. They also have practical applications in fields like architecture and chemistry.

3. How many regular polyhedra are there?

There are only five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. These are the only shapes that can have identical regular polygons as their faces.

4. What is the significance of regular polyhedra in mathematics?

Regular polyhedra have played a significant role in the development of mathematics, particularly in the study of geometry and symmetry. They have also been used as a basis for understanding more complex shapes and structures.

5. How are regular polyhedra relevant in modern science?

Regular polyhedra are relevant in many fields of modern science, including crystallography, geometry, and computer graphics. They also have practical applications in chemistry, physics, and engineering.

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