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Archimedean polyhedra

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Archimedes, a scientist from the Ancient Greece, discovered 13 types of polyhedra, called now as Archimedean solids, which are also referred as semi-regular polyhedra.

Each of them is limited by different types of regular polygons where the polyhedral angles and identical polygons are equal. Furthermore, the same number of equal faces meets at each vertex. In the same order each of these solids can be inscribed into a sphere.

Thirteen types of polyhedra.

All these 13 names make your head spin when you see them for the first time. However, it is possible to sort things out, understand and remember them.

First of all, this is how all Archimedean solids look like:

   1. Truncated tetrahedron Truncated tetrahedron Truncated tetrahedron
   2. Truncated octahedron Truncated octahedron Truncated octahedron
   3. Truncated hexahedron (or truncated cube)
truncated cube truncated cube
   4. Truncated dodecahedron Truncated dodecahedron Truncated dodecahedron
   5. Truncated icosahedron Truncated icosahedron Truncated icosahedron
   6. Cuboctahedron Cuboctahedron Cuboctahedron
   7. Rhombicuboctahedron Rhombicuboctahedron Rhombicuboctahedron
   8. Truncated cuboctahedron Truncated cuboctahedron Truncated cuboctahedron
   9. Snub cube Snub cube Snub cube
   10. Icosidodecahedron Icosidodecahedron Icosidodecahedron
   11. Truncated icosidodecahedron Truncated icosidodecahedron Truncated icosidodecahedron
   12. Truncated rhombicosidodecahedron Truncated rhombicosidodecahedron Truncated rhombicosidodecahedron
   13. Snub dodecahedron  Snub dodecahedron Snub dodecahedron

What forms the name of the polyhedron?

Pay attention to the fact that there is a word-basis for every type of polyhedron which forms their names. It is this word-basis that helps to classify the current polyhedra to one of the five regular polyhedra.



Truncated tetrahedron     


Truncated octahedron     



Truncated cuboctahedron     


Truncated cube     

Snub cube     


Truncated dodecahedron     


Truncated icosidodecahedron     

Truncated rhombicosidodecahedron     

Snub dodecahedron     


Truncated icosahedron


Which polyhedra forms the basis?

One of the five Platonic polyhedra forms the basis for these 13 types of polyhedra.

tetrahedron => Truncated tetrahedron        
octahedron => Truncated octahedron Cuboctahedron Rhombicuboctahedron Truncated cuboctahedron  
cube => truncated cube Snub cube      
Dodecahedron => Truncated dodecahedron Icosidodecahedron  Truncated icosidodecahedron Truncated rhombicosidodecahedron Snub dodecahedron
icosahedron => Truncated icosahedron        

What geometric figures can be used to construct Archemedean polyhedra?

All types of Archemedean polyhedra can be consisted of a certain set of regular
1. Truncated tetrahedron Truncated tetrahedron    =  4 3 +  4 6    
 2. Truncated octahedron  Truncated octahedron    =  6 4  +  8  6    
 3. Truncated hexahedron
(or truncated cube)
 truncated cube     =  8  3  +  6  8    
 4. Truncated dodecahedron  Truncated dodecahedron     =  20  3  +  12  10    
 5. Truncated icosohedron  Truncated icosohedron     =  12  5  +  20  6    
 6. Cuboctahedron  Cuboctahedron     =  8  3  +  6  4    
 7. Rhombicuboctahedron  Rhombicuboctahedron     =  8  3  +  18  4    
 8. Truncated cuboctahedron  Truncated cuboctahedron     =  12  4  +  8  6  +  6 8 
 9. Snub cube
 Snub cube     =  32  3  +  6  4    
 10. Icosidodecahedron  Icosidodecahedron     =  20  3  +  12  5    
 11. Truncated icosidodecahedron  Truncated icosidodecahedron     =  20  3  +  30  4  +  12  5
 12. Truncated rhombicosidodecahedron  Truncated rhombicosidodecahedron     =  30  4  +  20  6 +  12  10
 13. Snub dodecahedron 
 Snub dodecahedron     =  80  3  +  12  5    

The size of polyhedra

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To create a collection of polyhedra, we need to adhere to certain conditions, so that sizes will be comparable and models can easily be compared with each other.

One of the options is to create a model which fits into the sphere of the standard sizes. Then all 13 polyhedra will look like this:

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Another option is to set up the same length for the side of all the polygons which will be used in the model. The proportion of polygons with the same side length is:

- Triangle;

- Square;

- Pentagon;

- Hexagon;

- Octagon;

- Decagon.



This is how the collection of polyhedra with the same side length will look like:

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