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Archimedean Polyhedra

all archimedean solids
Archimedes, a scientist from Ancient Greece, discovered thirteen types of polyhedra, now called Archimedean solids, which are also referred to 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 meet at each vertex. In the same order, each of these solids can be inscribed into a sphere.

All these thirteen 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.

What all Archimedean solids look like

 

1. Truncated tetrahedron

2. Truncated octahedron

3. Truncated hexahedron (or truncated cube)

4. Truncated dodecahedron

5. Truncated icosahedron

6. Cuboctahedron

7. Rhombicuboctahedron

8. Truncated cuboctahedron
 

9. Snub cube
 

10. Icosidodecahedron
 

11. Truncated icosidodecahedron (Small rhombicosidodecahedron)
 

12. Truncated rhombicosidodecahedron
 

13. Snub dodecahedron

How is the name of the polyhedron formed?

 

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.
Name

  Word-basis

 Truncated tetrahedron      
tetrahedron

Truncated octahedron     

Cuboctahedron     

Rhombicuboctahedron     

Truncated cuboctahedron     

octahedron

Truncated cube     

Snub cube     

cube

Truncated dodecahedron     

Icosidodecahedron     

Truncated icosidodecahedron     

Truncated rhombicosidodecahedron  

Snub dodecahedron  

dodecahedron

Truncated icosahedron

icosahedron
 

Which polyhedra forms the basis?

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

Octahedron
Truncated octahedron
Cuboctahedron
rhombicuboctahedron
Truncated cuboctahedron
Dodecahedron
Truncated dodecahedron
Icosidodecahedron
rhombicosidodecahedron
Truncated icosidodecahedron
Snub dodecahedron
 

 

What geometric figures can be used to construct Archemedean polyhedra?

All types of Archemedean polyhedra can consist of a certain set of regular.
 
1. Truncated tetrahedron : 4 triangles + 4 hexagons
Truncated tetrahedron
= 4 3
+ 4 6
 
2. Truncated octahedron: 6 squares + 8 hexagons
Truncated octahedron
= 6 4
+ 8 6
 
 
3. Truncated hexahedron (or truncated cube): 8 triangles + 6 octagons
Truncated cube
= 8 3
+ 6 8
 
4. Truncated dodecahedron: 20 triangles + 12 decagon
Truncated dodecahedron
= 203
+ 1210
 
5. Truncated icosahedron: 12 pentagons + 20 hexagons
Truncated  icosahedron
= 125
+ 206
 
6. Cuboctahedron: 8 triangles + 6 squares
Cuboctahedron
= 8 3
+ 6 4
 
7. Rhombicuboctahedron: 8 triangles + 18 squares
rhombicuboctahedron
= 8 3
+ 184
 
8. Truncated cuboctahedron: 12 squares + 8 hexagons + 6 octagons
Truncated cuboctahedron
= 124
+ 8 6
+ 6 8
 
9. Snub cube: 32 triangles + 6 squares
Snub cube
= 323
+ 6 4
 
10. Icosidodecahedron: 20 triangles + 12 pentagons
Icosidodecahedron
= 203
+ 125
 
11. Truncated icosidodecahedron: 20 triangles + 30 squares + 12 pentagons
rhombicosidodecahedron
= 203
+ 304
+ 125
 
12. Truncated rhombicosidodecahedron: 30 squares + 20 hexagons + 12 decagons
Truncated icosidodecahedron
= 304
+ 206
+ 1210
 
13. Snub dodecahedron: 80 triangles + 12 pentagons
Snub dodecahedron
= 803
+ 125
 
 

The size of a polyhedra

all arhimed tela3

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:

 

all arhimed tela4

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 what the collection of polyhedra with the same side length will look like:

all arhimedеan solids

Where to find the nets of Archimedean solids

Nets for all thirteen Archimedes polyhedra can be found in the kit "Magic Edges":

Magic Edges № 18
- Truncated tetrahedron;
- Truncated octahedron;
- Truncated cube;
- Cuboctahedron.
  
Magic Edges № 19
- Truncated icosahedron;
- Icosidodecahedron;
_
_
  
Magic Edges № 21
- Rhombicuboctahedron;
- Truncated cuboctahedron
 
Magic Edges № 27
- Truncated dodecahedron;
- Truncated icosidodecahedron
 
Preparing for release:

Magic Edges № 29 (Snub cube, Snub dodecahedron);

Magic Edges № 31 (Truncated rhombicosidodecahedron).

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