Platonic and Archimedean solids
Models of every Platonic and Archimedean solid can be built with Geomag. If you want to refresh your memory, MathWorld pages Platonic Solid and Archimedean Solid have lots of information, including three-dimensional models, plane nets, formulae, etc. This page (in Spanish) is also a handy reference.
There are five Platonic solids and thirteen Archimedean ones. Adding the pentagonal prism and antiprism gives twenty convex uniform polyhedra. There are at least 60 non-convex uniform polyhedra which, as far as I know, cannot be built with Geomag (see the MathWorld article Uniform Polyhedron).
In the following picture all eighteen solids appear, ordered by size. Building some of them is very quick and straightforward, while others are more involved and require auxiliary structures.
Body edges are made of blue rods, and auxiliary structures, required for building hexagonal, octagonal, and decagonal faces, are made of silver rods. One can tell polygons which are true faces of the bodies from auxiliary ones in this manner:
- Pentagons and squares: we use red panels for true faces, yellow panels for auxiliary ones.
- Triangles: we use green panels for true faces, no panels for auxiliary ones.
- Rhombuses: all of them are yellow and part of hexagonal true faces.
Platonic and Archimedean solids
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The solids are named as follows, front row to rear row, left to right (links go to individual pictures, below in this page):
The following four pictures show more detail. The provided higher resolution versions of the last three pictures use about the same scale as the first one:
Tetrahedron and truncated tetrahedron
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Snub cuboctahedron and icosidodecahedron
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Cube and octahedron subfamily
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Dodecahedron and icosahedron subfamily
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Finally, individual pictures of each body. When no auxiliary structures are required, ball, rod, and panel quantities match those of vertices, edges, and faces. When there are auxiliary structures, quantities don't match. In the last cases, we specify in brackets the quantities of vertices, edges, and faces.
For instance, (12v, 18e, 4f6, 4f3) means: 12 vertices, 18 edges, 4 hexagonal faces, and 4 triangular faces.
Platonic solids:
Regular tetrahedron
14 pieces: 4 balls, 6 rods, 4 triangles (65.08 g)
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Cube or regular hexahedron
26 pieces: 8 balls, 12 rods, 6 squares (136.12 g)
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Regular octahedron
26 pieces: 6 balls, 12 rods, 8 triangles (112.96 g)
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Regular dodecahedron
62 pieces: 20 balls, 30 rods, 12 pentagons (349.00 g)
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Regular icosahedron
62 pieces: 12 balls, 30 rods, 20 triangles (256.60 g)
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Archimedean solids:
Truncated regular tetrahedron
68 pieces: 16 balls, 36 rods, 12 rhombuses, 4 triangles (326.56 g)
(12v, 18e, 4f6, 4f3)
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Truncated cube
130 pieces: 32 balls, 72 rods, 18 squares, 8 triangles (651.32 g)
(24v, 36e, 6f8, 8f3)
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Truncated regular octahedron
122 pieces: 32 balls, 60 rods, 6 squares, 24 rhombuses (602.68 g)
(24v, 36e, 8f6, 6f4)
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Truncated regular dodecahedron
482 pieces: 120 balls, 270 rods, 12 pentagons, 60 squares, 20 triangles (2.46 kg)
(60v, 92e, 12f10, 20f3)
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Truncated regular icosahedron
302 pieces: 80 balls, 150 rods, 12 pentagons, 60 rhombuses (1.52 kg)
(60v, 92e, 20f6, 12f5)
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Cuboctahedron
50 pieces: 12 balls, 24 rods, 6 squares, 8 triangles (231.88 g)
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Icosidodecahedron
122 pieces: 30 balls, 60 rods, 12 pentagons, 20 triangles (588.40 g)
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Rhombicuboctahedron
98 pieces: 24 balls, 48 rods, 18 squares, 8 triangles (469.72 g)
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Rhombicosidodecahedron
242 pieces: 60 balls, 120 rods, 12 pentagons, 30 squares, 20 triangles (1.18 kg)
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Rhombitruncated cuboctahedron
314 pieces: 80 balls, 168 rods, 42 squares, 24 rhombuses (1.59 kg)
(48v, 72e, 6f8, 8f6, 12f4)
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Rhombitruncated icosidodecahedron
782 pieces: 200 balls, 420 rods, 12 pentagons, 90 squares, 60 rhombuses (3.98 kg)
(120v, 180e, 12f10, 20f6, 30f4)
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Snub cuboctahedron
122 pieces: 24 balls, 60 rods, 6 squares, 32 triangles (519.16 g)
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Snub icosidodecahedron
302 pieces: 60 balls, 150 rods, 12 pentagons, 80 triangles (1.31 kg)
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- A smart way to build the truncated regular icosahedron without panels.
- A rhombicosidodecahedron can be internally reinforced in a very smart way (see Construction of the reinforced rhombicosidodecahedron, method 1 and Construction of the reinforced rhombicosidodecahedron, method 2).
- There is another way of building the auxiliary structures in a rhombitruncated icosidodecahedron which results, as if by magic, in a concentric rhombicosidodecahedron (see Concentric icosidodecahedra). In a similar way, the rhombitruncated cuboctahedron hides an interior rhombicuboctahedron (see Concentric cuboctahedra).
