What is it?Each face of a Platonic solid is a regular polygon and all of the faces which are arranged to create the solid convex polyhedra are congruent. From the five Platonic solids Archimedes found that there are exactly thirteen semi-regular convex polyhedra. A solid is called semi-regular if its faces are all regular polygons and its corners are alike. These thirteen polyhedra are aptly called the Archimedean solids:
- Truncated Tetrahedron (3,6,6)
- Truncated Cube (3,8,8)
- Cuboctahedron (3,4,3,4)
- Truncated Octahedron (4,6,6)
- Rhombicuboctahedron (3,4,4,4)
- Truncated Cuboctahedron (4,6,8)
- Snub Cube (3,3,3,3,4)
- Icosidodecahedron (3,5,3,5)
- Truncated Dodecahedron (3,10,10)
- Truncated Icosahedron (5,6,6)
- Rhombicosidodecahedron (3,4,5,4)
- Truncated Icosidodecahedron (4,6,10)
- Snub Dodecahedron (3,3,3,3,5)
Archimedean solids are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.
Archimedean solids are convex figures that can be made up of two or more types of regular polygons. All edge lengths of the polygons must be equal, and all of the vertices must be identical, meaning the polygons that meet at each vertex do so in the same way. The same polygons appear in the same sequence, e.g., hexagon-hexagon-triangle in the truncated tetrahedron.
Here are the possibilities as to what can appear at a vertex. The notation (3, 4, 3, 4) means each vertex contains a triangle, a square, a triangle, and a square, in that cyclic order.
Truncation is the process of removing all the corners from a figure in a symmetrical way. So a 1/3 truncation involves removing 1/3 of the lengths of all the edges that meet at each vertex. The process of cutting off identical lengths along each edge emerging from a vertex adds a new face to the polyhedron, and replaces n-sided polygons with 2n-sided ones, e.g., octagons instead of squares.Truncation of a figure changes the faces of the original figure and adds n sided polygons at each vertex where n edges meet.
- Number of faces of truncated figure = number of faces of original figure + number of vertices of the original figure.
- Number of vertices of truncated figure = number of vertices of vertex figure = (number of edges at each vertex or valency of original figure) * (number of vertices of original figure)
Different measurements of truncation cause the original faces to change into different faces. Truncating a particular figure using different measurements separately results in different truncated figures. For example, 1/3 truncation of the cube changes the original square faces to eight sided irregular polygons.
1/2 truncation of the cube does not change the original square faces.
Both 1/3 and 1/2 truncations form equilateral triangles at the truncated corners.
Let A and B be the number of faces and vertices of the original figure, respectively. For 1/3 truncation:
Number of edges of truncated figure = [(number of edges of changed original faces * A) + (number of edges of vertex figure * B)]/2
= [(number of edges of each original face * 2 * A) + (valency of original figure * B)]/2
For 1/2 truncation, Number of edges of truncated figure = [(number of edges of changed original faces * A) + (number of edges of vertex figure * B)]/2
= [(number of edges of each original face * A) + (valency of original figure * B)]/2
|Platonic solid||1/3 truncation||1⁄2 truncation|
Archimedean solids obtained by truncating Platonic solids
- Truncated Cube
- Truncated Tetrahedron
- Truncated Octahedron
- Truncated Icosahedron
- Truncated Dodecahedron
- Cuboctahedron Icosidodecahedron
Archimedean solids obtained by truncating other Archimedean Solids
- Rhombitruncated Cuboctahedron
- Rhombitruncated Icosidodecahedron
Archimedean solids obtained by "snubbing" Platonic Solids
- Snub Cube
- Snub Dodecahedron
Truncated CubeThe truncated cube is created by taking a cube (which is a Platonic solid) and cutting off the corners to create eight equilateral triangles. The truncated cube has one triangle and two octagons around each vertex (3,8,8).This creates a form with eight faces of three sided figures (8 equilateral triangles), and six faces of eight sides (6 regular octagons). This has an interesting relationship with the original cube of 6 faces, 8 vertices, and 12 edges. The vertices and edges of the Truncated Cube are three times the number in the original cube while there are just 8 more faces (which was the number of cuts made).
- Open SketchUp
- Select the Rectangle Tool (R) and click and start to drag a square on the ground plane. Type 100,100 and press ENTER/RETURN
- Control click on the square and select Zoom Extents from the Context Menu
- Press the Space bar to select the Selection Tool. CTRL+click on the right line and select divide:
- Move the mouse in or out to make three segments
- Divide the top line into three segments
- Divide the bottom line into three
- Divide the last line, then connect the endpoints:
- Continue dividing and connecting:
- Delete the corner pyramids and color the faces:
Truncated TetrahedronAnother Archimedean solid created from a Platonic solid is the Truncated Tetrahedron. This solid is created by cutting the vertices off the tetrahedron. At each vertex of the truncated tetrahedron is an equilateral triangle and two regular hexagons. The symbol is then (3, 6, 6) or 3.6 2. The Tetrahedron has 4 faces, 4 vertices, and 6 edges, while the Truncated Tetrahedron has 4 equilateral triangular faces and 4 regular hexagonal faces which totals 8 faces, 12 vertices, and 18 edges. There are 4 more faces on the Truncated Tetrahedron due to the four cuts made. The number of vertices and edges are multiplied by three due to the addition of equilateral triangles (the same relationship that existed for the truncated cube).
- Open SketchUp
- Select the Polygon tool
And type the number 3. It will show up in the lower right hand corner. Press RETURN
- Change the view to top by selecting Camera>Standard Views>Top
- Draw a triangle so that one of the vertices falls along the green axis. Start at the origin and drag up along the green axis:
- Select the triangle, CTRL+click and choose explode Curve. This process separates the edges from the faces:
- Bisect the triangle
- Bisect the triangle again:
- Orbit the triangle, Select the segment connecting the last two points along the green axis. Choose the Rotate tool. Move the cursor until the protractor is red. Move the protractor to the top of the screen, when it is red, hold down SHIFT and move it to the endpoint. While still holding SHIFT rotate the protractor until the dotted blue line appears
- Press OPTION and rotate the protractor again to rotate the segment along the blue axis:
- Connect the vertical line to the intersecting point:
- Rotate the hypotenuse of this triangle up vertically. Use the protractor, if you need to adjust the angle type 45:
- Delete the hypotenuse and adjacent edge:
- Connect two corners to the top of the rotated line:
- Delete the rotated line and connect the last vertex to the top:
- By connecting the midpoints to the vertices, you end up with more tetrahedrons:
- CTRL+click on each edge and divide into 3:
- Connect the vertices
- Delete the pyramids