Exploring 3D space with a computer – Part 3: building models from plans
Adrian Oldknow
Icosahedron Summary
So we can summarise some of the results we have found about the icosahedron:
It has:
- 20 equilateral triangles as faces;
- 12 vertices around each of which there are 5 faces;
- 30 edges each of which separates two faces ;
- a circumsphere whose radius is less than an edgelength (by a factor of 0.951);
- an insphere whose radius is less than an edgelength (by a factor of 0.756);
- a dual solid which has 20 vertices and 12 pentagonal faces.
You can find out more at: http://mathworld.wolfram.com/Icosahedron.html
The answers to the radii are at: http://whistleralley.com/polyhedra/icosahedron.htm
Can you follow the ideas in the section to show that there are also solids made up from:
- four equilateral triangle faces at each of eight vertices (octahedron);
- three equilateral triangles faces at each of four vertices (tetrahedron);
- three square faces at each of eight vertices (cube);
- three regular pentagon faces at each of twenty vertices (dodecagon)?
Project - find out as much as you can about each of the five Platonic solids. Which are duals of which? How are their inradii, circumradii and edge lengths related? How many planes of symmetry does each have? Which can be stellated, and how? What shapes can you make by truncating them? Which shapes will fill space on their own, and which sets of shapes together can be used to fill space? Are any of these shapes naturally occurring as crystal structures? Is Cabri 3D a useful tool to help you? Find other sources of interesting resource materials on polyhedra. For examples try the following links:
Solids: http://lgfl.skoool.co.uk/viewdetails_ks3.aspx?id=551
Planes of symmetry: http://lgfl.skoool.co.uk/viewdetails_ks3.aspx?id=564
Thought experiment 1
Place a cube on a table (in your mind's eye).
Colour one of the vertices above the table with a red dot.
Surround it with three more cubes on the table.
Place another layer of four cubes on top so that your red dot is right in the middle.
What shape do the eight cubes fill up?
Could you fill the room with cubes in this way without leaving any gaps?
Thought experiment 2
Place a tetrahedron on a table (in your mind's eye).
Colour the vertex above the table with a red dot.
Can you surround the red dot with tetrahedra to make a bigger solid with no gaps?
If so, how many tetrahedra will you need, and what shape will they fit into?
(Hint - how many different platonic solids have equilateral triangles as faces?
Do any of these have the same circumradius as edge length?)
And now, the culmination of any traditional piece of geometry - a theorem !
Theorem : regular tetrahedra do not fill space
Proof : Consider any point P in space and construct a regular tetrahedron having P as a vertex. Start to surround P by adding similar regular tetrahedra having P as one vertex, and sharing a common face with one of the existing tetrahedra. If P was completely surrounded by an integer number n of such tetrahedra then the resulting object would be a regular solid consisting of n equilateral triangles with P as centre. But there are only three such possible shapes: icosahedron ( n =20), octahedron ( n =8) or tetrahedron ( n =4). We have already shown above that the circumradius of the icosahedron is (slightly) less than its edge length - so n is clearly not 20.
Clearly neither 4, nor 8, regular tetrahedra come anywhere near completely surrounding P , so we have exhausted all the possibilities and the theorem is proved.
Finally here is a landscape of Platonic solids for you to play with. You can manipulate them here, or download the file 
Platonics.cg3.
