Made with Virtual flower
What types of shape we can make with perfectly symmetrical triangles, squares, pentagons, and such like faces?
Let's start by using just one type of face for each one, and lets have the same number meeting at any vertex.
With triangles, you can make the
Three triangles meet at every vertex
Four at every vertex
five triangles at every vertex.
If we try six triangles at a vertex we find they lie flat, and any more than that won't fit together at all. So that's the complete list for triangles.
One might also want to check if these shapes actually work perfectly - can the triangles meet together in this way, or might there be tiny gaps here and there too small to see in the models?
It's clear that the tetrahedron and octahedron work (as two square based pyramids one on top of the other). For the icosahedron, you need to find a way to construct it, then prove that the edges are all the same length in that construction. One way is to divide the edges of an octahedron in the ratio of the golden section and join those together - see the end of my volumes page .
Squares next - these make a cube of course, and if you try to fit four at each vertex they will lie flat, so that's that
Next, try pentagons, with three meeting at each vertex to make a dodecahedron
The dodecahedron has three faces meeting at each vertex, and each face has five sides - while the icosahedron has five faces meeting at each vertex, each with three sides. The reason is that if you join together the faces of a dodecahedron, you will construct an icosahedron, and vice versa. They are said to be duals of each other. Similarly, the cube and octahedron are duals. The tetrahedron is self dual - when you join together the centres of the faces of a tetrahedron, the result is another tetrahedron.
Now if you try hexagons, they will lie flat. If you try more than six sides you can't fit more than two at each vertex, so that's it and we are finished.
Now lets try truncating them - this means, to cut off the corners, to make a new face at the vertices.
You can either truncate it a little way in to make a new shape with more sides, or truncate it right in at the half way point - each method makes a new shape with regular faces.
If you truncate the tetrahedron at the mid points you get an octahedron (to put that another way, if you add tetrahedra to alternate faces of an octahedron, you make a larger tetrahedron), so we can skip that one.
If you truncate the octahedron at the mid-points you get the
- one of the figures with equatorial polygons.
This next one
may look more familiar in two colours:
It's a soccer ball shape
If you truncate the icosahedron at the edge mid-points you get
This is another one with equatorial polygons ("equators").
I wonder if you've noticed - each of these shapes has exactly the same pattern of faces at every vertex.
Here, each vertex has two triangles and two pentagons, alternating.
Finally the cube gives the cuboctahedron again, which we may as well show again in colour too
(give you a rest from those anaglyph specs :-))
There are many more shapes you can make with regular shaped tiles.
See George' Hart's Virtual Polyhedra encyclopedia to find out more.
To view and turn the anaglyphs in 3D you need to install the Cortona VRML client or another VRML plug in, and then click on the pciture.
The anaglyphs are optimised for a distance to the screen of twenty times the eye separation. As you look at them from closer or further away, the figure will flatten or stretch.
Windows users can make anaglyphs like these in my Virtual flower program.
The Five Platonic Solids and the Archimedian Solids in George Hart's Virtual Polyhedra on-line encyclopedia
I have a page about how to find the volumes of the platonic solids