The Excavated Truncated Icosidodecahedron
Some Geometry of the Truncated Octahedron or Mecon
based on a talk first given at the BSRLM’s New Researcher’s Day at Loughborough University on 9th August 2016
Consider an Octahedron with its surfaces removed and each edge connected to the centre:
What is a truncation? A truncation is a slice at right angle to an axis of rotation through a vertex, such that the resulting faces are regular.
What happens when we truncate the octahedron? We will obtain a square face and a hexagonal face.
What is the angle at the centre subtended by the edge? To which well-known triangle does that angle belong?
Let the Octahedron have an edge length of 3√10
Consider the face of the Octahedron. From the simple geometry of a hexagon inscribed in an equilateral triangle, we can see that the edge will be divided in three by the truncation.
Consider the ‘petal’ of the Octahedron. It is an isosceles right angle triangle. AC = CB ,Angle ACB = 90 degrees, AB = 3√10.
By Pythagoras, the lengths of the sides enclosing the right angle will be 3√5. AB = AC = 3√5
Then we can find the length of the perpendicular from the centre to the edge CF = (3/2)√10
Now we can find the radius of the truncated octahedron by Pythagoras’s theorem CD = CE = 5.
Consider the 5:5: √10 triangle. Drop a perpendicular DG from a base vertex to the opposite side.
Use similar triangles to find the length of the perpendicular!
It should now be clear that the central angle of the truncated octahedron is the most acute angle of a very old friend, the 3:4:5 triangle.
This fact can be used to construct the nets using compasses and straight-edge.
This solid, the Mecon, is one of the two Platonic and Archimedean solids which tesselate 3D space, the other being the Cube. It is a question whether the presence of the 3:4:5 traingle has anything to do with this property.
3D space may also be tesslated by a combination of octahedra and tetrahera.
Developments
One can begin from first principles and construct everything with ruler and compass. This excellent for the simple cases, but as the snub dodecahedron has 92 faces, it may be impractical for anyone but an extreme-geometer to use straightedge and compasses to construct all 92 nets for the pyramids.
So the nets have been printed out on A4 coloured card. Copying the nets files from Geometer’s Sketchpad to Word gives sufficient accuracy for good models to be made.
So far, using Geometer’s Sketchpad, the 5 Platonic Solids and the 15 Archimedean Solids (if you count the laeval and dextral forms of the two snubs as distinct) have been made. As has Kepler’s Rhombic Dodecahedron which also tesslates 3D space.
Bibliography
Sinclair N., Mathematics and Beauty which is a study of the part which aesthetic appreciation plays in the learning of Mathematics. Sinclair believes that aesthetic appreciation is neglected as a factor in motivating mathematical learning. Interestingly, all the research she undertakes concerns students working with computer programs designed to support the teaching and learning of Mathematics.
Skemp R., The Psychology of Learning Mathematics which is a very readable classic. Skemp is interested in how schemas–organised collections of mathematical ideas–are built up in the mind, and in human beings’ intrinsic motivation to study mathematics.
Watson A., Jones K. and Pratt D., Key Ideas in Teaching Mathematics: Research-Based Guidance For Ages 9-19. This is a compendium of resources for teaching all branches of Secondary Mathematics with a thought provoking section on Geometry. Geometry is the “branch of Mathematics which studies visual phenomena” in all their glory and richness. (p. 114)
