Puzzles
21 December
To get today's number, multiply the number of modules needed to make a dodecahedron by half the number of tube maps used to make a stellated rhombicuboctahedron.
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One module is needed for each edge of the dodecahedron. Therefore 30 modules are needed.
48 tube maps are needed for the stellated rhombicuboctahedron.
Therefore today's number is 720.
Folding tube maps
Back in 2012, I posted instructions for
folding a tetrahedron from tube maps. When tube maps are used, the sides of the tetrahedron are not quite equal. What ratio would the rectangular maps need to be in to give a regular tetrahedron?
Show answer & extension
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Once the map is folded, it will look like this:
For the final tetrahedron to be regular, the red lengths must be equal. Let each red length be 2 (this will get rid of halves in the upcoming calculations). By drawing a vertical line in we can work out the width and height of the rectangle:
The width of the rectangle is 3 (one and a half red lengths). Using Pythagoras' Theorem in the blue triangle, we find that the height of the rectangle is \(\sqrt{3}\). Therefore, the ratio of the rectangle is \(\sqrt{3}:3\) or \(1:\sqrt{3}\).
Extension
If the ratio of the rectangle is \(1:a\), what is the ratio of the lengths of the sides of the tetrahedron?
![](javascript:showlimage("tetra1.png"))
![](javascript:showlimage("tetra2.png"))