To the right is shown a dodecahedron - the classic polyhedra with 12 equal sides. If we were to take 4 equidistant corners of the dodecahedron and connect them with lines, the result would be a pyramid (a tetrahedron) inscribed in the dodecahedron. This is illustrated below.

This tetrahedron has 4 corners, and the dodecahedron has 20 corners
total. Thus we could inscribe 5 distinct tetrahedra inside a dodecahedron! The
result of doing this is shown below.

So here is the task: Given this wildly complex-looking structure, how do we make it out of modular origami units? Well, if we could make a modular tetrahedral frame we'd be 90% there, right? I mean, in theory at least, if we could make a tetrahedral frame of sufficient thinness then all we'd have to do is make 5 of them and figure out how to weave them together. Yes, that last part (the "weaving") is the hardest step, but before we can even think about that we need to find a tetrahedral frame unit.

And what do ya know? A good perusal of the origami literature reveals a perfect unit for our task! In the December 1986 issue of the British Origami Society Magazine (No. 121, p. 32) we see Francis Ow's "60 degree Unit". This unit is made from a 1x2 piece of paper and produces a frame that's too thick for our purposes. That is, the width of the frame is too thick to allow 5 tetrahedra to be woven together as per the previous page. But if we instead fold Francis Ow's unit from a 1x3 piece of paper we'll be in buisness!

Making one tetrahedron frame requires six 1x3 pieces of paper. In other words, it will take two squares which then must be cut into 1x3 strips. To make the full 5 intersecting tetrahedra model you'll need to make 5 of these tetrahedra - that's a total of 10 squares of paper. To make each tetrahedron a different color, as in the picture above, you'll thus need 5 different colors and 2 square sheets per color.

To build on this tripod you've just made, add two units to one of the tripod's legs to make another "joint". Then the last unit can be added to complete the tetrahedron.

There is a very strong symmetry behind the formation of this structure, and understanding this symmetry can aid you in the construction. The finished object should have the following property: any two tetrahedra are interwoven with one corner poking through a hole of the other and vice versa, kind of like a 3-D Star of David but slightly twisted. (This is what we tried to describe above.)

The important part, though, is that

Again, completing this model is a challenging puzzle, and the difficulty of this challenge is reflected in the fact that the finished model is nothing less than stunning. People's first reaction, when being shown the object, is usually to stop and stare at it for a few hours in fascination. Try it!

(2) Think about this "woven 5 tetrahedral frames" object for a moment. If the frames are too thick, the model is impossible to make. But if the frames are too thin, the tetrahedra will fall loose around each other and look like a mess! Between these extremes there's a certain frame width that is perfect, that is, will make the units fit snuggly together. When made from 1x3 pieces of paper, Francis Ow's unit makes frames that are 1/12th as thick as the edge of the tetrahedron. Is this the "perfect" width? Or is it just "close enough"?

(3) Think about the "5 intersecting tetrahedra" object that we looked at before turning the tetrahedra into tetrahedral frames (shown again on the right). Wouldn't it be cool to create a modular origami unit that produces this object? Try it!

These pages Copyright 1997 Thomas Hull

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