There’s a certain beauty in the *inevitability* of the 5 Platonic solids – we can say with certainty that they are **all** of the possible regular tilings of the sphere, and that they are the **only** possibilities.

There are also many *neat* and beautiful relationships between them (such as the dualities, the space filling packing of cubes, or of tetrahedra and octahedra).

So far, so tidy, but let’s take a look at the tetrahedron. Take 2 and join them face to face…take another and join it so they all share an edge…then another…and another…you are back to the beginning – full circle…*almost!*

If it wasn’t for this funny little **gap** then you could fit 20 of them together to make an icosahedron – linking the solid with fewest faces with the one with the most faces, making a complete loop. It would also lead to lots of other nice relationships – octahedra would fit together with icosahedra to fill space, and dodecahedra would fill space on their own.

But the reality is *they simply do not fit*. With physical models you can squeeze them together so it looks like they do*, and Aristotle seems to have believed that they did. It’s an easy mistake to make if you dont have a computer because the difference is so **tiny** – If the dihedral angle of the tetrahedron was only about 1.4712 degrees (or **0.004** of a circle) bigger then it would fit. Of course in the world of mathematics an *almost* doesn’t count for much, but it seems odd to me. It looks a bit like a mistake – What is this *almost* doing in the Platonic world of perfection and exactitude?

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*Actually this is what drew my attention to this idea in the first place – In my first year of architecture I was playing around with paper octahedra, joining them in this way, it worked fine for a while but when the packing got beyond a certain size it kept pulling apart. At the time I put it down to the thickness of the paper. It wasnt until I modeled it on the computer that the real reason became clear.

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July 26, 2007 at 11:13 am

I recently found out that John Conway (inventor of the Game of Life) corresponded with Martin Gardner (the famous mathematics writer) about this packing of tetrahedra. See the paper here :

March 31, 2009 at 7:56 am

I love the idea that this is an error, the universe, in fact all universes that allow geometry have a flaw. Luckily in this case the flaw can be used, like the fact that three pentagons do not quite fit round a point. The pentagons can instead be folded up to make a dodecahedron. Similarly the tetrahedra fold into a fourth dimension and loop round to eventually give a regular figure with 600 tetrahedra: the 600-cell.

July 27, 2009 at 1:49 pm

Great site and contribution to this discipline.

JMP

January 24, 2010 at 7:08 pm

Is Im(log z) a true helicoid?

January 24, 2010 at 7:09 pm

Obviously I meant to post that question elsewhere. Oops.

January 24, 2010 at 11:43 pm

Hi Anton, yes it is a true helicoid. Thomas Banchoff has written a bit about this here:

http://www.geom.uiuc.edu/~banchoff/script/CFGExp.html

May 7, 2013 at 3:00 pm

Exploring how to pack tetrahedrons lead me to this model showing a spiral form: http://www.shapeways.com/model/396262/cracked-spiral.html?li=productBox-search

(I broke the tetrahedrons into 4 equal sections and only used those sections that make contact in the middle to highlight the symmetry of the figure.)

I have also played with how to unfold the resulting construction of packed tetrahedrons in such a way that the flat pattern has no gaps- a third generation of radial stacking results in a nice parallelogram with internal cuts, but folding up as one contiguous piece of paper.

June 26, 2013 at 11:47 pm

Reminds me of the Pythagorean “circle of fifths” whereby a musical “fifth” (= exactly 2/3 the frequency of the fundamental) repeated 12 times comes back full circle to (a higher octave of) the original fundamental that started the circle, ALMOST! But not quite, it is off by the “Pythagorean Comma”

http://en.wikipedia.org/wiki/Pythagorean_comma

(see the 12 pointed star in the figure). I think there is something fundamental going on here.

February 20, 2018 at 12:20 am

I made this shape with my kids’ magnetic blocks. It works (no gaps). I used 10 isosceles triangles. In other words I don’t have the fourth wall of the tetrahedra. But all edges of the plastic blocks touch perfectly. I wish I could post a pic here.

February 20, 2018 at 2:25 pm

Hi Mark,

Yes -that is a pentagonal bipyramid. By leaving out the other faces of the tetrahedra it allows the angle to change and the shape to close up.