# Geometric Shapes #2: Truncated Icosahedron (without the math)

 Blog entry by rance posted 10-25-2011 04:25 AM 20743 reads 14 times favorited 18 comments
 « Part 1: Geodesic Domes Part 2 of Geometric Shapes series no next part

I am again intrigued by another …hedron shape. Is it an Icoso…, a Dodeca…, or a Tetra…, they are all mumbo jumbo to me. I just want to make one, I don’t want to learn the math. Maybe I should just go buy a Soccer ball and be done with it. Naaaaaaaaw. I want one made of wood.

For the design, my tool of choice is Google SketchUp…..... again. Maybe I cheated, well, I probably did. Sure I did. I went out to ‘3D Warehouse’ and downloaded a soccer ball and proceeded to tear it apart. I removed all the smoothed surfaces and was left with just a few lines. I then rebuilt the surfaces enough to get the angle between the surfaces.

The Truncated Icosahedron is made up of 12 Pentagons and 20 Hexagons. The Pentagons are surrounded on ALL sides by a Hexagon. Thus, the angle of all sides is the same. The Hexagons, however, are surrounded by alternating Pentagons and Hexagons. This means that 3 of its edges are at one angle, and the other three are at another angle.

The angles between the Pentagon and Hexagon are(from my cheating) 37.4 degrees. For a segmentation(which is what this becomes), you take half off of each side. So the Pentagon edges are all cut at 18.7 degrees. Likewise, half of the Hexagon edges are cut at that same 18.7 degrees. The other three edges of the Hexagon are joined to other Hexagons but that angle is 41.8 degrees. Half for each one brings that to an angle of 20.9 degrees.

Next was to rebuild it from scratch, and using those discovered angles, construct my own Soccer Ball. I set an abstract of having the edges at 2” long. With that, I built the two components I needed.

The ‘H’ and ‘P’ notations were indicators to let me know what mating piece would be adjacent to that piece. H for Hexagon, and P for Pentagon(doooooh). With these two components, it was infinitely easier to build them using triangles. Eventually I’ll eliminate those extra lines.

Next was to bend them at my previously discovered angles. Make more copies, then keep bending until I got all 32 pieces in place.

Well, it seems to work with the angles I used. Next will be the manufacture of the actual pieces.

I searched for ‘Icosahedron angles’ on LJ and found Sam and Paul’s works. I’ll be using some of Paul’s techniques for cutting the shapes on the TS. Sam and Paul have both been an inspiration to me with their work. I hope this work here somehow helps someone else in attaining their goals.

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Update:

And with some judicious rotating, I am able to place a Pentagon at the top(and bottom) and have a straight line in case I want to open it up like a clamshell.

It only takes cutting 5 of the Hexagons in half. Wow, that was easy. I like easy. And with it open…

Or an alternate separation(similar to Paul’s):

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[Note: For reference, Sam says “Hexagons have to be cut at 69.2 degrees and pentagons at 75.3 degrees.”.]

-- Backer boards, stop blocks, build oversized, and never buy a hand plane--