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

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Blog entry by rance posted 1030 days ago 10326 reads 13 times favorited 18 comments Add to Favorites Watch
« 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.

- – - – - – - – - -


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):




[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--

18 comments so far

View PurpLev's profile


8476 posts in 2272 days

#1 posted 1030 days ago

just be careful playing with this soccer ball, I’ve heard of tree huggers, but not tree kickers – could be painful.

-- ㊍ When in doubt - There is no doubt - Go the safer route.

View lanwater's profile


3076 posts in 1558 days

#2 posted 1030 days ago

This guy:
has similar finding.

His angles and yours are few minutes off. If he is right that could add up…

I think Sam has the answer.

I sure will be following.

-- Abbas, Castro Valley, CA

View rance's profile


4128 posts in 1784 days

#3 posted 1030 days ago

lanwater, I agree completely. Thanks for the link. If I’m off, then I’ll know it when I am gluing 85 joints in one swale foop. :) Hey, this is an experiment at this point. If it actually works, THEN I’ll call it a project. :D

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

View Mark Shymanski's profile

Mark Shymanski

5077 posts in 2337 days

#4 posted 1030 days ago

Intriguing…now my brain hurts. Thank you for a captivating post!

-- "Checking for square? What madness is this! The cabinet is square because I will it to be so!" Jeremy Greiner LJ Topic#20953 2011 Feb 2

View Michael1's profile


403 posts in 1284 days

#5 posted 1030 days ago

Very creative design. I am anxious to see how the project turns out. Thanks for posting.

-- Michael Mills, North Carolina,

View SPalm's profile


4783 posts in 2506 days

#6 posted 1030 days ago

See, I was right. You are a techno weenie.
Nice study.
Bugger to build I would think.


-- -- I'm no rocket surgeon

View rance's profile


4128 posts in 1784 days

#7 posted 1029 days ago

You’re probably right Steve. :)

I Googled more and found a real Scientific-Looking site( that states the mathematically-calculated angles. It looks like I’m just mere minutes away from my final answer. They publish the Dihedral angles as follows:

  • 138 degrees, 11 minutes for the hex-hex angle
  • 142 degrees, 37 minutes for the hex-pent angle

So what’s a minute? A minute is 1/60 of a degree.

So… (for the Hex – Hex) 11 minutes = 11/60 of a degree, or 0.18333 degrees. And the dihedral is stated from a flat surface. So… 180 – 138.18333 = 41.81667. Divide that by 2 to get the miter angle and I get 20.908335 degrees. I believe my 20.9 degrees agrees with that.

Now… (for the Hex – Pent) 37 minutes = 37/60 of a degree, or 0.61667 degrees. And converting from the dihedral, 180 – 142.61667 = 37.38333. Divide that by 2 to get the miter angle and I get 18.691665 degrees. I believe my 18.7 degrees should be fine there too.

Of course this all assumes that ‘’ knows what they are doing. I sure hope so. I trust their math more than my SketchUp.

I believe that cutting the pieces out will be simplified using Pauls method on the TS.

However, the glue-up could be a bear with 90 joints all at once. One way would be to use the Tape-Hinge method like what I use with mitered boxes. The tape gets it close(if not right on) and the clamps shmooze it into exact position. The key to this is the Net:

I got this one from Wikipedia. It is better than most as it centers it all around the base piece. Other ‘Nets’ can be found elsewhere and are typically used when making folded paper models.

Lastly, I’m sure a glue with a looooong open time would work best.

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

View lanwater's profile


3076 posts in 1558 days

#8 posted 1029 days ago

Why do you have to glue all at once?
That may prove unmanageable

Can you just start with a pentagone and add all the surrouding hexagone? That might keep it tight.
Once that is done then add more to the first subassembly?

I was thinking CA glue for very fast drying time and the glue does not “creep”. For clamping, small paper clips like those:

Just ideas.

-- Abbas, Castro Valley, CA

View Joe Lyddon's profile

Joe Lyddon

7674 posts in 2676 days

#9 posted 1029 days ago


You have done a wonderful job in studying, researching, & great use of SketchUp,

to SIMPLIFY a complex (to me anyway) problem!


I LOVE your idea of using the Taping Method, as used for gluing Miter joints, for gluing up your domes!
Hinge-type devices could be used for putting larger dome pieces together… (just thinking out loud)...

You are doing some GREAT work here… Keep it up!

You have peaked my interest again in this geometric subject!

Thank you very much!

After reviewing Part 1, I realize that you SOLVED the slight ZIG ZAG (bottom) problem! YES?
I knew there had to be a way to do it! I’m glad your brain was Up To the task… mine wasn’t… LOL

-- Have Fun! Joe Lyddon - Alta Loma, CA USA - Home: ... My Small Gallery:"

View rance's profile


4128 posts in 1784 days

#10 posted 1029 days ago


Yes, it could turn out to be unmanagable. This is one of those situations where any error in gluing one joint becomes accumulative to the next. Actually it almost compounds itself logarithmically. By building something like this all in one glue-up, any slight errors would be schmoozed out as it is pulled together.

This would be similar to gluing up an octagonal box, but with a 3rd dimension. The key to a straight Octagon is having the miters cut darned near perfect, and using a band clamp. Gluing up one joint at a time would be analogous to gluing one picture frame joint at a time. When you get to that last corner, any errors from the other 3 would be magnified at the last corner.

Of course, in building a separated lid, some of the edges would simply not be glued. This kind of a project SCREAMS for Planning. Something that gets left by the way side with many a project.

Hey, I’m talking out the side of my head here. Your suggestion could possibly work just fine. :)

One more bizarre solution might be to drill a hole in one segment, apply all the (slow setting)glue, quickly wrap it all up, and apply a vacuum to it. A weightless chamber would come in handy for this method. It would be a hoot if it actually worked. I’d be afraid it’d suck all the glue out of the joints though.

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

View Joe Lyddon's profile

Joe Lyddon

7674 posts in 2676 days

#11 posted 1029 days ago

If you did a Dry Glue-Up using Tape, you could hold off on the glue until you got it Right ahead of time…

Then, do another Dry fit up… to be sure…

Then, go for it with Glue!


-- Have Fun! Joe Lyddon - Alta Loma, CA USA - Home: ... My Small Gallery:"

View BertFlores58's profile


1646 posts in 1546 days

#12 posted 1029 days ago

Someone made it already…. just recently posted on this project

Rance, Thanks for posting this technique… I like to make one of this too! I am planning to use 2 hollow circular jigs clamping the layers of the joint pieces per latitude. But it will be variable circle diameter with the right angle (you already mentioned it above) of radii. Hard to explain… hope I can make it.\

-- Bert

View Alexey Khasyanov's profile

Alexey Khasyanov

158 posts in 1497 days

#13 posted 975 days ago

Hi, LJ’s!

Here is info for You. In One place.

-- Rev 22:21

View Joe Lyddon's profile

Joe Lyddon

7674 posts in 2676 days

#14 posted 974 days ago

That site will put your brain out to lunch! LOL

Thank you.

-- Have Fun! Joe Lyddon - Alta Loma, CA USA - Home: ... My Small Gallery:"

View waleedwaheed2013's profile


134 posts in 509 days

#15 posted 507 days ago

Hi dear , Thanks for your data , Can you tell me How to fixed them by glue.


-- Waleed Waheed

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