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 March 8th, 2018, 08:37 PM #1 Newbie   Joined: Mar 2018 From: USA Posts: 3 Thanks: 0 Building Platonic solids Hi Everyone, This is a question I just kind of asked myself one day. I have been thinking about it, trying to figure it out, but I cannot seem to get at it completely. At first, I was just thinking about a tetrahedron. But then I realized the same question could be applied to any platonic solid. Here is how it goes: How would you cut 2x2 boards so you could put together the platonic solids? Technically, it would really just make the frame of a solid. Here is what I have determined so far: To make any one of the solids, you would need the right amount of boards. Each board would be an edge of the solid. Tetrahedron: 6 edges Cube: 12 Octahedron: 12 Dodecahedron: 30 Icosahedron: 30 Each board would need to be the same length and each board would need to be cut identically at the ends. If you do it right, they should all fit together. I seems like there are actually two ways to cut them to fit together. They could be cut so that the boards taper down to their edge and the edges meet at each vertex. On the other hand, they could be cut so that the boards taper down to the middle of their faces to meet. My curiosity will give me no rest. I am almost ready to go buy wood and a saw just to try to figure it out. If any of you could teach me how to do the pertinent calculations and/or point me toward some learning resources, I would be most grateful. Last edited by skipjack; March 9th, 2018 at 05:31 AM.
 March 9th, 2018, 05:44 AM #2 Global Moderator   Joined: Dec 2006 Posts: 19,535 Thanks: 1750 I would consider a cube first. I suspect that there are infinitely many ways. However, I don't fully understand the description. What does "taper down to the middle of their faces to meet" mean? Why can't you use a board for each face? What does "identically at the ends" mean?
 March 9th, 2018, 11:40 AM #3 Newbie   Joined: Mar 2018 From: USA Posts: 3 Thanks: 0 In that part of the post, I was calling a flat side of the board a face. I was not referring to the face of the shape itself. I understand the confusion. When I say the boards will be identical at the ends I mean that every individual board will be the same exact shape. Since every vertex of a platonic solid is identical, the right number of boards, all properly and identically cut, should fit together to make the frame of the solid. I an not much of an artist but I just made an illustration with microsoft paint that might make things easier to visualize. Different colors represent different boards and they meet on the black lines. I found two pictures that look kind of like what I am going for. I laid some lines over one to show where the boards would meet. The two picture examples show what would be constructed if the boards taper down to their edges and the edges meet at each vertex. Math1.jpg Math2.jpg Math3.jpg
March 14th, 2018, 02:18 PM   #4
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 Originally Posted by skipjack I would consider a cube first. I suspect that there are infinitely many ways.
Depends on symmetry.
The edges are 2x2 - i. e. possess square symmetry before cutting.

Consider a vertex - three square prisms meeting at right angles.
The three prisms overlap in a cube. Meaning you must cut the small cube in three parts, each of which remains attached to one of the three faces of the small cube.

All three parts must be equal. This binds you to a threefold axis of symmetry along diagonal.

Are you keeping mirror symmetry, or no?
If yes, I think you have a unique way of cutting a cube in three along a diagonal.
If no, you have infinite number of options.

 March 15th, 2018, 10:53 AM #5 Newbie   Joined: Mar 2018 From: USA Posts: 3 Thanks: 0 I think I have figured out how to make a tetrahedron with the board edges meeting at the corners. It took a lot of thinking, sketching and some spreadsheet calculation. I think I will refrain from explaining the whole process because I would probably just make it really long and confusing. I made a couple of diagrams showing how the boards would be cut. Just two cuts would need to be made on each end. If I got it right, after making six boards the same length with those cuts, you should be able to glue them together to make a tetrahedron like the one in the picture I found and posted previously. When I get the time, I will try to figure out a cube, I think it will be easier since more things line up at that 90 degree angle so I can get at some measurements more directly. Below are two diagrams. One is a front view and the other has a side view. The side view just has a few measurements that would be used for marking the board. These assume the board has 2 inch sides. Many boards are called 2x2 but are actually smaller. Front-view.jpg Side-view.jpg
 March 22nd, 2018, 12:34 PM #6 Global Moderator   Joined: Dec 2006 Posts: 19,535 Thanks: 1750 Sorry about delay.
 March 22nd, 2018, 12:56 PM #7 Senior Member   Joined: Jun 2015 From: England Posts: 853 Thanks: 258 Three books of interest Mathematical Models Cundy and Rollett instructions for making just about any of these and many more besides, including a strong bibliography of the subject. Mathematical Recreations and Essays Rouse Ball Things to make and do in the fourth dimension Matt Parker Quite a bit of original material modern here
March 25th, 2018, 11:20 PM   #8
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 Originally Posted by snorkack Depends on symmetry. The edges are 2x2 - i. e. possess square symmetry before cutting. Consider a vertex - three square prisms meeting at right angles. The three prisms overlap in a cube. Meaning you must cut the small cube in three parts, each of which remains attached to one of the three faces of the small cube. All three parts must be equal. This binds you to a threefold axis of symmetry along diagonal. Are you keeping mirror symmetry, or no? If yes, I think you have a unique way of cutting a cube in three along a diagonal. If no, you have infinite number of options.
Note that the whole symmetry argument is the same when the solid is a tetrahedron rather than a cube - because a tetrahedron also has 3 edges meeting at each vertex, just at different angles. For the same reason, the symmetry argument still applies in case of dodecahedron.
The symmetry argument still applies in case of icosahedron... because although each vertex has 5 edges meeting, it still is odd.
The only case with different symmetry is an octahedron, because a vertex has 4 edges, which is even.

Now, what happens if you give up mirror symmetry?
Suppose one edge of a tetrahedron has a cut surface on the right at which you leave a knob.
Then you need to cut a hole in the left cut surface of the next edge, to fit the knob.
And they fit.
How about all 6 edges being identical?
You can ensure the 3 edges at the vertex being identical... because each of the 3 has a knob on the right and a hole on the left. They can be freely interchanged.
And then you ensure that the matching knobs on the right, holes on the left are also made on the opposite ends of the edges, and in the ends of the remaining 3 edges.
You have 6 identical edges. A symmetry what they do not have is mirror symmetry.

The precise shape of the knob is arbitrary, within the limits of fitting. And of course the reflection of the assembly is simply another legal solution.

Is that a legitimate solution of your challenge?

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