# Cutting a Tetrahedron



## rrich (Jun 24, 2009)

This is a safe method to cut a tetrahedron.

Set the table saw blade to 60° and leave it for the entire process. I would suggest using an angle box to set the blade.

Take your board and make a bevel cut completely through the board. Rotate the board as if on a Lazy Susan. Make a second bevel cut so that this bevel intersects with the first bevel cut. When looking at the end of the board it should be an equilateral triangle.

Use your miter gauge set to a square cut (90°). 

Put the board with the two bevel cuts on the table and against the miter gauge. Cut one end off with the blade at 60°. For the next cut, just slide the board into position and make a second cut. The object of this second cut is to make the surface against the table form a square. Make a second identical piece.

Now the really tricky part. Take the two pieces and put glue on the square surfaces. Stick the square surfaces together, square to square. If the object is not a tetrahedron, rotate one of the pieces 90° and you will have a tetrahedron.

Don't give me credit for this technique. It is from a puzzle. I just looked at the puzzle and adapted it to the table saw.


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## cabinetman (Jul 5, 2007)

Sounds simple enough. A good project would be for you to do a WIP for your method. Take a few before and after pics of each set up and cut. A tutorial like this is a great learning process.












 





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## cabinetman (Jul 5, 2007)

Have you started on it yet? I'm anxious to see.












 





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## phinds (Mar 25, 2009)

Rich,

I'm completely flummoxed by what you posted. I can't visualize your directions, so that part is my fault, but you talk about 60 degrees and 90 degrees and a regular tetrahedron does not anywhere have either one of those angles, so I just don't get it. Are you sure you aren't referring to some other shape?

Paul


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## rrich (Jun 24, 2009)

Paul,

I've been ill for the past few days and haven't been able to get into the shop.

I dug out the puzzle where I realized how it could be an instruction tool for cutting a regular tetrahedron. 

Let's start with a definition. A regular tetrahedron is a four sided, solid figure where all sides are the same. The sides are triangles, equilateral triangles. Every angle is 60°.

The first picture is the assembled tetrahedron.

The second picture is the two pieces before they are assembled. The two squares are faced together and rotated to form a tetrahedron.

The third picture is an end view of the two pieces. It is kind of misleading because the ends have been cut to 60°.

So when cutting... (Assuming that you're starting with a 2x4.)

Set your saw blade to a height of 1-3/4". Set the angle of the blade to 60° measured from the table. (I'm assuming a left tilt saw because not everyone is a right tilting weirdo like me.) Cut the 2x4 so that the kerf actually goes through the top of the 2x4. Discard the off fall.

Rotate the 2x4 so that the top of the previous cut is now against the fence. Set the fence to approximately 1-47/64". Rip the 2x4. Looking at the end of the 2x4, the shape should be very similar to the first picture. (i.e. an end view of an equilateral triangle.) It may be safer to just flip the 2x4 and cut, keeping the off fall. If the off fall method is used, then the fence settings would not be accurate.

Set your fence at 2-47/64". Clamp a 1" spacer to the fence. Use your miter gauge at 90° and cut pieces to look like the second picture. 

Take two pieces and glue the "square" sides together to form a tetrahedron.

Like I said, don't give me credit for figuring this out. I'm only adapting a wooden puzzle to solving the problem of cutting a tetrahedron safely.

I know that it seems a bit obtuse but it works.


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## cabinetman (Jul 5, 2007)

Sorry to hear you weren't feeling well. Using your method, make one from a single piece. With pictures it would make for an informative post. 












 





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## phinds (Mar 25, 2009)

Well, I certainly blew it in saying no 60 degrees. I still can't see how it works, but I believe you if you say it does, since you have the definition right.

I agree w/ cMan, would love to see it put together / taken apart.

I'm also sorry to hear you weren't feeling well. Hope that's past you.

Paul


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## SteveEl (Sep 7, 2010)

It must be catching.... I don't see it either. Congrats for getting well and for figuring it out.


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## Gary0855 (Aug 3, 2010)

47/64" that's funny.......oh that's not the joke?


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## billral (Dec 17, 2015)

*Cutting a tetrahedron*

The previous posters have it right, but maybe another set of instructions with some photos will help. Here goes.

It took me a while to figure out how to make the two-piece tetrahedron puzzle, but this method works very well. Be sure the angle between blade and table is exactly 60 deg for all cuts. Use the crosscut guide set at 90 deg to make third and fourth cuts from opposite sides of blade, keeping the " bottom" on the table for both cuts. Make the second cut so the flat bottom makes a perfect square. In this way, make two identical parts. Assemble as shown in other posts. Note that the assembled tetrahedron has almost twice the altitude as the thickness of the stock.


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## PSM (Aug 1, 2019)

*The correct dihedral angle is 70.53°*



rrich said:


> This is a safe method to cut a tetrahedron.
> 
> Set the table saw blade to 60° and leave it for the entire process. I would suggest using an angle box to set the blade.
> 
> ...


This is almost, but not quite right. It certainly is the easiest and safest way to cut a tetrahedron, but by using an angle of 60°, you get something that almost looks like a tetrahedron, but isn't. Measure all four edges and you will see that they are not all the same.

That is because the dihedral angle (the angle between two intersecting faces) of a tetrahedron is 70.53°, not 60°.

If you adjust the above instructions by @rrich slightly, by *setting the table saw blade to 70.5°* and leaving it there for the entire process, you will end up with a beautiful, perfect tetrahedron. The only downside will be the visible joint, but this is by far the best method for cutting a solid tetrahedron.



> The object of this second cut is to make the surface against the table form a square


Also, to help you cut this second cut, note that the length of the edge of the tetrahedron is exactly double the length of the edge of the squares - the ones that you are gluing together.

Thanks for the tip, @rrich


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## Tool Agnostic (Aug 13, 2017)

Our woodworking club has a booth at the local county fair, staffed by volunteers from the club. They have several of those two-part pyramids for visitors to try. Most people struggle to assemble a pyramid from the two pieces. Eventually they give up and one of us shows them. If I am too busy to demonstrate the solution, I tell them:

"Put the two square faces together, then rotate them 90 degrees while holding them face-to-face."


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## PSM (Aug 1, 2019)

For a great in depth tutorial on how to make these tetrahedral pyramid puzzles, see Next Level Carpentry's video


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