Woodworking geometry question

[Elrond Hubbard]

Calling all geometers:

I want to make a tetrahedron out of square cross-section sticks. I want a corner of the stick to define the edge of the tetrahedron - i.e., a plane through the diagonal of the stick will pass through the center of the tetrahedron. (I've already made one where a flat on the stick defines the edge - the math is easy on that one).

What I can't figure out (and I've spent a frustrating day in the shop trying) is:

What are the angles on the face of a square stick that work?

The critical information:

The central angle is 109.48 degrees. That is, the angle from the center of the tetrahedron to any two vertices is 109.48.

Any help would be much appreciated!

Scott

[Jay Pique]

I believe the answer is "42".

JP

[Ed Edelenbos]

[Elrond Hubbard]

"Ed Edelenbos" wrote in news: [e-mail]:

Lots of info there, but none that answers my question. I know what the mating surfaces of the sticks have to look like - my problem is projecting those surfaces onto the faces of a square stick.

['lektric dan]

I've done similar projects with geometric solids. What really helped me was to make very accurate models out of heavy cardboard. I made them 3 to 4 inches on a side (some were as large as 6 inches on a side); make yours to the same scale as your wooden one. Then you can measure angles directly off of the model. You will probably need to tweak the measurements slightly because your paper model won't be absolutely perfect.

[Larry]

Elrond Hubbard wrote in news:Xns9B51A2A78805Foldshoe@216.151.153.22:

Being geometricly (is that a word?) challenged myself, I would draw it up in a CAD program and then let the software figure it out. Just a thought...

Larry

[Elrond Hubbard]

Jay Pique wrote in news: [e-mail]:

That is the right answer - but just not to this particular question ;)

[Robatoy]

Do you want each corner to come out to a point?

[MB]

This PDF has formulas for making pyramid for "n" sides. I think a tetrahedron would be a 3 sided pyramid. It has all the derivations, but scroll to the end to see the answer.

Mitch

[Puckdropper]

Elrond Hubbard wrote in news:Xns9B51C980D4146oldshoe@216.151.153.45:

What is the question, and what are those two mice doing with Robatoy's brain?

Puckdropper

[Elrond Hubbard]

 

Yes.

If it helps, here's another way to describe it -

Picture a square stick, viewed from the side. Roll the stick on its axis so that it is now resting on one edge. One diagonal of the stick is now vertical, one horizontal.

Make an angled cut at 35.26 degrees from the horizontal plane. This is one corner of the triangle formed by the central angle and the two ends of the stick.

Draw a line down the center of the plane formed by this cut, from the short point on top to the long point on the bottom. Cut material away on both sides of this line so that the resulting angle, with this line as the vertex, is 120 degrees. In other words, cut a 30 degree slice from each side, through this line. This completes one end of the stick.

The question I have is: What are the resulting angles, as measured on the flats of the original square stick?

[Elrond Hubbard]

Puckdropper wrote in news:000a9b1f$0$25939 $ [e-mail]:

Going hungry, I suspect :P

[David G. Nagel]

The true answer is: 9.

[Bored Borg]

I'd make an accurate model in Sketchup, roll the finished thing around and either read the angles off directly or print a 1:1 view and use it as a cutting template.

Google for the links.

[MikeWhy]

Assuming that we know what a tetrahedron is, what do you mean "angles on the face of a square stick"? (I understand square sticks, too, but the angles are all 90 deg.)

The "simple" answer of the first part, the diagonal of the stick through the center, is to first layout the tetrahedron on a cube. All edges of a tetrahedron lie on the diagonals of its enclosing cube. It won't be easy to describe in words, but here goes:

Say two squares stacked atop each other describes the cube. Name the vertices of the top square a, b, c, and d. Name the vertices of the other square e, f, g, and h. (Sketch it up so the names go clockwise, e directly below a, f below b, etc. Your sketch will then match mine.) The cube encloses two tetrahedons. The vertices a, c, h, and f are vertices of one tetrahedron described by the cube. The remaining vertices define the second tetrahedron. The center of the cube is also the center of the tetrahedron(s). The diagonals of your sticks, then, are perpendicular to the face of the cube.

If you're asking about the compound miter at the vertices... :D That will have to wait for the second pot of coffee.

(Hmmm. Here's a picture that matches:

[Elrond Hubbard]

"MikeWhy" wrote in news:8YhSk.7465$ [e-mail]:

If you re-read my opening paragraph, you'll see that I know what a tetrahedron is and how to make one.

What I want to do is make another one, with the sticks rotated 45 degrees along their axes. The "angles on the face of the square stick" are what I need to know to lay out the sticks.

[Elrond Hubbard]

"MikeWhy" wrote in news:8YhSk.7465$ [e-mail]:

That's exactly what I'm asking :)

[Dan Coby]

Hwy do any more math at all?

Make a jig that holds the sticks rotated 45 degrees. (I.e. cut a V groove in a board.) Then use the jig to hold the sticks and cut them just like you did for your first tetrahedron.

[Elrond Hubbard]

Dan Coby wrote in news:jM6dnRRz7LAIqofUnZ2dnUVZ [e-mail]:

That's a good idea, and would certainly work. I've also considered taking one of my original sticks and cutting the corners off then measuring the resulting angles - same result. But part of me would really like to get there by calculation rather than by analog means.

[MikeWhy]

If you haven't already, sketch it out on a cube as I described earlier. The geometry at each vertex is surprisingly simple, and not entirely obvious. Looking at one corner of the cube, the tetrahedron edges form the prime, 45 degree diagnonals on each of the three faces meeting at that corner.

If you've already done that, I won't be able to help further. Compound miters might as well be rocket science for me. I'm guessing, for a radial arm saw, swing the arm 22.5 degrees to split the plan view 45 deg; tilt the blade 67.5 degrees (probably wrong) to split the 45 degree rise. Repeat for the other side. Maybe someone else can take it from there...

I'll lay it out in solidworks later, see if any more insight forms. It sounds interesting enough to lay out on a small stick and whittle it.