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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

I believe the answer is "42".

JP

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

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

Puckdropper

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.

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.

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

Maybe your finished piece needs to have the faces co-planar with the edges, but mine needs to be made from square sticks. The faces are not, nor do they need to be, in the same plane as the triangle formed by the edges. When I'm done, the faces of the sticks will be proud of the plane of the edges. Not a problem - that's what I'm after.

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

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.

www.swcp.com/~awa/images/pdf%20files/Angle.pdf

Mitch

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?

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: http://en.wikipedia.org/wiki/Image:Compound_of_two_tetrahedra.png )

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.

That's exactly what I'm asking :)

#### Site Timeline

- posted on November 9, 2008, 8:21 pm

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

- posted on November 9, 2008, 8:51 pm

I believe the answer is "42".

JP

- posted on November 10, 2008, 12:48 am

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

- posted on November 10, 2008, 1:44 am

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

Puckdropper

--

If you're quiet, your teeth never touch your ankles.

To email me directly, send a message to puckdropper (at) fastmail.fm

If you're quiet, your teeth never touch your ankles.

To email me directly, send a message to puckdropper (at) fastmail.fm

Click to see the full signature.

- posted on November 10, 2008, 1:56 am

Puckdropper <puckdropper(at)yahoo(dot)com> wrote in

Going hungry, I suspect :P

Going hungry, I suspect :P

- posted on November 10, 2008, 6:53 am

Puckdropper wrote:

The true answer is: 9.

The true answer is: 9.

- posted on November 14, 2008, 9:48 pm

I agree with the cardboard model, and makig it wider to gain accuracy,
ut instead of cardboard, on our shop we use a lot of clear acetate
sheets of .020 "and .040" thickness. It breaks and peels off cleanly on
a knifed llne. Then after dry-fitting those pieces, you can trace the
angles onto your wood pieces. Thin styrene works well, too.

- posted on November 9, 2008, 8:53 pm

- posted on November 9, 2008, 8:59 pm

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.

- posted on November 9, 2008, 10:08 pm

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.

- posted on November 11, 2008, 12:23 pm

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.

Google for the links.

- posted on November 9, 2008, 10:16 pm

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

- posted on November 12, 2008, 6:46 pm

"Projecting onto faces of a square stick" is your problem.

The "sticks" would form the edges of the tetrahedron and, by definition, the faces of those sticks would be the same as the faces of the tetrahedron relative, one to the other. It would seem that you would start off with "sticks" that were triangular, not square. The image at: http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/120px-Tetrahedron.png

shows one built out of "sticks" joined at little balls (as if some sort of kids construction toy) and ignores the edges as such.

In your case, the end of your (triangular) stick, would be cut at the same angle as the faces of the tetrahedron and each stick "end" would form a third of the tetrahedron in miniature. Sticks of equal length, glued together at end points so cut, should create the figure/shape you asked about.

But square sticks won't cut it as the edges of the finished piece need to be co-planer with the face(s) of the figure.

The "sticks" would form the edges of the tetrahedron and, by definition, the faces of those sticks would be the same as the faces of the tetrahedron relative, one to the other. It would seem that you would start off with "sticks" that were triangular, not square. The image at: http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/120px-Tetrahedron.png

shows one built out of "sticks" joined at little balls (as if some sort of kids construction toy) and ignores the edges as such.

In your case, the end of your (triangular) stick, would be cut at the same angle as the faces of the tetrahedron and each stick "end" would form a third of the tetrahedron in miniature. Sticks of equal length, glued together at end points so cut, should create the figure/shape you asked about.

But square sticks won't cut it as the edges of the finished piece need to be co-planer with the face(s) of the figure.

- posted on November 13, 2008, 1:25 am

Maybe your finished piece needs to have the faces co-planar with the edges, but mine needs to be made from square sticks. The faces are not, nor do they need to be, in the same plane as the triangle formed by the edges. When I'm done, the faces of the sticks will be proud of the plane of the edges. Not a problem - that's what I'm after.

- posted on November 10, 2008, 1:14 am

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

- posted on November 10, 2008, 1:34 am

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.

www.swcp.com/~awa/images/pdf%20files/Angle.pdf

Mitch

- posted on November 10, 2008, 1:54 am

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?

- posted on November 11, 2008, 3:57 pm

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: http://en.wikipedia.org/wiki/Image:Compound_of_two_tetrahedra.png )

- posted on November 12, 2008, 12:47 am

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.

- posted on November 12, 2008, 12:57 am

That's exactly what I'm asking :)

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