# Tetrahedron Questions



## PhelpsRB (Oct 15, 2010)

Hi, Ok here is the problem, I am in a 3D college class, and I am trying to cut a tetrahedron out of wood. I created one by first cutting a prism, and then using a 30 degree angle cut the other sides, but the triangles were not the same size as the base, so what I need help with is figuring out how I can cut it out, and have all the sides the same size. Anything you can suggest would be a big help, the wood shop at my school has every thing I need to do anything I want. And to tell the truth this has my teacher stumped, he tried and cut one out like I did. So thank you for any help.


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## b00kemdano (Feb 10, 2009)

Does it have to be cut from one piece of wood?


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## woodnthings (Jan 24, 2009)

*Difficult.....*

Please don't ask me to do it! :laughing:
The definition is as follows:
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle *(any of the four faces can be considered the base), *so a tetrahedron is also known as a *triangular pyramid*.
From wikipedia: http://en.wikipedia.org/wiki/Tetrahedron
The calculations are way beyond my level of math and I actually made one of these in folded paper back in 3D design class, about 50 years ago. I think geodesic domes are made with these if I recall. 
It seems that all the side are of equal length and all the angles are the same, unless I misread the description/math formulae. I wouldn't know where to begin. So if you end up making one take photos! :blink: bill


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## Itchy Brother (Aug 22, 2008)

Yeah,What they said,heck I thought it was some kind of Dinasaur!


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## GeorgeC (Jul 30, 2008)

THIS  will tell you all the information that you need.

George


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## phinds (Mar 25, 2009)

GeorgeC said:


> THIS will tell you all the information that you need.
> 
> George


 
Well, no, unless you're a math whiz, that article will just make your head hurt. BADLY. The math on this is conceptually simple but REALLY nasty to work through. You don't need trig, but you need to LOVE algebra.

The things to keep in mind are
(1) the slope of the sides relative to the base is NOT 30 degrees (or 60 degrees) which means that if you tilt the saw blade over 60 degrees and cut the base of the tetrahedron out of the base of a cube, then it won't work.

but

(2) It can be exactly contained in a cube, but NOT if you put the base flat on one surface of the cube. You have to make the edges of the tetrahedron be diagonals of the face of the cube. There are pics on the internet showing what this looks like. 
​How you line up and hold the parts while you cut it using the information in (2) would be really hard so the only reasonable way I can see to do it is either do the ugly math or, better, find an internet site that gives the angle from the base to the sides, cut the base out of a rectangle of wood, then use the angle to cut the sides.

Since the base anges ARE 60 degrees, once you know the slope of the side, it should be easy.

Paul

​


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## xelntchance (Jan 2, 2008)

how big do you want this to be and what tools are you using.
Fairly small it could be an off cut from a beveled strip. If it needs to be larger a card board template would help make individual sides.


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## frankp (Oct 29, 2007)

I don't believe 30 degrees is the right angle for the sides... I think it's closer to 50 degrees. There are a bunch of calculations for regular tetrahedrons that involve sines and cosines and such... I don't remember any of them off the top of my head. If it doesn't have to be a regular tetrahedron, just make one side different than the others.


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## rrich (Jun 24, 2009)

A tetrahedron is solid figure of four sides and all sides are equilateral triangles. 

ALL the angles are 60°. All the edges of the triangles are equal in length.

It's like that old puzzle. Take 6 tooth picks and make 4 equilateral triangles with each side being the length of the tooth pick.

When thinking about cutting one from wood... All I can think of is DANGEROUS.


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## SteveEl (Sep 7, 2010)

nevermind


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## SteveEl (Sep 7, 2010)

Actually it was a really


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## phinds (Mar 25, 2009)

One thing I should add to my previous post is that although my technique is likely the only reasonable way to do it, you WILL have to use hot melt glue or double-sided tape, or something, to hold the base down while you cut the sides. The angles will make it impossible to clamp them after the first one. If you don't do that, it will be, as rrich said, dangerous.

Also, if the sides you need are too big to clear the arbor of the saw, you're going to have yet another problem and the only way I can see to get past that one is to just do it with a handsaw, and good luck trying to make THAT come out right.

Paul


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## phinds (Mar 25, 2009)

I just re-read your original post, and I see that you've done the right start already, starting with a prism, so really all you have to do it, as I said in my first post, get the angle of the sloping sides. I think it's about 70 degrees, but my memory for these things isn't what it used to be and you need to get it exact.


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## woodnthings (Jan 24, 2009)

*Angle is ...*

Angle between two faces[2] 







*(approx. 70.528)

*
from the Wiki link. :blink: bill


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## cabinetman (Jul 5, 2007)

If I were to make one I wouldn't be looking up math formulas. I would start with a block of wood, mark off sizes on a side to start. Then cut what was convenient with a table saw, RAS, or band saw. Make positioning jigs to hold for other angles, and finally support it to do whatever carving or shaping that has to be done by hand. 












 





.
.


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## GeorgeC (Jul 30, 2008)

phinds said:


> Well, no, unless you're a math whiz, that article will just make your head hurt. BADLY. The math on this is conceptually simple but REALLY nasty to work through. You don't need trig, but you need to LOVE algebra.
> 
> The things to keep in mind are
> (1) the slope of the sides relative to the base is NOT 30 degrees (or 60 degrees) which means that if you tilt the saw blade over 60 degrees and cut the base of the tetrahedron out of the base of a cube, then it won't work.
> ...


We learn from "trigenometry" that the sides of an equlateral triangle are at 60 degrees to each other.

However, if you will look at the link that I provided you will see the actual angles (look past the formulas to the actual numbers that are posted) measurements. You do not have to do the actual mathematics, it is already done for you.

G


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## SteveEl (Sep 7, 2010)

This is an untested idea.... proceed to waste your time at your own risk......


If part of the exercise is to do this using only the tools in the shop (without math or references tables) I may have an untested idea that might, maybe, get you partway.

The idea keeps the saw blade straight up, and uses a crosscut sled and a series of test triangles to make miters while you sneak up on a measurement.

Make your prism.

Label one end Face 1, and the points of its triangle A B and C.

Essentially, the problem is to find mystery Point D somewhere in the body of your stock.

Lay the prism on the table with Face 1 toward you and B up. 

Now label the long face to your left Face 2, to the right Face 3, and the one on the table Face 4. The far end of your prism is all waste material.

Place a 30 degree wedge under point C so that Line AB is straight up. Position things so that line AB is on your sleds cut line. 

Place your first test-miter triangle between the fence and your assembled stock-plus-30-deg-wedge, so that the assembly pivots slightly left. Do whatever you're going to do to clamp all this stuff, and then cut. You just cut off the marking "Face 2" so remark it.

Slide your stock to the other side of the kerf, keeping B up, but move your 30 degree wedge under Point A. Now Line CB should be straight up on the cut line. Move your first test-miter triangle to the other side to pivot the same amount to the right. Clamp and cut again. Relabel that cut surface Face 3.

Look at the common edge between Face 2 and 3, starting at Point B, and ignore new faces that might have appeared..... just look at the straight edge starting at B. Wherever that straight edge cuts an angle, mark that as Point X.

Measure line BX. It should be longer than line AB. If so, repeat the miter cuts with a slightly larger test-miter triangle.

Unless I'm mistaken (very possible!) as you increase the size of that test-miter triangle eventually the length of line BX will equal line AB. If so, Point X = Point D, the other point of your finished product.

I'm not sure what happens next, except I better get back to work.


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## Mizer (Mar 11, 2010)

Itchy Brother said:


> Yeah,What they said,heck I thought it was some kind of Dinasaur!


I am with you Itchy.


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## AlWood (Apr 18, 2010)

phinds said:


> Well, no, unless you're a math whiz, that article will just make your head hurt. BADLY. The math on this is conceptually simple but REALLY nasty to work through. You don't need trig, but you need to LOVE algebra.
> 
> The things to keep in mind are
> (1) the slope of the sides relative to the base is NOT 30 degrees (or 60 degrees) which means that if you tilt the saw blade over 60 degrees and cut the base of the tetrahedron out of the base of a cube, then it won't work.
> ...


Paul, the angle "phi" between two adjacent planes of that pyramid is determined by the formula
sin (phi/2)=1/sqrt(3)
The calcs would give you approx. phi=70.53 (degrees). Notice it substantially greater than 60 degrees. -- Al


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## phinds (Mar 25, 2009)

"However, if you will look at the link that I provided you will see the actual angles (look past the formulas to the actual numbers that are posted) measurements. You do not have to do the actual mathematics, it is already done for you."

Good point, George.

I saw all that math up front and remembered from about 40 years ago when I was studying Analytic Geometry and went through every calculation imaginable on a tetrahedron and what a PAIN it was, and I quickly closed that page lest my head explode and overlooked the actual result that Bill points out:










because I was looking for just NUMBERS (that is, directly useable answers, not equations)

Steve, I don't see why you want to go to all that trouble when the method I explained is very simple and straightforward if you start with a prism, as he is doing, and you have the slope angle to cut the sides. It's 3 simple cuts, complicated only by having to hold the base down but that's easy w/ hot-melt glue or double-sided tape. You are WAY overcomplicating this.

Paul


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## joesdad (Nov 1, 2007)

Phelps, can't you just build a cool bong out of beer cans like every other college student?...wow my head hurts. I hope we get to see this thing finished.


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## SteveEl (Sep 7, 2010)

phinds said:


> Steve, I don't see why you want to go to all that trouble when the method I explained is very simple and straightforward if you start with a prism, as he is doing, *and you have the slope angle....*


You certainly did come up with the fast way, Paul, but we all find fun in different aspects of what we do. Some woodworkers ignore electricity for fun, I ignored the trig for the same reason.

(I agree it is a rather cumbersome approach)


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## phinds (Mar 25, 2009)

I do NOT blame you for ignoring trig ... I've been doing my best, farily successfully, to NOT use it for almost 40 years. :icon_smile:


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## rrich (Jun 24, 2009)

See new thread.


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## phinds (Mar 25, 2009)

rrich said:


> See new thread.


 
huh ?


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## kgstew (May 26, 2016)

*Heres the solution*

Read this post today looking for answered to the same question. Found this later.

http://sbebuilders.blogspot.com/2013/03/making-wood-polyhedron-or-wooden.html


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## Steve Neul (Sep 2, 2011)

OK, if you had one, what would you do with it?


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## m.n.j.chell (May 12, 2016)

Since this thread has been revived from 2010 ... I just had to comment on the information. I read the article that went with the pictures, on the site kgstew linked. Pretty neat, that he was able to build that structure ... nice little piece of art that it is. It's a better use of wood than some other projects I've seen.

But then, the author states it would've been impossible for da Vinci to have built one.

I don't know what he thinks Leonardo da Vinci was capable of, but just look at the sculptures ... 
Wood is extremely easier to work with than stone. I've no doubt in my mind that an artist (especially one of da Vinci's caliber) could make such things with hand tools, perhaps even easier than he did with power tools.

Apologies ... just had to vent on that point.


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## BigJim (Sep 2, 2008)

mikechell said:


> Since this thread has been revived from 2010 ... I just had to comment on the information. I read the article that went with the pictures, on the site kgstew linked. Pretty neat, that he was able to build that structure ... nice little piece of art that it is. It's a better use of wood than some other projects I've seen.
> 
> But then, the author states it would've been impossible for da Vinci to have built one.
> 
> ...


Old thread but interesting, I do agree with you, the people of the olden days of da Vinci were much much smarter than people of today think they were. Pit todays thinkers with Aristotle, Plato, Diogenes of Tarsus, and others of that era the list is endless. I dare say very very few of today could stand their ground with anyone of these. Why would one think someone of that time couldn't do what we do and without our tools of today.


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## m.n.j.chell (May 12, 2016)

There's also a difference of "need", BigJim.
I don't actually believe the ancients were "smarter" ... if anything, we've evolved some since then. BUT, they had fewer items of convenience, less items of comfort, and almost zero items that alleviated the NEED to think. Everything still needed to be discovered, to be imagined and built. 
They didn't have the luxury of sitting around, watching TV, researching online, or even in encyclopedias.

Today's thinkers (smarter than ever) exist ... they've just never had to. And we keep dumbing down education systems.

In another 3000 years, if we're still around at all, I think H.G. Wells' "Time Machine" is closest to that truth. Eloi and Morlocks ... neither race needing to think anymore.


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