## Thursday, March 28, 2013

### Making A Wood polyhedron or Wooden Polyhedra

This model is the Tetrahedron (3 equilateral triangles at each vertex) . See my last article on the geometry of the Platonic Solid Tetrahedron for the angles of the Platonic Solid.

I was trying to see if it was possible for Leonardo da Vinci (1452-1519) to have made wooden polyhedra models  having solid edges for his illustrations in Luca Pacioli's 1509 book The Divine Proportion. Conclusion, no. He might have made wooden polyhedra models out of twigs, but to have cut all the edge pieces exactly would have been beyond his tool-man-ship. This Platonic Solid Tetrahedron has 6 edge pieces. That's 6 x 4 = 24 compound cuts on the wooden model. In his Dodecahedron model / drawing it would been 30 x 4 = 120 compound cuts on the wooden model and all of them would have to be exact. If the  compound cuts on the wooden model are not exact, then you'll never be able to put all of the pieces together. This isn't even counting the 30 x 2 = 60 edge bevels, hip rafter backing.

Leonardo da Vinci's Polyhedrahttp://www.georgehart.com/virtual-polyhedra/leonardo.html

1: Rip 6 pieces of wood to the same dimension. In this model I had 4x4 redwood post left over from a job and I ripped them to 2 1/2 x 2 1/2.

2: Cut a scrap of wood to 35.2°. This is half of the Dihedral angle.

3: Use the scrap of wood to set the table saw blade tilt to 35.2°.

4: The hip rafter backing angle is 54.73572°.  Since the table saw blade tilt is limited to 45° we're ripping  the material with the blade tilt at 35.2° , half of the Dihedral angle, on both sides of the material to form the Dihedral angle on each edge piece of the Tetrahedron. This is the same as backing out the hip rafter.

5: Check the Dihedral angle. It should be 70.5°.

6: Set the blade tilt of your miter saw to 45° and the miter to 45°. These two angles are for cutting the edge piece on the top edge of the material. If we were cutting the edge piece on the side of the material, like we do with hip rafters, then we would use 54.73561° as the miter angle and 30° as the saw blade bevel tilt. It would result in a compound  bevel angle of 45.00°.

7: Cut one end of each of the 6 edge pieces of the Tetrahedron.

8: Setup a stop on the miter saw. All the edge pieces of the Tetrahedron must be exactly the same dimension.

9: Reverse the edge piece and cut the other end using the stop.

10: Assemble the 6 edge pieces. Assemble the 3 bottom pieces first. This will tell you if your miter angle and blade tilt were exact.

In this photo you can see one edge of the tetrahedron model does not fit perfectly. This is due to the blade tilt or miter angle not being exactly 45°.

These pictures make it look easy to build the platonic solid, but it's not. Assembling the 6 edge pieces perfectly is harder than cutting the 6 edge pieces.

### Platonic Solid Icosahedron

With the material laying flat on the compound miter saw
Your miter angle is = 31.71747°

or you can cut the material on edge in your compound miter saw using
Saw Miter Angle = 58.2825°
Saw Blade Bevel Angle = 18.00°

#### Left and right miter angle jigs for Icosahedron.

The saw miter and saw blade bevel angles for the Icosahedron are beyond the capabilities of most compound miter saws.  You will have to build a jig to cut the edge pieces of the Icosahedron.

It's been a couple of years since I built the Icosahedron, but I think the jig triangles are 30° - 60° triangles and the saw miter angle would be 58.2825° - 30° = 28.2825° with a saw blade bevel angle of 18.00° for cutting the edge pieces on edge in the compound miter saw.

Roof Framing Angles
for the
Great Pyramid

Hip Rafter Backing Angle
33.78620°
also used for the plywood

1. HOLY MOLY. You just made my life so much easier! thank you for sharing this, and explaining it so clearly.

2. I'm wanting to attempt to make a polyhedron for a chandelier like those from restoration hardware any chance you have a tutorial on that.. Or can spell it out to someone who is a visual leaner and bad with math lol.. It would be much appreciated. I looked over some of the others but didn't understand all the terms

3. http://www.restorationhardware.com/catalog/product/product.jsp?productId=prod640007&categoryId=cat1701013

1. Is the chandelier a Platonic Solid Icosahedron?
http://sbebuilders.blogspot.com/2013/03/platonic-solid-stereotomic-descriptive.html
With the material laying flat on the compound miter saw
Your miter angle is = 31.71747°

or you can cut the material on edge in your compound miter saw using
Saw Miter Angle = 58.2825°
Saw Blade Bevel Angle = 18.00°

2. This comment has been removed by the author.

3. This comment has been removed by the author.

4. so i have been studying your diagram.. and i think im understanding maybe a tiny bit.. my question is since the angles its calling for are greater than what my saw can cut to i think it goes to 45.. how do i do that. i was reading in step four for this one you took half of the dihedral angle to cut both sides but on the Icosahedron half of 138.16038 is 69.08019 still more than what my saw can do how do i account for that?

5. It's been a couple of years since I built the Icosahedron, but I think the jig triangles are 30° - 60° triangles and the saw miter angle would be 58.2825° - 30° = 28.2825° with a saw blade bevel angle of 18.00° for cutting the edge pieces on edge in the compound miter saw.

6. thank you so much!

4. Hello, I am planing to do the same chandelier, so any help on the angles would be great!!!

5. Hello, thank´s alot for share this. It´s a beatiful job you made. I need to ask you about the cube, as a way to star a bit "more easy" to me to understand HOW i must cut edges to joint.
I´m gonna buy this days a miter saw with 45º inclination of the plate, and 45º inclination of the head ( where is the blade ).
Here in Argentina - maybe in all spanish language - terms as "gauge" "rafter" "bisel" are difficult to understand.
Can you make for me some kind of reference to show me which one is which cut.
F.E.: to joint as i say the three edges of a corner of the cube. Thanks!!!

6. Truly the best blog I never got such information before this thanks.Masterbuild Roofing

7. Hi Sim,
I´ve been trying to build the Icosahedron with your guidelines, but I must be making a mistake or something, I´m unable to make the triangles meet at their ends, I´ve set up the miter angle with the jig 30-60 and add the 28.2825, set the saw bevel angle to 18, but I´m unable to do it.
I also want to ask you how the jig works because I can cut one side with the jig as the fence, but to cut the other side, I have to flip the piece and put it behind the saw, which I don´t think it´s right or safe,
any suggestions?
Thanks,
Mario

1. It’s been a while since I built the Icosahedron. Are you cutting the material on edge? The miter of 28.2825 is only for cutting the material on edge. For the material lying flat use
Your miter angle is = 31.71747°

Yes, I did cut the material on the edge, but I must be doing something wrong.
If I cut the material lying flat, how do I set up the saw blade angle to 54? do I have to use a wedge or something similar?
is there a way to send you some pictures of what I´m doing?
By the way, you do a fantastic work, congrats!

8. I want to make a tetrahedron out of concrete, instead of just using a cardboard box and setting it with one point down and filling it with concrete to predrawn lines, I would rather cut the "face peices" and use a wooden box so I would be able to pour 4 tetrahedron peices at a time.I tried 30 degrees and that wasn't enough, then built a jig to cut the edges of a wooden triangle at 60 degrees, that was way to much, the final product would be with each of the four faces all the same size and the lengths of each side all the same also, any chance you know the angle I need to cut the edges of my triangle to have it fit flush inside a boxes corner giving me a mold I can reuse ?

9. To better simplify what blade angle I'm looking for (using my table saw) imagine using a peice of plywood and cutting out a triangle with all 3 sides measuring the same, then cutting each of the 3 sides at the correct angle as if you were simply going to glue a base on your tetrahedron with the plywood edges continuing flush with the tetrahedron to the bottom/base

1. Sean, layout the 60-60-60 triangle and use a 54.76 saw blade bevel angle to cut the concrete forms.

10. Thanks so much, very greatfull for your help and knowledge :). Now just one more question, I've been making scaled down replicas of the great pyramid for over 15 years and although I've been fairly successful, I've never actually validated the exact degree of the blade angle for the side cuts of the faces-(just that it's extremely close to 34 degrees) - now upon making ones with the intention of using them for casting resin molds I've come to notice that the closer I can get to perfectly flush the less hassle I will have making them the most efficient. I have made them out of wood for gifts and my own uses.
I have most the angles given from Google memorized but I will start with what might make you not have to research, the triangle faces are 58.3-63.7-58.3 (58.3 being the bottom left and right of it) - the ending face angle when finished is (51.85). So for instance the bottom blade angle cut I use is 38.15 degrees. And like I mentioned I've been "give or take around 34 degrees for the sides" which although sometimes very flush using cabinet grade plywood I still don't know what it is supposed to be legitimate wise as for the math of it goes, and just recently upon making some out of plexiglass the percentage of error is very noticeable, and it would be awesome to know the "exact" intended blade angle finally after so many years. I only found it once being advertised as "55°"/"35°" - (trial and error have gotten me so close, but just using the degrees on the sticker on my table saw is far from exact) - thanks again so much for the 60-60-60/ 54.76° angle as its worked perfectly :) - i know I write to much when trying to ask a simple question so thanks also for bearing with the "novel" -

11. Are there any other dimensions or angles you need me to find out for you in order for you to figure out the blade angle for the "sides of each triangle piece 58.3,63.7,58.3 that will make the scaled down version of the Great Pyramid ?" I forgot to mention the only other angle on the internet that was for the sides of each face (plywood flat on table saw) was 55 degrees and it wasnt at all better then my original 56/34 something ive been "fudging" so far over the years. It would be so nice to finally know the "correct blade angle" Thank you again for the blade angle for the 60-60-60 :)

12. Sean,
I need the slope angle and eave-plan angle of the pyramid.

13. For the blade bevel use one of these
arccos (sin Plan Angle ÷ sin Sheathing Angle)
arccos (cos Common Rafter Pitch Angle ÷ cos Hip-Valley Pitch Angle)
arcsin (sin Common Rafter Pitch Angle × cos Plan Angle)

14. Well I'll have to Google eave plan angle but the face "flat face angle" - the main "sacred" angle of the GP is 51.85 degrees, thank you again :)

15. http://www.cheops-pyramide.ch/khufu-pyramid/great-pyramid/pyramid-dimensions.GIF

16. The 5 angles of the Great Pyramid are:
edge to edge of face at apex =76:17:13.2 (degrees - minutues - seconds),
edge to diagonal edge at apex = 96:3:0.0, dihedral or
face to face parallel to base = 112:25:39.4
edge to base = 41:59:50.5
face to base = 51:51:14.3.
The face to base angle is the angle of the casing stones.

17. I don't know how to comment pictures my normal paste doesn't work, but the above address shows where the 41.9 and other angles are shown on a picture diagram..it's the whole comment ending with .gif as a link to the picture

1. I really wish I new more/anything as far as the hip and valley and actual carpenter/roofing angle descriptions are labeled, I've been pondering the GP angles for 15 plus years but never learned how to distinguish there relative angles into the dialogue of roofers and carpentry in general, very exciting and fustrating at the same time..

18. Great Pyramid
Angle of face - (51°50'40")
Inverse Slope of face - (.785667)
Angle of edge - (41°59'15")
Inverse Slope of edge - (1.111101)

19. Sean,
Take a look at the drawing I just added to this post. The saw blade bevel angle should be 33.78620° for the plywood.

1. I've been looking for the correct blade angles to make a cubeoctahedron and can't find any using a table saw for the life of me. Even in an Archimedean Solids picture of the shapes and miter/blade angles given I am stumped, If you could help me out that would be so great, thank you- also with the miter and bevel angles given I think they are just if your using a compound miter saw, as the miter angles I think need to be doubled for using a table saw, I have searched a lot without results (I know it is made of (8)-"60-60-60" triangles and (6)- squares, if I just had the blade tilt angles I'd be ready to cut, happy holidays :) hope to hear from you soon ! :)

20. Awesome !!! Great picture and thank you so so much I'm so sincerely greatfull for you helping me ! - so many many years I can't believe I now have the most pondered over and intreging blade angle ! So appreciative of your instant responses I commend you very much for the help you provide :) !!! Thanks again I am in awe of your skills and now having the exact angle opposed to "around 34" I can only blame my own skills for my future creations of which I highly dedicate myself to as a perfectionist :) also thanks for bearing with me not knowing the appropriate terms along with the picture you provided me to help me learn them for myself ! much easier then from scratch ! THANKS AGAIN SO MUCH SIM !!!!

21. So I'm wondering if I csn cut 4 triangles to form s tetrahedron using my table saw, knowing the blade angle is limited at 45° does that mean I would need to make a "wedge piece/riser" to have the triangle pieces not laying flat but on an angle ?
for instance if it is 54 degrees for the edges could I set my saw blade to 35.26 and build a "riser" at 19.47 degrees to lay my triangle pieces on ? combining both angles equalling 54 degrees ? - do you think that would work or do you have a easier method ? (I'm using plywood and out of all the platonic solids the tetrahedron is the only one I think I need to make an angled platform to end up with the correct edge angles so 4 triangles will meet with flush edges creating the tetrahedron) I know the dihedral angle is 70.4 or close but unlike the other shapes dividing that in half just gave me a 4 sided pyramid with 60•60•60 triangles and the blade at 35.26, like the top of an octrahedron..., I have found it very confusing being I'm only using a table saw and not a compound miter saw..which I think most of your angles for the solids are refference too. Thank you for your help, and also thanks again as I finished the cubeoctahedron you helped me with just a week or two ago.

22. Hey there! We will built a woodenTetraeder with based on this guideline tommorrow. It will be 4m high! Thanks for this awesome blogpost!!! ⚠️

23. You will find photos of the action on our facebook page: https://www.facebook.com/TetraederCrew/