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My Three-Sided Dice by DonSimpson My Three-Sided Dice by DonSimpson
Construction of three-sided dice of my design.

There are many ways of making three-sided dice (or "D3s"), and a Google image search on "three sided dice" will show some of them. In the old days, people just used six-sided dice and subtracted three from larger numbers, but many found the idea a design challenge. I came up with this solution to the problem around forty years ago, but I didn't make any until one evening at a poetry reading when I was trying to describe the shape to Professor Nathaniel Hellerstein, and cut one out of a cheese cube from the buffet table. Since then, I've made many of them, mostly of wood, and sold them at conventions and festivals. The shape is difficult to visualize without actually holding one in your hand.

Professor Hellerstein, a professional mathematician, has made many also, using a plastic foam material with glued-on paper faces. He has developed several marking systems (I usually just put notches in the edges of my wooden ones to mark them), and came up with the names "S" and "Z" for the two mirror-image forms this shape can be made in. He has also proposed two questions: If the volume of the cube the die is cut from is 1, what is the volume of the die? What is the shape of one face of the die if it is unrolled to lie flat (he currently approximates this with two joined quarter-circles)?

(Professor Hellerstein also makes "magic" dice, sets of three six-sided dice numbered so that die A statistically beats die B, die B statistically beats die C, and die C statistically beats die A. Yes, this is possible. He has made dice games that use these dice, amazing card games based on similar principles, systems of multi-valued logics, etc.)

I did the views on the left with a free 3D modeling and animation program called Blender, that I am not good at yet. Some better views, done in a program called Rhino, have been created by bladedancer190: bladedancer190.deviantart.com/…

UPDATE 14 April 2014: looking at the current Google images for "three sided dice" I find my design in the top row, someone selling dice of my design on Shapeways (Louis Zocchi of GameScience wrote to me when he was about to start producing his own design, which is different from mine, but maybe this was invented independently), and someone else's patent on a version I rejected as trivially obvious. Also, bunches of other stuff, some of which I think is very interesting.

UPDATE 20 June 2014: Another D3 using this principle is now on Shapeways. It is very stylish, using just the edges (which also saves material costs), and the designer has other art worth looking at: www.shapeways.com/shops/Nvenom…
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:iconposhfrosh:
PoshFrosh Featured By Owner Oct 21, 2016
This is a really neat idea. I've been able to produce the 3d file for this design, but I am not going to post it on shapeways because there is a patent which looks very similar:
www.google.gm/patents/US604211…
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:iconposhfrosh:
PoshFrosh Featured By Owner Oct 30, 2016
Hello, thanks for your reply. Interesting. Well, here is what I came up with following your proposed method. I labelled it as a fudge/fate die (with +, -, and 0) instead of 1,2,3 and it is posted here:
shpws.me/MQl1
It is a nice looking shape, but it seems like it would have to be labelled on the edges since it doesn't land with a face up.
I have come across some other shapeways instances of this shape or something at least somewhat similar. Some seem to have been able to manipulate the shape so that it can be labeled on the faces. I think a user named Magic posted one recently.
I like collecting dice with novel shapes. It sure would be nice to see this shape mass produced.
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:icondonsimpson:
DonSimpson Featured By Owner Dec 16, 2016
Hi! You have a lot of fun and interesting designs for dice.
As I mentioned above, I generally used edge notches (one, two, and three) on the three-sided dice I made. Your +-0 set can be read as scissors, paper, rock; which I like. Triangle, square, circle works similarly. If the edges were wide instead of sharp, and rounded so that a die couldn't balance on them, they could be edge-marked in many different ways, but wouldn't be purely mathematical like my original design. Professor Hellerstein's markings included colors, which might require using a multi-color material or having snap-in color parts.
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:icondonsimpson:
DonSimpson Featured By Owner Oct 22, 2016
The die in that patent has three sides and is tapered, but it's made by cutting a three-sided prism with parabolic curves normal to the prism faces. Mine is made by three quarter-circle cuts to a cube, with the cuts at three right angles.
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:iconposhfrosh:
PoshFrosh Featured By Owner Oct 31, 2016
In addition to the fudge/fate version i mentioned, I've also posted a regular pipped version (shpws.me/MRGq). I've linked back here from both shapeways pages. Would be nice to see this design injection molded.
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:iconkevincook:
KevinCook Featured By Owner Nov 25, 2010
Don .. have you ever considered 3D printing your design?

I believe that companies like Shapeways(.com) can use blender images ... I KNOW they can use Google Sketchup
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:icondonsimpson:
DonSimpson Featured By Owner Nov 25, 2010
I've considered it, but I haven't investigated it yet; other things have had priority.
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:iconkasz:
Kasz Featured By Owner Oct 13, 2010
Now that I look back at this.... how do you tell which face is up after a roll?
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:iconkevincook:
KevinCook Featured By Owner Nov 25, 2010
This photo may help .. this is one of Professor Hellerstein's designs

[link]
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:icondonsimpson:
DonSimpson Featured By Owner Oct 13, 2010
After a roll it has an edge upward, and I generally mark the edges with one, two, and three notches, but there are lots of possible ways to mark the edges, or the parts of the sides near the edges (colors, words, numbers, etc.).
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:iconkasz:
Kasz Featured By Owner Aug 13, 2010
I'm assuming Professor Hellerstein's D6s are weighted to offer the statistics described. What kind of games did he come up with to work with them? This actually sounds like a lot of fun.
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:icondonsimpson:
DonSimpson Featured By Owner Aug 14, 2010
No, they are normally weighted. It's which numbers go on which faces that does the trick. Hellerstein calls his three dice Red, Yellow, and Blue, and each uses three numbers, with each number being on two of the six faces:

Red: 3 3 5 5 7 7
Yellow: 2 2 4 4 9 9
Blue: 1 1 6 6 8 8

Red beats yellow, which beats blue, which beats red, each with a probability of 5/9 (five throws out of nine).
Note that each number from one to nine is used, and no die has a number used on one of the other dice. I'd have to dig out the game rules I got from him, and I don't know offhand where they are.

One of his marking systems for three-sided dice uses red triangles ("scissors"), yellow squares ("paper"), and blue circles ("rock"), so that each possible throw beats one of the others, and is beaten by one. Another of his systems just marks the edges with the three colors, instead of having two colored shapes on each side, next to the edges.
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:icondeath-pony7:
Death-Pony7 Featured By Owner Aug 13, 2010  Professional General Artist
I thought this was impossible, but you have proved me wrong. Amazing design! I need to show this to one of my D&D friends.
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:icondonsimpson:
DonSimpson Featured By Owner Aug 14, 2010
There are some things that sound impossible, but aren't, like the Szilassi polyhedron (I have a model of that, made by cutting out the faces from plastic sheeting. I used to do a lot of polyhedron models that way.)

Lots of game stores carry D3s made in other shapes, which I gather work fine, though I consider them mathematically less elegant. Lou Zocchi, of GameScience, wrote to me to be sure his D3 design didn't conflict with mine. He has created some interesting dice. His seven-sided die is a length of five-sided prism proportioned to be statistically correct by testing on a die-throwing machine. And he has a D100 he calls the Zocchihedron.
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:icondeath-pony7:
Death-Pony7 Featured By Owner Aug 14, 2010  Professional General Artist
My mind has officially been blown!
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:iconbladedancer190:
BladeDancer190 Featured By Owner Aug 11, 2010  Hobbyist Photographer
Huh, that's an interesting die. The idea of projecting a 3 sided die on a 6 sided net is odd, but I think I can see how it folds up.
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:icondonsimpson:
DonSimpson Featured By Owner Aug 11, 2010
It's not a projection, it's a prescription. :) If you mark an actual cube of something that way, and make the three cuts, _then_ you get the die.
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:iconbladedancer190:
BladeDancer190 Featured By Owner Aug 12, 2010  Hobbyist Photographer
Huh, OK. So, the trick is changing the curve of the cut correctly as you go along?
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:icondonsimpson:
DonSimpson Featured By Owner Aug 12, 2010
No, it's three identical, simple, quarter-circle cuts. The trickiness is that when you have made the first cut, you have a quarter-cylinder, and then you have to cut that as though the whole cube was still there, and then you have an odd shape and have to cut _that_ as though the whole cube was still there. If you were here, I could demonstrate it in a couple of minutes, but trying to _describe_ it is a pain. If I can just get better with Blender, the program I used to make the views on the left, I will be able to do 3D renderings of the intermediate steps, which should help. I hope. The only way I could get it across to Nathanial was to slice it out of a cube of cheese right in front of him.
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:iconbladedancer190:
BladeDancer190 Featured By Owner Aug 13, 2010  Hobbyist Photographer
Huh, OK. I think I get what you're saying. It's obviously hard to visualize, without the aid of cheese (both delicious and useful) but I think I get it. Heck, I could probably take a stab at rendering it in Rhino.
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:icondonsimpson:
DonSimpson Featured By Owner Aug 13, 2010
I hadn't heard of Rhino, but I Googled it and it looks like it is about a hundred times as powerful as would be needed to do the job. :) I used Blender because its free. Please let me know the results.

I made a cube of side 1, and then a cylinder of radius and length 1. I put the axis of the cylinder on an edge of the cube and did a boolian to get a quarter-cylinder, did a couple of duplicates of that, and then rotated and moved them and did boolians to get the die.
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:iconbladedancer190:
BladeDancer190 Featured By Owner Aug 14, 2010  Hobbyist Photographer
Yup, that's about what I just did. It looks pretty cool, really. All spirally.
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:icondonsimpson:
DonSimpson Featured By Owner Aug 14, 2010
I want to see your Rhino pictures.
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(1 Reply)
:iconstormis:
stormis Featured By Owner Aug 11, 2010  Hobbyist General Artist
Interesting. the S and Z diagrams, they could be used as a guide to sculpt the die from a cube?
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:icondonsimpson:
DonSimpson Featured By Owner Aug 11, 2010
That is indeed what they are for. Three quarter-circle cuts (making the cube into a quarter-cylinder), each from a different direction, produces the die. the first cut is easy, then it can get tricky, but I can manage it, and hope others can, too. :)
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