Truncated Spheres

Discussion in 'My Work In Progress' started by Magic, Dec 21, 2010.

  1. Orangery
    Orangery Member
    Wow, that D33 is the weirdest die I have even seen. I thought my eyes were going funny when I first saw it. I also looked at the D9 and D11... they are also mad things. You seem to be on a roll :laughing: .

    I painted the D10's but I got in a mess with them. I didn't have a gel pen. Will try a different technique.

     
  2. Magic
    Magic Well-Known Member
    :)
    Thanks again.
    If you like my designs (in particular the ones you received), do not hesitate to rate them!
    Good luck with your painting!
     
  3. Magic
    Magic Well-Known Member
    The long awaited Truncated Sphere D20 is now available!
    D20.jpg
    This is the intersection of an icosahedron with a sphere. As you can see, due to the fact that a vertex is surrounded by 5 triangles, the rounded area is larger than in other Truncated Sphere dice. So the faces are smaller and the numbers too.
    The distance from one face to the opposite one is 2 cm. This die is hollow, with a thickness of 1.5 mm (suitable for Alumide for instance).
     
  4. Magic
    Magic Well-Known Member
    I also designed a D10 for percentage and a new version if the D6 (circle faces nearly in contact and wall thickness of 1.5 mm only).
    As a consequence the Truncated Sphere Dice Set is now available !
    SphereDiceSet.jpg
    This set is composed of 7 regular Truncated Sphere Dice:
    - D4
    - D6
    - D8
    - D10
    - D%
    - D12
    - D20
    Ideal for dice collectors and RPG players!

    I will probably create later another set for unusual Truncated Sphere dice (D9, D11, D17 etc.)
     
  5. Orangery
    Orangery Member
    I wish these dice could turn out exactly as the 3D CAD drawings (not a mark on them), :).

    I think there is only the Rhombic Dodecahedron left. I'm assuming it will have different properties than the regular shape?

    The two D10's you made (not the one with the 14 sides) roll really well and for about the same length of time (on average) even though they are constructed quite differently.
     
  6. Magic
    Magic Well-Known Member
    I have seen outstanding results in metal after sanding and polishing, but my post-production skills stop at inking and varnishing plastic... :)

    The Rhombic Dodecahedron is definitvely different from the regular Dodecahdron and will come with the Rhombic Triacontahdron (D30).
    By the way, the Rhombic Dodecahedron is from the same family as the D9 and the future D15 (because 9=3+3+3, 12=4+4+4 and 15=5+5+5)

    I plan also to make a different D8.

    Concerning the D10s, you are speaking of the D10 with 4-fold symmetry (polyhedron with squares and pentagons) and the D10 with 5-fold symmetry (derived from a regular dodecahedron), right?
    Is there a problem with the 3rd one (D10 with 3-fold symmetry, derived from the D14) or is is just that you did not test it?
     
    Last edited: Apr 11, 2011
  7. Orangery
    Orangery Member
    Re D9 family of dice, does this mean there is a D3 (1+1+1)?

    Going off on a tangent, I thought I had read somewhere that the D11 was constructed using the '1+n+n+n+1 formula (1+3+3+3+1). Same with the D17 (1+5+5+5+1). Does this mean that a D5 might be possible (1+1+1+1+1) and also the 'different' D8 you mention (1+2+2+2+1)?

    As for the D10 (fourteen flat sides), there is no problem with it. I didn't mention it simply because it rolls for longer than the other two (as expected). My favourite D10 is the one made up of squares and pentagons.
     
  8. henryseg
    henryseg Well-Known Member
    Have you looked at circle packings on the sphere? That is, finding the arrangement of n identical non-overlapping circles on the sphere so that the circles have maximum radius.

    This website looks like it has some data you might be able to use (although not much in the way of pictures):
    http://www.buddenbooks.com/jb/pack/sphere/intro.htm
     
  9. Magic
    Magic Well-Known Member
    @Orangery No, unfortunately, n cannot be 1 or 2. Or more precisely when it is 2 the polyhedron is degenerated.

    For instance, 1+3+3+3+1 implies that the face of the poles are triangles (because n=3). If you put n=2 then the poles degenerate into simple edges (a polygon with two vertices is a "double-edge") and what you actually get is 2+2+2 which is a simple cube (D6)...

    Same thing for 1+n+n+1 (n=4 gives your favorite D10 :))
    For n=3 you get the regular octahedron (D8) and for n=2 you can consider that the poles degenerate into perpendicular edges and you obtain the regular tetrahedron (D4).

    The new D8 will actually be a 4+4, like the regular octahedron, but with a twist of 45° on the four lower faces.

    I also have some ideas for a D3... ;)

    @Henryseg Yes, I had a look at them and at the repulsion force polyhedra of Martin Trump, but basically I found it difficult to take advantage of this information (maximum angle for the cones or coordinates of the center of the points). So, sometime I use them to fin a good approximate position for the face but then I try to solve equations to maximize the radius of the circles.
     
    Last edited: Apr 13, 2011
  10. Magic
    Magic Well-Known Member
    As announced in the previous message, I am glad to present the Alternative D8 Sphere.
    It is not based on the regular octahedron but on an trapezohedron ("antidiamant" in French) that are dual of the antiprisms.
    D8Alt_poly.jpg

    It is a co-creator, so you can choose your numbering.
    With numbers on edges, the number will be on top of the die, when it lies on a horizontal plane. With the numbers on the bottom faces, you will have to look under the die to find out the result. Choose no numbers if you prefer contemplating the shape or if you would like to take care of the numbering by yourself!
     
  11. Orangery
    Orangery Member
    I like the look of this one. Could this configuration work for a D6 (instead of a cube?).
     
  12. 28396_deleted
    28396_deleted Member
    Very clever
     
  13. Magic
    Magic Well-Known Member
    @Orangery: I knew you would like it: it is like the D10 with 4-fold symmetry but without the top and bottom faces (and the angles are different). Unfortunately for the D6, the trapezohedron that maximize the diameter of the circles is precisely... the cube! So, this method gives nothing new. For n=5, I think you would get the D10 with 5-fold symmetry and for n=6 the rounded part would probably become too large (larger than a face).

    @Dizingof: thanks and welcome back!
     
  14. 28396_deleted
    28396_deleted Member
    Thanks Magic, good to be back ;)
     
  15. Magic
    Magic Well-Known Member
    I am glad to introduce the D15 Sphere:
    D15.jpg
    3 rows of 5 faces (5 at each tropic, 5 at the equator).
    The numbering follow the usual rule (as for the D9 for instance): the sum of each row is constant (it's 40 in this case), the sum of two numbers of the tropic symmetic to the equator plan, and the sum of two numbers of the equator circle symmetric to the "middle number" (8 here) is constant (it's16 in this case). Note that there must be an exception (all the numbers cannot go by pair since there is an odd number of face :)) and that is 8: 8 cannot have a symmetric since the sum of two symmetric numbers is 16 (and only 8+8 = 16).
     
  16. Magic
    Magic Well-Known Member
    This a a die I designed a long time ago and that I forgot numbering.
    So, after a long time on the drawing table, here is the D18 Sphere:
    D18.jpg
    As the number of faces is even, each number is exactly on a face (sometimes, these dice look very usual :)).
    Instead, the numbering is very interesting in this case.
    Opposite faces numbers sum to 19, obviously. As a consequence any group of 8 numbers of a large diameter sum to 76. A more unexpected consequence is that any group of 5 numbers follows this strange rule: if you sum the double of the central number to the 4 other surrounding numbers, you obtain 57.
    Example in the rendering image:
    ...4..
    9 18 5
    ...3..

    2x18 + 4 + 9 + 5 + 3 = 57
    But the most interesting property in this particular numbering is that the sum of the six numbers surrounding a spherical zone is always 57.
    Example here: 18 + 5 + 7 + 11 +13 + 3 = 57
    I had to use a computer to find the good combination (there are a lot of them, by the way).

    I could number this die from 1 to 9 twice (I found a way of doing it where the 8 numbers of the large diameters always sum to 40) or from 1 to 6 three time or even from 1 to 3 six times. Let me know if you want some particular numbering.
     
    Last edited: May 1, 2011
  17. Orangery
    Orangery Member
    I wasn't expecting a D18. I like the way it could be also set up to be a D9, D6 and D3... very interesting!

    Going back to a previous post, I now understand how a cube is made up of six trapezohedrons... but what happens if you stick two (squashed) tetrahedrons together?

     
  18. Magic
    Magic Well-Known Member
    Ah, yes, i understand what you mean. If you stick together two pyramids you obtain a "dipyramid" like this one from Friz. If you squash it and intersect it with a sphere you will indeed obtain a new D6. I had no plan to design this one though, because it is less optimal than the cube: for a given sphere radius, the circles would be smaller. So it is not a solution of the repulsion force polyhedra. But not all my Truncated Dice have as an underlying polyhedron the dual of a repulsion force polyhedron, so perhaps one day, you will see it coming... :)

    I am not forgetting the Rhombic Dodecahdron and the Triacontahedron, but I am currently working on a D24 and some other secret stuff...

    Stay tuned! :D
     
  19. Orangery
    Orangery Member
    I have sent along a couple of pictures of the D10's that I bought from you (3 sorts). The black and white one turned out pretty good but I hope to redo the other two (I didn't have a fine enough pen for the brown die).


    Dice 10.JPG
     
  20. Orangery
    Orangery Member
    As for the D32 football dice, well I must admit I got in a bit of a muddle with the numbering. Some are not even finished. I tried using rubber number stamps for one of them but the white ink did not cover very well and it was difficult positioning numbers :(.

    I used 'Uni Posca' pens for the base colours and the numbering (obviously hand written!).


    Dice 32.JPG