Timeline for Fair but irregular polyhedral dice
Current License: CC BY-SA 2.5
14 events
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| Apr 26, 2018 at 23:34 | comment | added | Joseph O'Rourke | @ZachTeitler: We did print various height pentagonal cylinders and collected statistics on rolls, but we never surmounted the problem of randomizing the die throws uniformly. So: No written report. But I still have the cylindrical dies. :-) | |
| Apr 26, 2018 at 21:10 | comment | added | Zach Teitler | @JosephO'Rourke Were you able to carry out the planned study in 2011? Do you have a report, or any results? | |
| Jan 1, 2011 at 2:10 | vote | accept | Joseph O'Rourke | ||
| Dec 16, 2010 at 12:02 | comment | added | Joseph O'Rourke | @Bill: One printer prints ABS (hard) plastic, which should suffice. We'll work in Mathematica and export in a format the printer can accept. | |
| Dec 16, 2010 at 3:48 | comment | added | Bill Thurston | @Joseph: What a nice idea! I bet the undergraduates will have some good ideas. Will they have software good for designing the shapes? How bouncy are the materials you can use for your 3D printers? | |
| Dec 15, 2010 at 22:43 | comment | added | Joseph O'Rourke | This is not an ideal forum to mention this, but because of the interest raised by Bill's impressive reflections on the problem: (1) I purchased and control two 3D printers, and (2) I intend to print and experimentally investigate (with undergraduates) candidate fair but irregular polyhedra in the (U.S.) spring 2011 semester. Any and all ideas/advice would be greatly appreciated! Thanks! <[email protected]> | |
| Dec 15, 2010 at 15:09 | history | edited | Bill Thurston | CC BY-SA 2.5 |
Added a bit on rolling modes.
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| Dec 15, 2010 at 14:38 | comment | added | Bill Thurston | A "cubical" die is of course not exactly cubical: the corners are rounded. How much rounding? One limiting case would be the intersection of 3 orthogonal cylinders, with corners shaped like small pieces of spheres. Now make the cylinders slightly squarish in cross-section, so the only stable configurations are on the faces. In this exaggerated form. Dice near the limit would seem to guarantee the phase space splitting as I described (with other orbits like rocking between corners). Another limiting case is an actual cube. The degree of rounding appears to make a difference. | |
| Dec 15, 2010 at 12:24 | comment | added | Joseph O'Rourke | @Bill: Your ruminations deepen and enrich the problem! Even thoroughly understanding the phase space and dynamics of a cubical die is suddenly interesting and challenging. Thanks! | |
| Dec 15, 2010 at 9:36 | comment | added | Bill Thurston | @sleepless in beantown: The partitioning of phase space is something you have some freedom of, in constructing a proof: the part important to the dynamics is what happens at the critical levels when connectedness changes. If you shake the die in your hand before throwing, then it's not obvious that a low energy level of the throw causes bias, but if the die is not symmetrical, it could, even if it is fair at higher energy levels. But: there should be a weaker set of conditions that are sufficient. In particular, the dynamics needs to be chaotic enough times, not always. | |
| Dec 15, 2010 at 6:34 | comment | added | sleepless in beantown | Perhaps I should use "fractal" instead of chaotic as the more appropriate word for the boundaries at higher energy levels. | |
| Dec 15, 2010 at 6:32 | comment | added | sleepless in beantown | (continuation) distributed (in regards to solid angle=steradians) phase-space of orientation alone, c) determine if there is then a bias of which faces are most likely to be the "face down" (or face-up) resting position. I like this phase-space decomposition break-down which you've suggested. I would conjecture that the partitioning of phase space may be "clean" and well defined at low energy levels, but may become chaotic at larger energy levels. Of course, this depends on the model's elasticity in collision and the coefficients of friction used (for static and rolling $\mu$) | |
| Dec 15, 2010 at 6:26 | comment | added | sleepless in beantown | @Bill-Thurston, of course, throwing a die with the minimal amount of energy such that it cannot overcome the threshold of "switching the most likely face after initially landing" is one of the key ways of cheating when throwing a die, regardless of whether it's fair or unfair. Every throw will have to have a key-axis of rotation such that there will be two (probably disjunct) sets of preferred/likely faces and non-preferred/unlikely faces. My thought on the problem is that: a) given a fixed "throwing schema" of translational and rotational velocities... b) evaluate over an evenly (cont.) | |
| Dec 15, 2010 at 3:10 | history | answered | Bill Thurston | CC BY-SA 2.5 |