Conclusive, from the horse's mouth?
Jul. 9th, 2007 07:07 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackWell, one or other end of the horse, anyway :)
This probably won't be the end. But I finally stopped thinking about actual instantiations, and stepped back to consider what minimal theoretical conditions the d7 I want must satisfy.
To be theoretically fair, you need (a multiple of) seven orientations, each of which is equivalent to any of the others. Whether those are faces, or edges, and whatever other features there are, the fact that any must be equivalent to any of the others means they must be a symmetry group.
I think a "kind of spherical" die could be defined as one that has more than one axis of >2 rotational symmetry. Some messing around makes this seem unlikely (if you have two, the second must map to six more under rotation by the first, and vice versa, and those do not look like they can overlap) .
But fortunately, this is a completely solved problem, witness an exhaustive list from wikipedia: http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions.
The only shapes with more than one genuine axis of symmetry correspond to the platonic solids. (Not just those, eg. a cube made up out of parallelagrams and other interesting dice qualify too.)
But there are no sevens to be seen. Thus, for 3d dice, without messing around with dynamics, the only ones with a multiple of seven can be around one axis, eg:
* The pencil, with ends flat, or curved to a point, or with seven-sided pyramid points.
* A seven-sided dipyramid, a la the octohedron for 4. (Seven edges round the middle, forming seven triangular faces with the points at the top, and the same at the bottom.)
* A twice-seven-sided Trapezohedron, a la d10. (Fourteen zig-zag sides round the middle, forming seven kite-faces with the point at the top, and the same at the bottom.)
* Gyroelongated_dipyramid, one of those padded out with more triangles round the middle. This probably looks most spherical, I'd like one of these.
NB: The actual 3d solution used is a pentagonal prism, which is nearly fair but not theoretically so.
This probably won't be the end. But I finally stopped thinking about actual instantiations, and stepped back to consider what minimal theoretical conditions the d7 I want must satisfy.
To be theoretically fair, you need (a multiple of) seven orientations, each of which is equivalent to any of the others. Whether those are faces, or edges, and whatever other features there are, the fact that any must be equivalent to any of the others means they must be a symmetry group.
I think a "kind of spherical" die could be defined as one that has more than one axis of >2 rotational symmetry. Some messing around makes this seem unlikely (if you have two, the second must map to six more under rotation by the first, and vice versa, and those do not look like they can overlap) .
But fortunately, this is a completely solved problem, witness an exhaustive list from wikipedia: http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions.
The only shapes with more than one genuine axis of symmetry correspond to the platonic solids. (Not just those, eg. a cube made up out of parallelagrams and other interesting dice qualify too.)
But there are no sevens to be seen. Thus, for 3d dice, without messing around with dynamics, the only ones with a multiple of seven can be around one axis, eg:
* The pencil, with ends flat, or curved to a point, or with seven-sided pyramid points.
* A seven-sided dipyramid, a la the octohedron for 4. (Seven edges round the middle, forming seven triangular faces with the points at the top, and the same at the bottom.)
* A twice-seven-sided Trapezohedron, a la d10. (Fourteen zig-zag sides round the middle, forming seven kite-faces with the point at the top, and the same at the bottom.)
* Gyroelongated_dipyramid, one of those padded out with more triangles round the middle. This probably looks most spherical, I'd like one of these.
NB: The actual 3d solution used is a pentagonal prism, which is nearly fair but not theoretically so.
no subject
Date: 2007-07-10 10:09 am (UTC)Only really been glancing at these not thinking about it fully, so not really sure if this has already been mentioned.
Just a thought (not as interesting as a 3D D7):
In some childs games (especially cut out and put together ones) you used to get spinning tops instead of dice, which in effect reduces the problem to a 2D one. Wouldn't, therefore, a heptagonal spinning top give a simple solution to the problem?
no subject
Date: 2007-07-10 01:08 pm (UTC)