Musings on rigidity, the d30, a.k.a. Rhombic Triacontahedron, and the Buckyball / Soccerball
Inspired by a comment I made elsewhere.
The talk of rigidity is very important in these toymag structures. Only triangles are rigid. I believe the rigid triangle is the basis for much of Buckminster Fuller's work -- I intend to read a lot more about his work here (looks so fascinating!). So any structures with faces that are nontriangular will need bracing of some sort -- either enough crossbars to turn the face into a series of trianges, or with structures build on those faces consisting of triangles, or with a solid face (like the "panels" available for Geomags).
otherbill wrote: The d30 is an interesting structure.
Oh, this object is perhaps my favorite polyhedron! It's the rhombic triacontahedron. I've gone on about it in two recent posts in my LJ (here and here). The first of those two posts was devoted to creating the rhombic triacontahedron (and/or something like it) in Geomags. I knew this d30 wouldn't be stable ('cause of the rhombic faces -- only triangles are stable). I wanted to see how close it was to a 60-sided figure that I'd pictured in my head for years (similar to the top center diagram on the right) that I thought would be close to that d30 -- a dodecahedron with pentagonal-based pyramids attached to each of its pentagonal faces. Ohhh, it turned out cool!
But of course it wouldn't work as a die, since the structure isn't convex. But if you didn't make the pentagonal pyramids out of equilateral triangles (like using Geomags forces you to do), you could choose sides for the pyramidal edges (the edges that connect to the apex of the pyramid) that were shorter -- pick the right length and adjoining triangles from adjacent pentagonal pyramids could be made to be coplanar, yielding the d30 (rhombic triacontahedron). And a lesser length than that would give you a convex polyedron that could be used as a d60.
Funny that otherbill mentioned the soccerball / buckyball / 32-sided object. Its "math name" is truncated icosahedron. On New Year's Eve, I brought my Geomags to the party I went to. I was rebuilding that augmented dodecahedron I mentioned above, when the guy who threw the party asked me to make a buckyball instead. I refused, since it would take too long, and finished my littler project. But I was curious, as you apparently otherbill is too: could the geomags handle the weight of a full buckyball construction?
So a couple of days after New Year's Eve, I decided to build it myself. I used the exact same technique you mentioned, a soccerball pattern with pentagons replaced by pentagonal-based pyramids (I made them point inwards towards the center in my construction), and each hexagon (mostly) stabilized by adding six internal struts to form a mesh of six triangles. And the cool thing is: It is a stable Geomag construction -- it doesn't actually collapse under its own weight! These Geomags are so strong and cool! It is pretty tough to pick up without damaging it (best to use two hands, cupping it), but it does work! Later, I found a website that shows someone else built it exactly the same way too (with the inward pyramids as well): picture here. There's lots more stuff on his website devoted to polyhedra construction! The truncated icosahedron is pictured on his Archimedean Solids page.
Pictures soon... Real Soon Now....
The talk of rigidity is very important in these toymag structures. Only triangles are rigid. I believe the rigid triangle is the basis for much of Buckminster Fuller's work -- I intend to read a lot more about his work here (looks so fascinating!). So any structures with faces that are nontriangular will need bracing of some sort -- either enough crossbars to turn the face into a series of trianges, or with structures build on those faces consisting of triangles, or with a solid face (like the "panels" available for Geomags).
otherbill wrote: The d30 is an interesting structure.
Oh, this object is perhaps my favorite polyhedron! It's the rhombic triacontahedron. I've gone on about it in two recent posts in my LJ (here and here). The first of those two posts was devoted to creating the rhombic triacontahedron (and/or something like it) in Geomags. I knew this d30 wouldn't be stable ('cause of the rhombic faces -- only triangles are stable). I wanted to see how close it was to a 60-sided figure that I'd pictured in my head for years (similar to the top center diagram on the right) that I thought would be close to that d30 -- a dodecahedron with pentagonal-based pyramids attached to each of its pentagonal faces. Ohhh, it turned out cool!
But of course it wouldn't work as a die, since the structure isn't convex. But if you didn't make the pentagonal pyramids out of equilateral triangles (like using Geomags forces you to do), you could choose sides for the pyramidal edges (the edges that connect to the apex of the pyramid) that were shorter -- pick the right length and adjoining triangles from adjacent pentagonal pyramids could be made to be coplanar, yielding the d30 (rhombic triacontahedron). And a lesser length than that would give you a convex polyedron that could be used as a d60.
Funny that otherbill mentioned the soccerball / buckyball / 32-sided object. Its "math name" is truncated icosahedron. On New Year's Eve, I brought my Geomags to the party I went to. I was rebuilding that augmented dodecahedron I mentioned above, when the guy who threw the party asked me to make a buckyball instead. I refused, since it would take too long, and finished my littler project. But I was curious, as you apparently otherbill is too: could the geomags handle the weight of a full buckyball construction?
So a couple of days after New Year's Eve, I decided to build it myself. I used the exact same technique you mentioned, a soccerball pattern with pentagons replaced by pentagonal-based pyramids (I made them point inwards towards the center in my construction), and each hexagon (mostly) stabilized by adding six internal struts to form a mesh of six triangles. And the cool thing is: It is a stable Geomag construction -- it doesn't actually collapse under its own weight! These Geomags are so strong and cool! It is pretty tough to pick up without damaging it (best to use two hands, cupping it), but it does work! Later, I found a website that shows someone else built it exactly the same way too (with the inward pyramids as well): picture here. There's lots more stuff on his website devoted to polyhedra construction! The truncated icosahedron is pictured on his Archimedean Solids page.
Pictures soon... Real Soon Now....