Polyhedral Modeling, Using Steel Balls and Cylindrical Magnets

Many commercial products are available to model polyhedra, such as Zometools, Stella 4d, Polydrons, Astro-Logix, and magnetic spheres which can be assembled into polyhedral shapes, sometimes with brightly-colored struts for the edges of the polyhedron. The first three tools, I can recommend without reservation (and I simply haven’t tried Astro-Logix, yet), but there is a problem with using rare-earth “ball magnets” to model polyhedra: the magnets don’t last long, for, while their magnetic fields are powerful, the neodymium-iron-boron alloy used to make these magnets is not durable, and such spherical magnets break easily.

For this reason, I decided to try a variation of the “ball magnet” idea, and instead obtained some (non-magnetic) steel balls, along with small, cylindrical rare-earth magnets to go between them, thus serving as polyhedral edges, while the steel balls serve as polyhedral vertices. With the steel balls keeping these cylindrical magnets separated (rather than smashing into each other), the magnets are more durable, and the steel balls, of course, do not have a durability problem. Here’s what I was able to produce when I attempted to make a set of Platonic solids, using this method:

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The icosahedron, cube, octahedron, and tetrahedron shown above were easy to make, but attempting to construct a dodecahedron from these materials was an exercise in frustration. Forming one pentagon of this type is easy, but pentagons of this type lack the rigidity of triangles, or even the lesser rigidity of squares, and I was never able to get twelve such pentagons formed into a dodecahedron without the whole thing collapsing into a big ferromagnetic glob, which isn’t what I wanted at all.

Every polyhedron-modeling system has advantages and disadvantages, and the weakness of this particular system was made apparent by my failed attempt to construct a dodecahedron. I next tried adding triangles to pentagons, hoping the rigidity of the triangles would stabilize the pentagons, and allow me to construct an icosidodecahedron, the Archimedean solid which combines the twenty triangles of an icosahedron with the twelve pentagons of a dodecahedron. This method of combining triangles with pentagons did work, and I was able to construct an icosidodecahedron.

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A major advantage of this medium for polyhedral modeling is that it is incredibly economical, compared to most specialized-purpose polyhedron-building tools. The materials are readily available on eBay. Non-magnetized steel balls are much less expensive than their magnetic counterparts; also, small cylindrical magnets are inexpensive as well, especially in large quantities. These will not be the last polyhedra I build using these materials — but they are suited for certain polyhedra, more so than others. With this system, the more equilateral triangles a given polyhedron has as faces, the better, for the rigidity of triangles adds to the overall stability of triangle-containing polyhedral models.

Proposed Mechanisms for New and Different Types of Novae

Theoretical New Type of Nova

The picture above shows a proposed model for the production of a sudden increase in the brightness of a star — or rather, what is apparently a single star, optically, but would actually be a suddenly-produced binary stellar system.

The yellow object is a star, the system’s primary, and it has high mass (at least a few solar masses), when its mass is compared to those of the brown dwarfs in the two highly elliptical orbits shown in blue. These brown dwarfs aren’t quite stars, lacking enough mass to fuse hydrogen-1, which requires 75 to 80 Jupiter masses, but one of them (the larger one) is close to that limit. The smaller brown dwarf has perhaps half the mass of the larger brown dwarf. Their high orbital eccentricities give them very long orbital periods, on the order or 100,000 years. In a very small fraction of orbits, both brown dwarfs will be near perihelion (closest point to the primary) at the same time, and, during those rare periods, the two brown dwarfs become much closer to each other than they are to the primary.

When the two brown dwarfs become close enough to each other, matter from the smaller one could be drawn, by gravity, into the larger brown dwarf, increasing its mass, at the expense of its smaller sibling. At some point, in such a system, the larger brown dwarf’s mass could then reach the threshold to begin fusing hydrogen-1, and “turn on” as a true star — a red dwarf. From Earth, this red dwarf would not be distinguishable from the system’s most massive star, shown in yellow, until much later, when the two moved further apart. There would, however, be a sudden increase in luminosity from the system as a whole. Unlike other types of novae, this increase in luminosity would not fade away quickly, for red dwarfs have very long lifespans. This would enable them, upon discovery, to be distinguished from other single-brightening stellar events. Confirmation could then come from resolution of the new red dwarf component, as it recedes from the primary, making detection easier.

For a variation on this mechanism, the primary star could be somewhat more massive, and the two large brown dwarfs could be replaced by two large red dwarf stars. The larger red dwarf could draw matter from the smaller one, until the larger red dwarf became large enough to cross a higher mass threshold, and brighten substantially, with its color suddenly changing to orange or yellow.

A problem for this model:  no such events are known to have happened. If they do happen, a likely explanation for their rarity is the likelihood that such orbits would be unstable, in a large fraction of similar cases, preventing the stellar-brightening event from having time to happen — in all but a few cases, none of which humans have (yet) both seen, and understood. If one of these things goes off nearby, though, we will learn about it quickly, for it will make itself known.

For another possible mechanism, there is another option:  remove the primary altogether, and let the two objects of near-threshold mass orbit their common center of mass directly. They could then create a new star, or brighter star, by the mechanism described, one which might even produce a detectable accretion disk. A actual merger of the two brown dwarfs, or red dwarf stars, would be a variation of this idea, and would presumably be more likely if the two objects had masses very close to each other, so that neither would have an advantage in the gravitational tug-of-war.

By Request: The Compound of Five Rhombic Dodecahedra, with Nets

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By Request:  The Compound of Five Rhombic Dodecahedra, with Nets

I’ve been asked by a reader of this blog to post nets for this polyhedral compound. Printing nets with Stella 4d is easy, and I’m happy to post them here, in response to that request. Warning, though: there are many nets needed for this compound.

Each of these smaller images may be enlarged with a single click.

Cuboctahedra 5 net one

Here’s the first net type needed (above). You’ll need thirty copies of this net. The gray parts show, and the white parts are tabs to help put it together. Below is the second type needed, of which you need sixty copies.

Cuboctahedra 5 net two

There’s also a third type of net, and these last two types may need to be rescaled before you print them, to fit the net of the first type, also. You’ll need sixty copies of this third net (below) as well, It’s the mirror-image of the net of the second type.

Cuboctahedra 5

Finally, here’s a non-rotating image of the completed polyhedron, to help with the construction:

Cuboctahedra 5

I recommend using card stock or posterboard, and trying to get as much tape as possible on the inside of the model, making an uncolored version — and then painting it with five different colors of your choice, after the model is assembled. Happy building!

[Software credit:  I used Stella 4d:  Polyhedron Navigator to create all these images. It’s available at http://www.software3d.com/Stella.php. Downloading and trying a trial version is free, but you have to buy the fully-functioning version to print nets, or to make these rotating .gif files I post all over this blog.]