come_to_think

My success with the tensegrity icosahedron reminded me of another geometrical amusement I heard about a long time ago: If you inscribe a pentagram in each face of a regular dodecahedron, the resulting 60 line segments form 10 regular hexagons centered on the dodecahedron. I made such a thing out of 10 colors of construction paper about 40 years ago & gave it to my mother, but it struck me that if I made the dodecahedron edges out of transparent drinking straws & the hexagons out of 10 colors of chenille stems, you'd be able to see all of each hexagon at once. I calculated that if I bought 3 of the mixed-color packets of stems, I'd have enough for 10 different hexagons, so I went to Windsor Button, and damn! they were out of those packets, and wouldn't have any for at least a month; and they didn't have enough different packets in single colors. So I went on the Web & ordered 10 different-colored packets from an outfit in Florida that shall remain nameless. It sent me 7 of them & refunded me for the rest on the grounds that the remaining 3 were too little to backorder. (There was boilerplate on the invoice that contradicted that, but a telephone call yielded the information that I was supposed to ignore that.) Bad attitude. So today I went back to Windsor Button, and, mirabile dictu, they had packets of 3 colors other than the 7 I already had.

On top of that, I went to the Fens, and several men came on to me, and I got off & got to feel useful as well.

However, despite much trying, I did not manage to fine any clear drinking straws. The Web doesn't look promising either.

Nevertheless, it was a good day, as days go. I lit the candle at dinner to celebrate.

COMMENTS:

come_to_think 68.160.178.193
2009-10-16 08:45 pm (local)

I got some clear plastic tubing that does the job.

However, despite much trying, I never managed to get the thing put together. I got confused again & again. Finally, after making a careful drawing, I managed to convince myself that it was impossible. But that is crazy, because I made a model about 40 years ago by making 10 hexagons of different-colored construction paper, cutting them up sufficiently, and gluing the pieces together. Many things have changed in the last four decades, but surely not solid geometry.

come_to_think 68.160.178.193
2009-10-18 10:18 pm (local)

I had the idea wrong. The pentagram is inscribed in the face of the dodecahedron with its vertices at the midpoints of the edges, not at the vertices of the dodecahedron. Now it looks to me as if I can make the whole thing by weaving together 10 chenille stems.

come_to_think 68.160.178.193
2009-12-16 09:49 pm (local)

It took a long time, but I managed at last to put the thing together out of 10 different-colored chenille stems, tying joints together with thread. I was stalled for a long time because I wanted to make all the pentagrams elegant, with consecutively overlapping segments; but I kept getting confused, and eventually gave up & let the intersections be random.

However, the result is not impressive. One can see the pentagrams if one looks for them, but the pentagons in which they are inscribed, which make up the dodecahedron, are not visible. What is more, there is no easy way to make them visible. That is because (as I mentioned discovering) the vertices of the pentagrams are not the vertices of the pentagons, but the midpoints of their sides. So I will have to find some thin, stiff stuff to make the dodecahedron out of & then tie the midpoints of its edges to the existing vertices.

come_to_think 68.160.178.193
2010-01-05 06:29 pm (local)

The thin, (fairly) stiff stuff turned out to be in one of my hellboxes: copper wire with the insulation stripped off. I bent & soldered it into a dodecahedron around the hexagons. It does frame the pentagrams nicely.