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Lectures On The Icosahedron
It's all the mods' fault. One of the pics in Camp
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sick_wilson Pic Tac Toe Challenge, which looked to me like a billiard ball, turned out to be a Magic 8 Ball. Something whose existence I was totally and blissfully unaware of.Wikipedia informed me that inside the Magic 8 Ball there's just a simple, 20-faced, icosahedral die.
If you throw a usual, cubic die you have six faces, and six possible results. So why twenty? Good number, but could we make it with 15, or 33 faces?
The answer is no, at least if you want a honest die: in other words, if you want a so-called regular polyhedron.
When we draw on a piece of paper we can produce regular polygons with as many sides as we want: an equilateral triangle, a square, a pentagon, an exagon… after that the words become complicated (octagon and decagon I can still spell easily), and the question of how to make the drawing gets more complicated (the answer, as usual, is Google Images) but there's no limit to your fancy, and you can produce a regular polygon with 2000 sides if you want - except of course, it would like like a circle to me, and probably to you as well. Regular, by the way, means "seriously democratic": all sides look precisely the same (yes, there's a technical definition, but I don't want to write it).
Once we go to three dimensions, triangle and square have easy equivalents. The triangle becomes a tetrahedron (tetra means four, four vertices, four faces) which is a pyramid with base an equilateral triangle and whose sides are also equilateral triangles. The square becomes a cube, and I hope you all know what a cube is (although most broth cubes I know aren't cubical). It turns out that these two kinds of democratic arrangements exist in any number of dimensions - whatever that's supposed to mean.
But then things get iffy. There's only three more
Not only long and diligent search has failed to produce other examples, but we know that there are none. I learned how to prove this from a book at age 17 (the proof is easy once you know the trick) but the fact in itself was known already in ancient Greece, and influenced Greek philosophers no end: the popular name for all the five regular polyhedra is Platonic solids, because Plato liked geometry and got as much of a kick out of these five guys as I do, and probably more.
So why I am telling you that? Just because an icosahedron is something you can easily build if you have a magnetic construction kit. It's totally easy, and looks cute. You start by assembling a triangle. It becomes equilateral naturally, all the bars have the same length. Then you build another one with a side in common. Then you build a fan, with three equilateral triangle sharing a vertex. Then you add a fourth. And then you make a pyramid with a five-sided basis, just by connecting the free corner of the first triangle in the fan with the free corner of the last. Then you keep going, repeating the same pattern, and lo and behold, when you have used 12 balls and 30 bars, you're done and you get a cute and very stable structure, with no effort on your part.
No thinking required. [If you start with a fan with three triangles, and add a bar to make a pyramid with square basis, you have half an octahedron - to make a full one, add another pyramid on the opposite side of the basis.]
Of course in principle you can build an icosahedron, or any regular polyhedron, with any material you can easily cut: I once figured out a homework assignment in solid geometry by carving a cube of pecorino cheese - once I got the solution I had a very tasty snack. A cool thing you can do is make an icosahedron of some black material, paint it white, and then cut off all the sharp vertices (the tips where the balls would be if it were a magnetic model) so that some black shows up. And you would get…
PS: The title is from a very famous book, Vorlesungen über das Ikosaheder (1884) written by a very, very important German mathematician, Felix Klein (1884). The current school system keeps your mathematical education more than a century behind reality, sorry.
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In season one's episode "Love Hurts", House does play with a Magic 8 Ball. It is fitting because those silly things were a staple of American childhoods for many years. In the 70's and 80's every kid had one, sort of like a slinky.
Ooohh, that gives me an idea.....
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Yes. The only numbers you can get are 4, 6 (the usual die), 8, 12 and 20. I wish they would teach this stuff at school instead of trigonometric inequalities.
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I wish high school mathematics which teach basic statistics and probability. And then follow that with a financial literacy course.
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Still, a little bit of beauty has to be taught as well.
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There's a proof already in Euclid :-)
And did you know that the soccer ball was a cut icosahedron? I certainly didn't learn that at school!
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That's precisely what you're supposed to do! Mathematics is like music, drawing or sports - you can be a spectator and it's pleasant, but there's a lot of fun in doing it yourself. [Or at least, so I'm told about music, drawing, and sports./tmi]
Sorry about the missed work hours :)
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(I'm still not sure what the pink bird in the middle is - flamingo?)
Congratulations on becoming a Wilsoneer :)
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They had a brief return to fashion in the 1980s (that decade was to blame for a lot), and some people still have a weird affection for them. There even are places online where you can buy outfits for your lawn flamingos to wear. People have been known to kidnap other people's pink flamingos and then send pictures from the place/s the flamingos allegedly "escaped to", with the flamingos posed prominently in the picture.
At this point, pink flamingos in the US are kind of a cliche, shorthand for "tacky, but cute".
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Theres a place in Western Australia where people take their garden gnomes and leave them, apparently there are thousands there now, on holiday...
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Thanks!
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Sorry for throwing mathematics at you early in the morning, hope it didn't spoil your breakfast :).
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It was nice to meet them all again! And I loved the soccer ball -- it had never occurred to me that it was an icosahedron in disguise.
PS -- I think the item you mention as "an exagon" would, in English, be "a hexagon". I also remember having to write out a list of all the regular two-dimensional figures up to ten sides: triangle, square, pentagon, hexagon, septagon (a.k.a. "heptagon"), octagon, nonagon, decagon. What use has this knowledge been to me? Well, I was able to use it in this comment -- and that's been about it! :^)
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it had never occurred to me that it was an icosahedron in disguise
It never occurred to me either until I was told. I have zero intuition for 3D geometry.
"a hexagon"
I'm past blushing about missing initial H letters in my English/French/German spelling. I'm proud enough that at least I remembered the X - in Italian it's un esagono and of course I learned about regular polygons (poligoni, note the lack of Y) before I learned English. Thanks, and glad for your knowledge to have finally found some use!
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That pic of the icosahedron is really cute. I've seen those in a lot of places.
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