Graph theory

[johnckirk]

I've recently been going through my LJ archive and "tagging" my old posts, and that is now basically done. There are some cases where I've introduced a new tag after referring to the subject a few times, so the previous posts now have an incomplete set of tags, but this is good enough for now, and should make it easier to find old posts

Comments

[pozorvlak]

I got it by random poking (drew first vertex, fanned out, always leaving myself some space) on the second go, but it felt like the answer should have been obvious for some larger reason. I'd wondered about the numbers 5 and 12, but hadn't made the connection with Euler's formula. Gah! It's the dual graph of the regular dodecahedron, which you can project onto a plane using stereoscopic projection from the sphere.

Stapler story: this is why you should always carry a penknife :-) Incidentally, if you're still wondering whether to buy one, I can make the "how many blades" question easy: you want an Explorer Swiss Army Knife. Best blade combination at a sensible size in their range. Try the linked site, or eBay usually have them. Or Scout Shops, which often give student discounts. Or if you're feeling a bit richer, the Leatherman Juice XE6 looks very nice... eBay could be cheaper, but they don't turn up all that often IIRC. The problem with most Persons of Leather is that the blades lock, so it's actually illegal to carry them around

[pozorvlak]

Incidentally, I used the wire-stripper on my Swiss Army Knife for the first time the other day. It's surprisingly effective...

[johnckirk]

It's the dual graph of the regular dodecahedron, which you can project onto a plane using stereoscopic projection from the sphere.

Yup, that sounds about right :) At the bottom of this page, it says "On the sphere, 4 vertices implies an average degree 3 (tetrahedron), 6 implies 4 (octahedron), 12 implies 5 (icosahedron)". I can certainly recognise the neat diagram that I drew for T3 as a tetrahedron, and it seems reasonable that my diagram for T4 is an octahedron (assuming a bottom face in 3D), although I'm not familiar enough with that shape to recognise it, and it didn't match any of the diagrams here. As for the icosahedron, I don't even recognise that word, although "dodecahedron" sounds familiar (in the context of pentagons/hexagons). We covered stereoscopic projection in the lecture course, although it does seem to hinge on choosing the right point on the sphere to project from.

As for the penknife, I do now have a small one (courtesy of princesslen) with one blade - I'm not sure what happened to my old Swiss Army Knife, unless it's

[pozorvlak]

Ah, of course, it is an icosahedron. An icosahedron is a polyhedron with 20 faces, all of which are triangles. It's dual to the dodecahedron, in the sense that if you put a vertex in the middle of every face of a dodecahedron and an edge crossing every edge of the dodecahedron, you get an icosahedron (and vice-versa). Similarly, the cube and the octahedron are dual to each other. The tetrahedron is self-dual. As for the octahedron: take the four vertices in the middle, and arrange them in a square. Now put the other two on top and underneath, to form a pair of pyramids joined at the base.