stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2008-11-15 10:59 am
stoutfellow) wrote2008-11-15 10:59 am
Entry tags:
I'm Not Going to Try to Explain This
Wednesday, at the seminar, Prof. W introduced an odd intriguing problem, involving finding shortest paths on the surface of a polyhedron. Thursday, Prof. T asked that we meet again on Friday; she'd found something interesting. We met in her office. Under the cut, some bits of the discussion.
We met in T's office because she has all the neat models, and some of what she wanted to show us was on her computer.
T is holding a skeletal model of a dodecahedron. There are a couple of pieces of string wrapped around it, at odd angles. The computer shows an exploded view. On her desk is what looks like an origami chrysanthemum, except that the petals are all pentagons. There are lots of them. W ruffles it with a finger. "Is there any kind of patterning here?" T changes the picture to an exploded view of the chrysanthemum. Pentagons all over the place, overlapping each other. W points at the screen. "For instance, does this area" - there are no pentagons there - "stay clear?" T plays with the mouse, and another pentagon appears where he pointed. "Rats."
"Are the vertices discrete, at least?" Prof P grunts negatively. After a moment, he says, "Look at the centers. Two flips looks like a translation, and in five different directions." Discussion continues. "Zeta plus zeta to the fourth is the golden ratio? I didn't know that." "Yeah, so it's going to be dense along the real axis, and in five other directions. Combine that, and it's dense all over." "Hmph."
:Later:
W has a skeletal model of a cube, and he's labeled the vertices. He's rolling it around on the desk. "I'm wondering how many orientations you can get on the bottom. No more than twenty-four...." "Can you get adjacent sides?" "Yeah." He demonstrates. "Then it's at least six. Six, twelve, or twenty-four... I wouldn't be surprised if it's twelve." "Why?" "It takes an even number of rolls to get back where you started. That smells like A4 to me." W cocks his head doubtfully. P: "Look. All the rolls are conjugate." "Why?" P waves his hands distractedly. "Because you can't tell the sides apart."
W pulls out a manuscript and flips through it. He points. "This is what you're talking about." "Yeah. Let's see: One. Four. Three. Two. It's a four-cycle, and so is the other one. So it's A4." "Have we proved that? I think we just proved that."
With T's permission, W clears part of the whiteboard and draws twelve parallel lines, then starts linking them with stretched-out X's. "You've got ramification at the vertices.... It could be nastier; this could go with that and that...." "Yeah, it's non-abelian, so...."
:Later; W has gone to get a flu shot.:
P is playing with a skeletal octahedron. "One, two, three, four, five, six. What just happened? I can't tell." T hands him a piece of tape, which he affixes to one vertex. "One, two, three, four, five, six. Did anything happen?" "No, I think the taped vertex was toward you when you started." "Huh. Then it's four.... I don't like that." (T chuckles.) "Oh. OH! You can't get adjacent sides!" "Right, right! It's four. It's the Klein four-group! You've got total ramification at all of the vertices. It's abelian!"
:Later; W isn't back yet, and P has a bus to catch.:
T: "This is interesting, isn't it? We've learned some stuff here." P grins and nods before hurrying off.
We met in T's office because she has all the neat models, and some of what she wanted to show us was on her computer.
T is holding a skeletal model of a dodecahedron. There are a couple of pieces of string wrapped around it, at odd angles. The computer shows an exploded view. On her desk is what looks like an origami chrysanthemum, except that the petals are all pentagons. There are lots of them. W ruffles it with a finger. "Is there any kind of patterning here?" T changes the picture to an exploded view of the chrysanthemum. Pentagons all over the place, overlapping each other. W points at the screen. "For instance, does this area" - there are no pentagons there - "stay clear?" T plays with the mouse, and another pentagon appears where he pointed. "Rats."
"Are the vertices discrete, at least?" Prof P grunts negatively. After a moment, he says, "Look at the centers. Two flips looks like a translation, and in five different directions." Discussion continues. "Zeta plus zeta to the fourth is the golden ratio? I didn't know that." "Yeah, so it's going to be dense along the real axis, and in five other directions. Combine that, and it's dense all over." "Hmph."
:Later:
W has a skeletal model of a cube, and he's labeled the vertices. He's rolling it around on the desk. "I'm wondering how many orientations you can get on the bottom. No more than twenty-four...." "Can you get adjacent sides?" "Yeah." He demonstrates. "Then it's at least six. Six, twelve, or twenty-four... I wouldn't be surprised if it's twelve." "Why?" "It takes an even number of rolls to get back where you started. That smells like A4 to me." W cocks his head doubtfully. P: "Look. All the rolls are conjugate." "Why?" P waves his hands distractedly. "Because you can't tell the sides apart."
W pulls out a manuscript and flips through it. He points. "This is what you're talking about." "Yeah. Let's see: One. Four. Three. Two. It's a four-cycle, and so is the other one. So it's A4." "Have we proved that? I think we just proved that."
With T's permission, W clears part of the whiteboard and draws twelve parallel lines, then starts linking them with stretched-out X's. "You've got ramification at the vertices.... It could be nastier; this could go with that and that...." "Yeah, it's non-abelian, so...."
:Later; W has gone to get a flu shot.:
P is playing with a skeletal octahedron. "One, two, three, four, five, six. What just happened? I can't tell." T hands him a piece of tape, which he affixes to one vertex. "One, two, three, four, five, six. Did anything happen?" "No, I think the taped vertex was toward you when you started." "Huh. Then it's four.... I don't like that." (T chuckles.) "Oh. OH! You can't get adjacent sides!" "Right, right! It's four. It's the Klein four-group! You've got total ramification at all of the vertices. It's abelian!"
:Later; W isn't back yet, and P has a bus to catch.:
T: "This is interesting, isn't it? We've learned some stuff here." P grins and nods before hurrying off.
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Something like "Interesting gossip you're privy to. I take it it would now be redundant for me to go to the archdivine and turn myself in for death magic?"
Anyway, it did.
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no subject