Mathematics

I would be shocked if it could be finitely presented. Hard to see how composing a finite number of rotations could give rotations through arbitrary angles around arbitrary axes.

GFauxPas

12:35 AM

The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.

Ted Shifrin

Yes, I know that result.

GFauxPas

I didn't but it makes sense

Ted Shifrin

Compute it in the plane first.

anakhro

It's a great theorem.

Ted Shifrin

But using reflections, my complaint still stands.

I actually don't know the Cartan-Dieudonné Theorem. What sayeth it?

GFauxPas

12:37 AM

Yeah it does seem unreasonable to expect a finite presentation

Let (V, b) be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2. Then, every element of the orthogonal group O(V, b) is a composition of at most n reflections.

Ted Shifrin

I would hazard a guess that it's not even countably presented, but someone might show me I'm wrong.

Oh, cool.

I see: That's the generalization of the physicists' Euler angles in $\Bbb R^3$.

Oh, no, never mind. That isn't right.

That would say you can write every rotation as a composition of ($n$) rotations about the coordinate axes.

I doubt that's true for $n>3$.

These are all good questions for an industrious undergraduate to investigate.

Daminark

12:54 AM

Hello nerds

Ted Shifrin

Hi nerd Demonark.

rschwieb

@KarlKronenfeld Just filling a gap in my database , and I couldn’t believe it hadn’t been filled earlier.

Daminark

How's everything going?

N. Maneesh

1:28 AM

1

Q: How does the Evolute of an Involute of a curve $\Gamma$ is $\Gamma$ itself?

How does the Evolute of an Involute of a curve $\Gamma$ is $\Gamma$ itself? Definition from wiki:-The evolute of a curve is the locus of all its centres of curvature. That is to say that when the centre of curvature of each point on a curve is drawn, the resultant shape will be the evolute of th...

differential-geometry

anakhro

I can't even parse that title.

N. Maneesh

I got this question from a lecture video.

Differential Geometry | Math History | NJ Wildberger

►

17:04

He is not explaining completely.

anakhro

1:44 AM

Did he actually say the title anywhere

Because I still can't parse it.

N. Maneesh

yes. !7:04

evolute of the involute is the curve itself

anakhro

Do you mean to ask "how is the evolute of an involute of a curve $\Gamma$ itself?"

N. Maneesh

yes

for evolute, we have a fixed point on the curve right?

then we consider tangent to each point right of that

Involute is the curve that is perpendicular to each tangent line.

sorry, for involute we have fixed point.

then we consider tangent to each point right of that
Involute is the curve that is perpendicular to each tangent line.

For finding the Evolute of this Involute, we need to find the centre of curvature at each point.

anakhro

I have never heard of an evolute or involute, sorry.

N. Maneesh

okay

Kenkar

6:57 AM

Z

Thanks @TedShifrin for answering

ÍgjøgnumMeg

7:34 AM

Mooornin'

Grešnik

Good morning, we are trying to solve some games on infinite chessboard in one room, I will post them here also, it could be that someone finds them interesting.

3

Q: $6$ bishops and a knight on an infinite chessboard

Player $A$ places $6$ bishops wherever he/she wants on the chessboard with infinite number of rows and columns. Player $B$ places one knight wherever he/she wants. Then $A$ makes a move, then $B$, and so on... The goal of $A$ is to checkmate $B$, that is, to attack knight of $B$ with bishop in ...

combinatorics game-theory chessboard infinite-games

11

Q: A simple game on infinite chessboard

Player $A$ chooses two queens and an arbitrary finite number of bishops on $\infty \times \infty$ chessboard and places them wherever he/she wants. Then player $B$ chooses one knight and places him wherever he/she wants (but of course, knight cannot be placed on the fields which are under attack ...

combinatorics game-theory chessboard infinite-games

Grešnik

7:57 AM

Here is a room:

https://chat.stackexchange.com/rooms/95333/discussion-between-peter-and-gresnik

Adam

11:43 AM

I get a 6 day ban for telling someone to shut up?

fair enough I am socially inept in the ways everyone else I've met are not, but they are also socially inept in the ways I am not

Albas

11:56 AM

The invariant formula for the exterior product, why would someone come up with something like that. I mean it looks really similar to the formula of the covariant derivative along a vector field for a tensor but otherwise I don't see why would it be something natural to come up with. The only places I have used it is deriving the poisson bracket of two one forms

Transcript for

$Mathematics$ Mathematics