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How many non-infinite plane curves with infinite reflectional symmetry?

Claim 1. Let $\ell_1,\ell_2,\ell_3$ be three line in the plane that do not intersect in a single point. Then the only bounded subset of the plane that is symmetric to all three lines is the empty set. ...
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What is a Coxeter Group?

I'll try to sketch the connections between your bullet points with an easy example of a Coxeter group. You probably want to study some basic algebraic knowledge to follow this. We examine the ...
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Reflecting a point by a line in $\mathbb R^3$

Given a line $\overline r+t\cdot\overline{v}$ and a point to be reflected $\overline p$, we must first find the closest point on the line to $\overline{p}$. Let $t_0$ correspond to that point. Its ...
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Reflection in Geometric Algebra

Here's a less convoluted version of your calculation that should make your error more immediately obvious: $$\mathrm{Ref}_{v-w}(v) = −(v−w)v(v−w)^{−1} = -\frac{(v−w)v}{v−w} = -v$$ In short, you've ...
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Unitary operators and a product of reflections

This is akin to a typographical error. As you note the stated result is not true in complex vector spaces, at least if "Householder transformation"/"reflection" is defined as a ...
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Here is a general guideline for $2 \times 2$ orthogonal matrices. They have one of the two forms $$\text{Either} \ \ R = \begin{bmatrix} a &-b\\[0.3em] b & \ \ \ a\\[0.3em] ... • 81.2k 4 votes Accepted What algebra is generated by \mathrm{O}(2)? Well, as you said, any "complex number" \begin{pmatrix} a & b \\ -b & a \end{pmatrix} is a linear combinations of rotations. This implies that any matrix of the form \begin{pmatrix} 1 & ... • 9,621 4 votes How many non-infinite plane curves with infinite reflectional symmetry? Here is a theorem along the lines you are asking for. To avoid worrying about what the definition of "finite curve" should be, I will simply consider arbitary compact subsets of \mathbb{R}^2. ... • 329k 4 votes Symmetry of point about a line in 3d Let \frac{x+1}{4}=\frac{y+1}{-3}=\frac{z-15}{16}=t and (x,y,z) be the needed point. Hence,$$\frac{x+t_x}{2}=-1+4t,\frac{y+t_y}{2}=-1-3t$$and$$\frac{z+t_z}{2}=15+16t,$$which gives,$$x=-2+...
Denote circumcircle of $\triangle ABC$ as $\Omega$ and circle with center $M$ and radius $MI$ as $\Gamma$. It's well-known that $\Gamma$ is the circumcircle of $\triangle BIC$. Let $\Phi$ be the ...