Monthly Archives: May 2016

So far we have seen geometries in 2D and 3D, which are the dimensions we are familiar with.  But mathematician have found interesting geometries in higher dimensions and it would be great if we could visualize them.  In particular a huge … Continue reading →

Stereographic projection \[\sigma: \mathbb{R}^n \to S^n \setminus \{\mathbf{n}\}\] where \[\mathbf{n}=(0,\ldots,0,1)\in\mathbb{R}^{n+1}\] is the northpole of $S^n$ is a special case of an inversion: Let us consider the hypersphere $S\subset\mathbb{R}^{n+1}$ with center $\mathbf{n}$ and radius $r=\sqrt{2}$ and look at the image of \[\mathbb{R}^n =\left\{\mathbf{x}\in\mathbb{R}^{n+1}\,{\large … Continue reading →

Let $M\subset \mathbb{R}^n$ be a domain. A smooth map $f:M \to \mathbb{R}^n$ is called conformal if there is a smooth function $\phi:M\to \mathbb{R}$ and a smooth map $A$ from $M$ into the group $O(n)$ of orthogonal $n\times n$-matrices such that … Continue reading →

Quaternions, $\mathbb{H}$ are a number system like real or complex numbers but with 4 dimensions. In particular, $\mathbb{H}$ is nothing but $\mathbb{R}^4$ together with a multiplication law. The identification of $\mathbb{H}$ and $\mathbb{R}^4$ is given by: $$ \mathbb{H} = \lbrace … Continue reading →

If $M\subset\mathbb{R}^2$ is a plane domain and the image of parametrized surface $f:M\to \mathbb{R}^3$ is contained in \[\mathbb{R}^2=\{(x,y,z)\in \mathbb{R}^3\,\,|\,\, z=0\}\] then the defining equations of a conformal map \begin{align*}\left|f_u\right|&=\left|f_v\right| \\\\ \langle f_u,f_v\rangle &=0\end{align*} imply that $\left|f_v\right|$ arises from $\left|f_v\right|$ by … Continue reading →