# Monthly Archives: May 2016

## Tutorial 5: Lawson’s minimal surfaces and the Sudanese Möbius band

In the last tutorial we constructed certain minimal surfaces in hyperbolic space. These hyperbolic helicoids were generated by a 1-parameter family of geodesics: while moving on a geodesic – the axis of the helicoid – another geodesic perpendicular to the axis was … Continue reading

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## The 3-Sphere

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

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## Tutorial 4: Hyperbolic helicoids

A ruled surface is a surface in $$\mathbb R^3$$ that arises from a 1-parameter family of straight lines, i.e. these surfaces are obtained by moving a straight line though the Euclidean space. E.g. a normal vector field of a curve defines such … Continue reading

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A closed discrete curve $$\gamma$$ is map from a discrete circle $$\mathfrak S_n^1 =\{z\in\mathbb C \mid z^n = 1\}$$, $$n\in \mathbb N$$, into some space $$\mathrm M$$. In some situations it is more convenient to consider the discrete circle just as $$\mathbb Z/n\mathbb Z$$,$\mathbb … Continue reading Posted in Tutorial | Comments Off on Tutorial 3: Framed Closed Curves ## Conformal maps III: Stereographic Projection 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 Posted in Lecture | Comments Off on Conformal maps III: Stereographic Projection ## Conformal Maps II: Inversions 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 Posted in Lecture | Comments Off on Conformal Maps II: Inversions ## Quaternions 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 Posted in Lecture | Comments Off on Quaternions ## Tutorial 2: Framed Discrete Curves A discrete curve $$\gamma$$ in a space $$\mathrm M$$ is just a finite sequence of points in $$\mathrm M$$, $$\gamma = (\gamma_0,…,\gamma_{n-1})$$. In case we have discrete curve $$\gamma$$ in $$\mathbb R^3$$ we can simply draw $$\gamma$$ by joining the points by straight … Continue reading Posted in Tutorial | Comments Off on Tutorial 2: Framed Discrete Curves ## Mandelbrot Set If you google “mathematics visualization”, the resulting wikipedia page shows the Mandelbrot set as a famous example. The Mandelbrot set $$M\subset{\Bbb C}$$ is defined as the following. For each $$c\in {\Bbb C}$$, consider the iteration $$z_{n+1}(c) = ({z_n}(c))^2+c$$, $$n=0,1,2,\ldots$$ and \(z_0(c) = … Continue reading Posted in Tutorial | Comments Off on Mandelbrot Set ## Conformal Maps I: Holomorphic Functions 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

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