# Tag Archives: torus

## vector calculus examples

the following hieroglyph

$\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} \cos V\cos W\\ \cos V\sin W\\ \sin V\end{array}\right)$

represent a mapping $\mathbb{R}^2\to\mathbb{R}^3$. It is enough to take

$-\frac{\pi}{2}< V<\frac{\pi}{2}\quad,\quad 0< W<2\pi$

to G.P.S -ing (almost) every point in $\mathbb{R}^3$ which are at distance one from the origin. That is the sphere.

Now the partial derivatives $\partial_1=\frac{\partial\phi}{\partial V}$ and $\partial_2=\frac{\partial\phi}{\partial W}$ are pictorially as:

These derivative are:

$\partial_1=\left(\begin{array}{c}-\sin V\cos W\\ -\sin V\sin W\\ \cos V\end{array}\right)$ and $\partial_2=\left(\begin{array}{c}-\cos V\sin W\\ \cos V\cos W\\ 0\end{array}\right)$

An easy calculation give that for the inner products $\langle\phi,\partial_1\rangle=\langle\phi,\partial_2\rangle=0$, so the $\partial_i$ are orthogonal to the position $\phi$. Then the product $\partial_1\times\partial_2$, which is orthogonal to the plane determined by the couple of vectors $\{\partial_1,\partial_2\}$, hence colinear to the position $\phi$. In fact, after normalization $N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}$.

A good exercise is to unfold the same program for the torus by employing the parameterization given by:

$\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} (2+\cos V)\cos W\\ (2+\cos V)\sin W\\ \sin V\end{array}\right)$

Filed under differential geometry, geometry, math

## torus in chains

more than thousand googolplex parole

Filed under math

## seven circles in a torus

this right graffiti uses the  initial GPS of a torus to mark 7 copies of $S^1$ (circles) and between them being parallel and/or transversal… how?

which is the maximum open domain to maintain injectivity?

What about sending $\left(\begin{array}{c}\!\!v\\\!\!2v\end{array}\!\!\right)$?.. Or $\left(\begin{array}{c}\!\!3v\\\!\!4v\end{array}\!\!\right)$?

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Filed under cucei math

## can you see a solid torus?

The image on the right is a solid torus with four lateral annulus in colours: orange, green, yellow and blue.

Solid tori are important elementary type of 3-manifolds. Also called orientable genus one handlebodies and can be fibered -in the sense of Seifert- by longitudinal circles, and in many different ways.

Among three dimensional technologeeks, they are customed to see a solid torus that can be fibered as the $M\ddot{o}\stackrel{\sim}\times I$, the twisted interval bundle over the Möbius strip.  In constrast $M\ddot{o}\times I$ is the genus one non orientable handlebody: the solid Klein bottle. Isn’t difficult to prove that they are the only two $I$-bundles over $M\ddot{o}$