Gerard's posts

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A one-page proof of a nice theorem: You can find the N-dimensional circumsphere of an N-simplex directly from its Cayley Menger matrix. In other words, you need only to know the distance between the points.

https://westy31.home.xs4all.nl/Circumsphere/ncircumsphere.htm#Coxeter

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**Modular to Elliptic**

Ever since the proof of Fermat’s Last Theorem, the relation between modular forms and elliptic curves sometimes captivates me. (Disclaimer: I am not an expert on this!)

Both modular forms and elliptic functions can be thought of as being composed of symmetrical tiles on the complex plane. Each tile has a pole and a zero, which together completely determine the complex function. Modular forms have the poles on the real axis, precisely at the rational numbers. Each tile of a modular form corresponds to exactly one rational number. Elliptic functions have the poles and zeros on a regular 2D lattice. (For pictures, check out my website https://westy31.home.xs4all.nl/Geometry/Geometry.html) Imagine the rational numbers on such a lattice, numerators and denominators being x-axis and y-axis respectively.

(For graphical purposes, the real axis of the modular form has been mapped to a circle.)

This animation morphs the tiles of a modular form to tiles such that the poles end up on a 2D lattice. The resulting tiles are actually not quite the tiles of an elliptic function, but perhaps interesting anyway. Firstly, only numerator/denominator points on the lattice that are relatively prime get a pole from the modular tiling, the others not. I marked these relative prime pairs as green circles, the other as white. Remarkably, the modular form ‘knows’ which integer pairs are relative prime. The tree-like structure which appears to emerge out of the modular tiling is called the Stern-Brocot tree. A cool property of this “Stern-Brocot tiling” is that no lattice point ever lies inside a triangle, they all lie exactly on a vertex, or if they are non-relative prime points, they lie outside the tree. Looking at a region close to the origin, this region outside of the tree gets squeezed into less and less available surface area as more generations of tiles are added, but the non relative prime points still remain outside the tree, on ever-narrowing “cracks” along directions (p/q). In the meanwhile, if you look at the entire tree, it gets bigger exponentially with the number of tiles, while the surface is filled up grows only as the square root of the number of tiles. Does this demonstrate that there are much more non-computable numbers than computable numbers?

The area of each tile in the lattice configuration is ½. This can be seen with Pick’s theorem.

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A lamp I saw in the hague central station. Looks cool.

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I managed to fix a new ultra-bright small LED into my stereographic projection lamp. This LED requires a heat sink, normally it is mounted on a printed circuit with conductive metal layers. But i thought thick copper wires will also act as a heat sink. It works!

I also use a flexible goosneck tube now. It's on Etsy:

https://www.etsy.com/nl/shop/GerardWestendorp

I also use a flexible goosneck tube now. It's on Etsy:

https://www.etsy.com/nl/shop/GerardWestendorp

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**Accessories for HTC Vive**

http://www.accessorieslists.com/accessories-htc-vive/

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This 3D printed Pythagoras tree switches the plane in which new generation bricks are formed, so that it becomes more truely 3D than most other variations seen on the web. These generally are 2D versions with rectangles replaced by bricks.

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A Christmas version of my stereographic projeciton lamp.

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This photograph is of the

This is following up on the idea that 'star polyhedra secretly are higher genus surfaces' in my previous post The small cubicuboctahedron,has 6 octagon slicing through it. It is related to the Klein Quartic, it has 24 vertices (KQ has 24 heptagons), and it also has genus 3.

I want to make an animation of the small cubicuboctahedron warping into the genus 3 surface.

**fundamental polygon**of the small cubicuboctahedron. If you fold the edges of this polygon together so that a closed surface is formed, it should be a surface of genus 3.This is following up on the idea that 'star polyhedra secretly are higher genus surfaces' in my previous post The small cubicuboctahedron,has 6 octagon slicing through it. It is related to the Klein Quartic, it has 24 vertices (KQ has 24 heptagons), and it also has genus 3.

I want to make an animation of the small cubicuboctahedron warping into the genus 3 surface.

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