G2 shall be my favorite group! Well, for now. For a while. Or maybe for longer. It's the smallest of the exceptional Lie groups. Found 1893 by Élie Cartan. It was he who suggested to think about it in terms of rolling balls.

https://en.wikipedia.org/wiki/G2_%28mathematics%29

Picture two spheres, one three times larger than the other. Imagine them rolling on another without slipping nor twisting. Rolling surfaces have their own branch of mathematics: contact geometry!

HaloSim is specialized raytracer (for windows) which can generate ice halos. It can even show halos as they might appear in the atmosphere of Saturn!

> Halos from octahedral ammonia crystals as might exist in the cold high level clouds of Jupiter and Saturn. The 42° circular halo has four associated sundogs. The inner halos are produced by rays reflected within the crystal.

https://www.atoptics.co.uk/halo/oworld.htm

I got here from watching sixtysymbols' latest, on ice halos:

Boole didn't use bit vectors, because vectors weren't officially invented yet! Instead, he used multi-variable polynomials. He also used F₂ writing multiplication as 'and', and calling addition 'or', while curiously disallowing 1+1. Picking neither disjunction nor xor, he left this fight to his successors.

wp> The house at 5 Grenville Place in Cork, in which [George] Boole lived between 1849 and 1855, and where he wrote "The Laws of Thought"

See here for a newer image:

https://en.wikipedia.org/wiki/George_Boole

raytracing

→ dynamical billiards (in a spheroid)

→ hamiltonians

→ symplectic geometry

→ differential forms

→ exterior algebra

→ weyl algebra

→ ...?

#maths journey

The seahorse's tail, ubiquitous in its home valley, well deserves the highlight and its own five minutes of fame. What is value, after all?

Show thread

Here’s another one with City Hall near the red UFO un the center, which is a reflection from the interior.

Show thread

As part of this year’s christmas market in Bonn there’s this tower/ufo attraction, like a carousel, but for old people, which slowly lifts you 100m in the air. From up there you have a terrific view. Here’s an snapshot my wife took just this evening. It’ll be here until christmas eve, so you have about a week left.

You can see the “Bonner Münster” on the left, and at the bottom right there is a Ferris wheel and a tiny bit of illuminated christmas market.

Sometimes there's a dominant structure turning out too strong. Try burying it in a noise colored Flokati. Nevermind that the wool will find its way all over the place.

Pure #Mandelbrot as knitted by my old friend #XaoS.

Show thread

I wanted stripes, but it's difficult to obtain a striped palette by pressing "p" in XaoS. So I went for binary decomposition. Fading from black&white via gray to white&black is what this picture aims to show off. It yields a kind of 3d look.

#Mandelbrot made wielding #XaoS

Show thread

Repost, forgot the image, sorry.

> In fact, Gaudi designed a church in Barcelona using a web of strings and weights to find correct shapes for the arches -- of course, the actual building would have a shape like the reflection in a horizontal mirror above the strings

Also from ch1 of that mechanics textbook:

http://galileoandeinstein.physics.virginia.edu/7010/CM_01_Intro_Statics_Catenary_Arch.html

Woot Gaudi!

2/

Show thread

> Let's define an ideal arch as one that doesn't have a tendency to fall apart sideways, outward or inward. This means no shear (sideways) stress between blocks, and that means the pressure force between blocks in contact is a normal force -- it acts along the line of the arch. That should sound familiar! For a hanging string, obviously the tension acts along the line of the string.

> […] the ideal arch shape is a catenary.

Source, ch 1 from a textbook on mechanics:

http://galileoandeinstein.physics.virginia.edu/7010/CM_01_Intro_Statics_Catenary_Arch.html

1/

Minimal surfaces minimal style, made from paper strips.

+Alison Grace Martin has left a trail of these, mostly triaxially woven, topological (or rather conformal) exercises here on g+:

https://plus.google.com/118320887831309218676/posts/TBbHVE6SGRP

Joel David Hamkins is a well known research-level logician, and he usually posts about models of set theory, programs that only halt after a multiply infinite amount of time, and about infinite chess.

But this post is different, and very practical. Given a greeky straightedge and compass construction, like for example the one for finding the midpoint of a line segment, how much do errors grow while propagating. And what will their final shape look like?

http://jdh.hamkins.org/error-propagation-in-classical-geometry-constructions/

The earlier Lagrangian mechanics involved an extra parameter for the time t, and it always uses the derivative of the first kind of coordinates as the second kind of coordinates.

https://en.wikipedia.org/wiki/Lagrangian_mechanics

The extra parameter t amounts to another coordinate dimension, making the resulting spaces odd-dimensional. This is called contact geometry and it is closely related to symplectic geometry.

https://en.wikipedia.org/wiki/contact%20geometry

9/

Show thread

- g+ blog archive
- https://refurioanachro.github.io/g-viewer/

Higher maths is cool – come and see invisible worlds with me!

Joined Apr 2017