Pinned toot

These nice folks have engaged with me, most likely on a maths topic. Simply starting to follow me, or reacting on followfriday posts counts just as well.

Y'all are so cool! Thank you!

An enjoyable problem (104.B) from the latest Mathematical Gazette: A regular 7-gon is inscribed in the unit circle, with one vertex at (1,0). Find the equations of the two parabolas, symmetric across the x-axis, which pass through the vertices of the heptagon as shown.

Estimating untested infections. Now there's a publication in Nature about an idea I first learned from @ColinTheMathmo.

nature.com/articles/d41586-020

@dredmorbius

*Programming in 2D text*

This evening I'm thinking of two ideas I've thought about several times before, but never together.

The first: Arjun Nair's github.com/batman-nair/IRCIS, an esoteric language (art for art's sake) that comes with a cool visualizer. Maybe all languages should.

The second is Dave Ackley's movablefeastmachine.org, a tiled processor for very finely grained distributed computation. Programming it is like playing with a cellular automaton.

Category theory is a very high level tool. Take, for example, the idea of 'universal constructions'. I'd link Wikipedia, but its article comes up with a cat-theoretic description of the idea, which needlessly complicates things. It's a basic technique: Given some constraints, find the most general object (algebra, ...) that satisfies them. It's about interpreting things. It's nonconstructive. Commutative diagrams aren' programs, they're more like equations to be solved.

This is the second time I read about a problem becoming easier in hyperbolic space. I forgot the other one, I'll have to hunt for it! I can see what's happening with TSP here - there must be more problems like this! Thanks for inspiring!

@11011110

Adam Ponting found a way to cover arbitrarily large (e.g. as measured by inradius) contiguous patches of the plane by distinct squares of sizes from $$1$$ to $$(2n+1)^2$$, for any $$n$$. Via demonstrations.wolfram.com/Pon and mathpuzzle.com/

@freemo \ x_{n+1} = r x_n(1 - x_n) \ is a simple enough population model. Lets try simulating it! Oh no...

new blog post summarizing the recent stuff on mating Julia sets that I've been playing with

mathr.co.uk/blog/2020-01-16_sl

I think I solved it!

Animated slow mating of p=-1+0i with q=-0.122+0.75i via inverting the pullback, which gives a series of functions like (az^2+b)/(cz^2+d); starting from the pixel coordinates in equirectangular projection I compute the initial z and apply all the collected functions in reverse order, colouring white if the final output |z| > R for some large R.

Next step will be trying to draw the Julia sets in the halves of the sphere.

Good code is a myth

Userland is an experimental graphical environment which can run shell commands (pipes, really) in a spreadsheet!

userland.org

Have a look at this thread for a demo video and a link to fork with a Dockerfile:

mastodon.cloud/@RefurioAnachro

@RefurioAnachro yes it's on my code.mathr.co.uk , needs a few of my other libraries too. but it's a bit work-in-progress, and I haven't pushed the most recent progress yet. only tested on debian

Turns out it was much simpler to just clamp the potentially huge wake image coordinates to +/-10 in mpfr_t before converting to lower-range double for cairo filling.

The image is roughly +/-1 in that coordinate frame, depending on aspect ratio - clamping may break appearance with very wide images, left a #FIXME note in the code for later.

Thinking Outside the Plane: metafilter.com/183649/Thinking

Interesting roundup of 3d solutions to 2d problems, starting with Tarski's plank problem: Can you cover a diameter-$$n$$ disk with fewer than $$n$$ unit-width strips?

Sadly, they missed the 3d proof of Desargues' theorem: en.wikipedia.org/wiki/Desargue

There's also a 2d-3d connection with Miquel's six-circle theorem but I think it goes the other way: 11011110.github.io/blog/2006/0

Automatic annotation progress update:
✅ child bulbs of a mu-unit
✅ filaments of a mu-unit (done, this post)
❎ child islands of a mu-unit
❎ embedded Julia set filaments (next on the list)
❎ embedded Julia set islands
❎ embedded Julia set hubs

Mu-unit is Robert Munafo's terminology: mrob.com/pub/muency/muunit.htm

I didn't know that Philip Glass had composed a piece of music for Sesame Street, and its about geometry!

The Peirce quincuncial projection[1] is a conformal map projection developed by Charles Sanders Peirce in 1879. The projection has the distinctive property that it can be tiled ad infinitum on the plane, with edge-crossings being completely smooth except for four singular points per tile. The projection has seen use in digital photography for portraying 360° views.

en.m.wikipedia.org/wiki/Peirce

Non-concentration of the chromatic number of a random graph, arxiv.org/abs/1906.11808, via gilkalai.wordpress.com/2019/06

Shamir & Spencer '87 (doi.org/10.1007/BF02579208) and Alon & Krivelevich '97 (doi.org/10.1007/10.1007/BF0121) proved that random graphs $$G(n,p)$$ with $$p=n^{1/2-\epsilon}$$ have almost surely only two possible chromatic numbers. But now Annika Heckel has shown that dense random graphs have a significantly wider spread in colors.

I find fascinating how these two products differ:

x+iy

numbers x+εy

A nice context to think about the Laplace transform:

Eugene Khutoryansky — Convolution and Unit Impulse Response youtube.com/watch?v=acAw5WGtzu

it clearly adds to a more planar presentation of the material in that it elegantly exploits the third dimension to put the dual aspects in plain sight.

Eugene Khutoryansky has been producing quite a few of these for a large variety of concepts! Watch them all!

Oh, and if you haven't read what Mike Shulman has to say about it here:

golem.ph.utexas.edu/category/2