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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.

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

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

@RefurioAnachro yes it's on my , 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.

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Thinking Outside the Plane:

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:

There's also a 2d-3d connection with Miquel's six-circle theorem but I think it goes the other way:

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:

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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.

Non-concentration of the chromatic number of a random graph,, via

Shamir & Spencer '87 ( and Alon & Krivelevich '97 ( 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:


numbers x+εy

A nice context to think about the Laplace transform:

Eugene Khutoryansky — Convolution and Unit Impulse Response

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:

Les Éléments de l'Art Arabe,

Lots of plates of pretty geometric girih patterns, from an 1879 book by Jules Bourgoin. Via a comment at

ε² = 0

It makes sense to think of ε as a first approximation for infinitesimals: smaller than any non-infinitesimal, but still greater than zero.

You might also notice that Grassmann's exterior algebra works in a similar way.

Grassmann's exterior algebra also appears as the wedge product in differential algebra. That smells like a kind of unification of the two interpretations above, but I don't know how to make that precise...

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Recall that quarternions are like complex numbers on steroids. They have three imaginary units i, j, k each of which squares to minus one, just like the complex unit:

i² = -1

To get dual quarternions, take two quarternions A and B and think of one of them as carrying a new kind of symbol ε:


ε also reacts in a funny way when squared:

ε² = 0


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Dual quarternions are an 8-dimensional algebra that can be used to represent not only rotation and scaling, but also translations in R³.

Using mere quarternions for rotations in R³ is a well-known trick. But translations work differently, so we need a new kind of , octonions simply won't work here.


How many years until General Direct Dialed telephony dies?


Archimedes came up with a number system that could count up to 10^(8*10^16), and after summarizing the history of measuring the size of our cosmos, Parallax Nick asks "what was he hoping to count with that number?"

The Planck Reckoner

One or two bugs aside, I really like Nick's style, and the format, too! Go and click through to his channel, and watch all of his series about water in the solar system!

@11011110 Things get particularly weird (and wonderful) when your fractional base is smaller than 1. You can see a problem about inserting things into a rotating queue that leads to these kinds of expansions in these slides:

#fractals #graphics #monochrome #art

An excerpt from the parameter plane of this formula iterated in a loop (p = 2):

z := z^p + c
z := (|x| - i y)^p + c
z := (x - i |y|)^p + c

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