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Another unicode mystery: the Unicode character ℇ, U+2107, is labelled "EULER CONSTANT", but nobody can work out what it's supposed to stand for. The base of the natural logarithm is normally written as 'e', and the Euler-Mascheroni constant is normally written as 'γ'.
A 2002 post on the Unicode mailing list seems to basically say it's a mistake: unicode.org/mail-arch/unicode-

of course, the nanomb2 algorithm (aka "super series approximation") does work sometimes, and when it does it is often much faster than the older series approximation algorithm, which in turn is much faster than perturbation iterations alone, which in turn is much faster than using plain iteration with high precision numbers for each pixel

the algorithm works by approximating the orbit near a periodic cycle (usually a minibrot island) by a polynomial in *two* variables representing small changes in C and Z. you end up with a polynomial that does P iterations at once (a "super iteration"), where P is the period of the cycle.

this polynomial is only valid while dC and dZ are small, but when they get big you can switch to a different polynomial corresponding to a nearby less zoomed in minibrot of lower period. one "super-iteration" takes longer than one perturbation iteration or one plain iteration, but you need far fewer, so it works out faster.

combined with interior checking, you can set the iteration count exceedingly high (100100100 is routine for me now) with little-to-no slowdown, and get super-crisp minibrot boundaries

#fractals #mandelbrot #math #maths

150. DISSECTING A MITRE. The figure that is perplexing the carpenter in the illustration represents a mitre. It will be seen that its proportions are those of a square with one quarter removed. The puzzle is to cut it into five pieces that will fit together and form a perfect square. I show an attempt, published in America, to perform the feat in four pieces, based on what is known as the "step principle," but it is a fallacy.
(1/4)

Not being satisfied with Cocke’s arguments, my Lord and I did oppose the strength of his arguments, which brought us to a great heat, he being a conceited man, but of no Logique in his head at all, which made my Lord and I mirth.

If I ask you to "decompose a fraction into partial fractions", what should you do?

Warning: I'm looking for edge cases and loopholes.

Wow, the statistical concept of turned 100 years old this July. That's so young in the mathematical history scale! It's hard for me to imagine statistics without variance.

en.wikipedia.org/wiki/The_Corr

What a treat! Over the weekend I received an unexpected email from a fan, containing proofs of the Riemann hypothesis, Fermat's last theorem, the Beal conjecture *and* the abc conjecture! Furthermore, they're all proved by the one proof!

Inspired by an article in @chalkdustmag, I wrote code to generate Truchet tiles for any even number of sides. Then I looked up which tilings of even-sided polygons exist, and here we are: somethingorotherwhatever.com/t

When you mark a piece of work named "Mine", and wonder which student thought to write that... Then realise it is your solutions!!!

Set:
A = 0
B = 1
C = 10
D = 11

Given a binary number, substitute letters for strings of digits, e.g. 1001 = CAB.

Can you write an algorithm to find the ShortLex minimal letter representation of any number? ShortLex: compare by length, and then alphabetically, e.g. CA<BAA, BC<DA.

@loke @pkra we probably are getting closer to a more consistent conventional notation, but abuse of notation is quite helpful, and sometimes it just sticks.

This is more what I was looking for: in the C3 book, they ask you to simplify \(\sin(\theta) \cos (\theta) (\sec (\theta) + \operatorname{cosec}(\theta))\). The answer they give is \(\sin(\theta) + \cos(\theta)\).
So do they mean "put it in terms of sin and cos"?

I'm looking in A-Level textbooks for examples of "simplification".

On the first page of content in this EdExcel C1 book, this made me so cross: you're not applying the rule \((a^m)^n = a^{mn}\), you first need \((ab)^n = a^n b^n\), but that rule isn't even listed!

smooth jazz, piecewise-linear jazz, topological jazz

Prompted by @11011110, I've had a go at making a Mathstodon emojo: :mathstodon:

It's a 𝕄 with an elephant's trunk. Helpful suggestions for my colourblind eyes welcome!

@christianp The space before the \(dx\) in an integral (unless you’re picky and write \(\text{d}x\)).

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Use \( and \) for inline LaTeX, and \[ and \] for display mode.