Semi-seriously, connected (orientable) double covers of Möbius strips? (Requires two strips of each colour per model.)

Glue or tape a red strip to a white strip, making a single double-thickness strip that is red on one side and white on the other. (Optionally colour the white side to look like a fruit rind, such as watermelon.)

Make a second unit of the same type.

Place two of these units with red sides facing, making a single quadruple-thickness strip that is white on the outside and red on the inside as if the paper has been cut in half "depthwise."

Treating this quadruple-strip as a unit, create a Möbius strip conventionally: Give a half-twist, and "attach the ends, respecting the laminar division."

Tell math-y people you made a Möbius strip, cut the thickness of the paper in half, and this is what resulted!

As for semi-serious: The model clearly has two sides (one of each colour), and nicely illustrates how "orientable double-covering" amounts to "painting a non-orientable surface and peeling off the paint," i.e., picking a germ of orientation and constructing its connected component in the sheaf of local orientations.

This is a #linocut of French #mathematician & #physicist Joseph Fourier (1768-1830) remembered for his work on Fourier series, Fourier transforms, Fourier’s law of conduction, Fourier analysis & harmonic analysis & their use to solve heat transfer problems & credited with proposing the greenhouse effect as early as 1824.

Being a commoner he could not seek a commission in the scientific corps of the army but took a military lectureship in #math. 🧵1/n

First was a great talk by Jordan Ellenberg on the importance of uncertainty and contradiction in math and why mathematics is actually part of the humanities at the Santa Fe Institute https://www.youtube.com/watch?v=L1pXQNaS9Oo (2/11) #math#statistics

For a long time I felt like I didn't really understand the Yoneda Lemma. I knew some things that people said about it ('we can understand objects by the maps into them' and 'the Yoneda embedding is full and faithful') but the statement 'Hom(Hom(A, -), F) = F(A)' itself was something I could only use as a symbolic manipulation without understanding.

On the other hand, I did separately know facts like 'In the category of quivers there are objects which look like • and •→•, such that the maps out of them tell you exactly the vertices and edges in your quiver' and 'In the category of simplicial sets there are objects which are just an n-simplex; maps out of them are the n-simplices of the object you are mapping into'.

Somehow I only recently realised that these examples are precisely the Yoneda Lemma. These objects are precisely presheaves of the form Hom(A, -), and the Yoneda Lemma tells you what you get when you map out of them.

In particular I think it would be useful to give the quiver example to students when they learn the Yoneda Lemma.

i just took out my ear lobe studs for the first time and i NEED to show off my klein bottles. Also I'm accepting suggestions for fun #math#earrings !! #MathCoffeeSelfie

Approaching the annual "running out of beekeeping equipment" date. 😬

(Bees follow the rule of exponential growth and decay: you either have WAY TOO MANY BEES or NOT NEARLY ENOUGH BEES -- in spring they grow by the power of two or more, but they also tend to drop in half in winter). #beekeeping

I genuinely think that a fanatical emphasis on real world uses has done more to harm mathematical literacy than any other academic policy.

Kids love weird shit, and teachers should 100% take advantage of that and roll into class bein' like "Ok nerds, here's the math that proves you're a donut"

Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.

Here's something I just learned: the lucky numbers of Euler.

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.

Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.

Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).

The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.

h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.

🐍 aprxc — A #Python#CLI tool to approximate the number of distinct values in a file/iterable using the Chakraborty/Vinodchandran/Meel’s (‘coin flip’) #algorithm¹.

The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].

Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].

## The Ultimate IB Math Coursework Handbook with Examples (snokido.uk)