#modular form

Iza Tarasewicz. Equilibrium in Meteors study, 2018.

Jacob Hashimoto. 

Centuries of Space Shifting Slighly Beneath This Labyrinth, 2017.

“ Find 5 examples of a module and/or modular form, if you don't know what one is do some online research in your specific discipline to bring in for class. These can be printed out and pasted in your sketchbook, pinned on Pinterest or Tumblr.”

The unexpected interconnectedness of everything!

Let me give some examples:

You’re beyond a doubt familiar with Fermat’s last theorem, stating the unsolvability of the Diophantine equation xn + yn = zn for n > 2. It took more than 350 years to actually prove this result, but more astonishing is how it was proved: by weaving together elliptic curves and modular forms, two seemingly unrelated objects in mathematics.

A less known mathematical gem is Monsky’s theorem: it is not possible to dissect a square into an odd number of triangles of equal area. The only known method to prove this geometrical curiosity uses properties of 2-adic integers (algebra) and Sperner’s lemma (combinatorics).

Another, more technical example is monstrous moonshine, describing a unexpected and deep connection between the Monster group and a specific modular form, the j-invariant.

Happy April Fools' Day!

Sorry for my lack of originality; the original hoax was made by Martin Gardner in his column Mathematical Games in 1975. Gardner claimed the constant on the left is in fact an integer, as conjectured by the Indian math genius Srinivasa Ramanujan. Simon Plouffe hence coined the constant "Ramanujan's constant". The hoax was surprisingly well-accepted, but people couldn't rely on Mathematica those days :)

Of course, Ramanujan's constant is not an integer. It is irrational (in fact, transcendental) and equals 262537412640768743.999999999999250…

As you can see, the error in the previously stated equality is of an order 10-13, remarkably small compared with the number itself, which is of order 1017!

The number being almost integer is not a mere coincidence as there are deep mathematical objects hiding underneath this seemingly innocuous delectation (modular forms, Heegner numbers, the j-invariant, q-expansions…) but the explanation is far from easy. If you're interested, you can read about it on Wikipedia.