No Stupid Questions

The Banach - Tarski Theorm is up there. Basically, a solid ball can be broken down into infinitely many points and rotated in such a way that that a copy of the original ball is produced. Duplication is mathematically sound! But physically impossible.

https://en.m.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Here's a fun one - you know the concept of regular polyhedra/platonic solids right? 3d shapes where every edge, angle, and face is the same? How many of them are there?

Did you guess 48?

There's way more regular solids out there than the bog standard set of DnD dice! Some of them are easy to understand, like the Kepler-poisont solids which basically use a pentagramme in various orientations for the face shape (hey the rules don't say the edges can't intersect!) To uh...This thing. And more! This video is a fun breakdown (both mathematically and mentally) of all of them.

Unfortunately they only add like 4 new potential dice to your collection and all of them are very painful.

Great thread. I'm just reading and watching stuff this afternoon now

I heard that Pythagoras killed a man on a fishing trip because he solved a problem first.

That's a pretty wild math tale!

How Gauss was able to solve 1+2+3...+99+100 in the span of minutes. It really shows you can solve math problems by thinking in different ways and approaches.

Not so much a fact, but I've always liked the prime spirals: https://en.wikipedia.org/wiki/Ulam_spiral

Also, not as impressive as the busy beaver, but Knuth's up-arrow notation is cool: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

To me, personally, it has to be bezier curves. They're not one of those things that only real mathematicians can understand, and that's exactly why I'm fascinated by them. You don't need to understand the equations happening to make use of them, since they make a lot of sense visually. The cherry on top is their real world usefulness in computer graphics.

Maybe a bit advanced for this crowd, but there is a correspondence between logic and type theory (like in programming languages). Roughly we have

Proposition ≈ Type

Proof of a prop ≈ member of a Type

Implication ≈ function type

and ≈ Cartesian product

or ≈ disjoint union

true ≈ type with one element

false ≈ empty type

Once you understand it, its actually really simple and "obvious", but the fact that this exists is really really surprising imo.

https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

You can also add topology into the mix:

https://en.m.wikipedia.org/wiki/Homotopy_type_theory

The Julia and Mandelbrot sets always get me. That such a complex structure could arise from such simple rules. Here's a brilliant explanation I found years back: https://www.karlsims.com/julia.html

Non-Euclidean geometry.

A triangle with three right angles (spherical).

A triangle whose sides are all infinite, whose angles are zero, and whose area is finite (hyperbolic).

I discovered this world 16 years ago - I'm still exploring the rabbit hole.

There are more infinite real numbers between 0 and 1 than whole numbers.

https://en.wikipedia.org/wiki/Countable_set

The 196,883-dimensional monster number (808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8×10^53) is fascinating and mind-boggling. It's about symmetry groups.

There is a good YouTube video explaining it here: https://www.youtube.com/watch?v=mH0oCDa74tE

As someone who took maths in university for two years, this has successfully given me PTSD, well done Lemmy.

The fact that complex numbers allow you to get a much more accurate approximation of the derivative than classical finite difference at almost no extra cost under suitable conditions while also suffering way less from roundoff errors when implemented in finite precision:

(x and epsilon are real numbers and f is assumed to be an analytic extension of some real function)

Higher-order derivatives can also be obtained using hypercomplex numbers.

Another related and similarly beautiful result is Cauchy's integral formula which allows you to compute derivatives via integration.

Incompleteness is great.. internal consistency is incompatible with universality.. goes hand in hand with Relativity.. they both are trying to lift us toward higher dimensional understanding..

The Monty hall problem makes me irrationally angry.

Integrals. I can have an area function, integrate it, and then have a volume.

And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.

One thing that definitely feels like "magic" is Monstrous Moonshine (https://en.wikipedia.org/wiki/Monstrous_moonshine) and stuff related to the j-invariant e.g. the fact that exp(pi*sqrt(163)) is so close to an integer (https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant). I hardly understand it at all but it seems mind-blowing to me, almost in a suspicious way.