No Stupid Questions

I find the logistic map to be fascinating. The logistic map is a simple mathematical equation that surprisingly appears everywhere in nature and social systems. It is a great representation of how complex behavior can emerge from a straightforward rule. Imagine a population of creatures with limited resources that reproduce and compete for those resources. The logistic map describes how the population size changes over time as a function of its current size, and it reveals fascinating patterns. When the population is small, it grows rapidly due to ample resources. However, as it approaches a critical point, the growth slows, and competition intensifies, leading to an eventual stable population. This concept echoes in various real-world scenarios, from describing the spread of epidemics to predicting traffic jams and even modeling economic behaviors. It's used by computers to generate random numbers, because a computer can't actually generate truly random numbers. Veritasium did a good video on it: https://www.youtube.com/watch?v=ovJcsL7vyrk

I find it fascinating how it permeates nature in so many places. It's a universal constant, but one we can't easily observe.

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

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

That's a pretty wild math tale!

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

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.

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

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

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.

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

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.

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.

Saving this thread! I love math, even if I'm not great at it.

Something I learned recently is that there are as many real numbers between 0 and 1 as there are between 0 and 2, because you can always match a number from between 0 and 1 with a number between 0 and 2. Someone please correct me if I mixed this up somehow.

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..

Szemeredis regularity lemma is really cool. Basically if you desire a certain structure in your graph, you just have to make it really really (really) big and then you're sure to find it. Or in other words you can find a really regular graph up to any positive error percentage as long as you make it really really (really really) big.

An arithmetic miracle:

Let's define a sequence. We will start with 1 and 1.

To get the next number, square the last, add 1, and divide by the second to last. a(n+1) = ( a(n)^2 +1 )/ a(n-1) So the fourth number is (2*2+1)/1 =5, while the next is (25+1)/2 = 13. The sequence is thus:

1, 1, 2, 5, 13, 34, ...

If you keep computing (the numbers get large) you'll see that every time we get an integer. But every step involves a division! Usually dividing things gives fractions.

This last is called the somos sequence, and it shows up in fairly deep algebra.