I want to strenuously hold that a fact can “make sense” and be mind-blowing at the same time. Indeed, *my* favorite definition of `e^x`

is “the unique function f(x) satisfying f’(x) = f(x) and f(0) = 1”

# Mind Blowing Facts

**alanza**#141

**alanza**#142

A book that is probably only tangentially related (since it covers science and not math) and not one I’ve actually read is A People’s History of Science, where the author argues against the notion (due maybe in part to Karl Popper?) that progress in science is made by Great Thinkers, and instead is more of a distilling of folk knowledge or reflecting on what a community already held to be true or assumed

**ht73**#143

One of the things I’ve always had difficulty understanding is the relationship of composition of trigonometric functions and their Fourier expansion in terms of Bessel functions (with Bessel functions as coefficients.) It comes up in FM theory.

Here’s the form it’s usually expressed:

It’s mind-blowing when you realize a similar kind of expansion follows trivially for sin(a sin(x)). But the “why” just totally escapes me. I can go through the mechanics of the proof, but the intuition, or indeed the deeper meaning behind Bessel’s differential equation, or what Bessel functions really *are* has always eluded me, because I don’t really have a background or interest in the physics, or I can’t relate the physics to the composition of sine functions, and so on.

**csboling**#145

Thanks for this, this is exactly the kind of “top down” perspective on signal processing I was seeking out a few years back. I feel like there’s this whole range of interesting generalizations of “traditional” LTI filter theory that is just fascinating. I’m put in mind in particular of Markus Püschel et al.'s work where IIRC they use algebraic methods to develop some efficient algorithms for DFTs on sampling spaces other than 1D linear ones. Michael Robinson also developed this awesome formulation of sampling/reconstruction in analogy to restriction/extension of sections of a sheaf, allowing the Nyquist condition to be expressed as a statement about sheaf cohomology, so we can generalize it to sampling spaces with topologies potentially very different from **R**^n. Definitely pretty mind blowing and something I recall only hazily now, need to go dig up my notes.

**alanza**#146

*Sheaf cohomology,* of all things, showing up in a DARPA tutorial… I guess I’ll have to eat my hat!

**pichenettes**#147

Since there are a few mathematically inclined people here, here’s a nice one about convolution.

What’s special about the Gaussian function (and distribution) is that it’s “designed” to be the Fourier transform of itself, and thus, it is stable under convolution. From this, the central limit theorem is kind of obvious: since adding random variables is convolving their distributions, it’s not surprising that if you repeat the process ad infinitum, you end up with something stable under convolution – the Gaussian distribution!

**ht73**#148

I’m having trouble drawing the conclusion of stability under convolution simply because the Gaussian is its own Fourier transform. Don’t you *also* have to assume it’s stable under multiplication, which happens also to be the case?

Periodic impulse trains are their own Fourier transform, but do not exhibit stability under convolution – the underlying Dirac sequences simply become non-convergent when convolved or multiplied. Nor do they develop as the limiting case of iterated convolution.

**pichenettes**#149

Yes, you also need that. Stability under multiplication and FT is how you can prove the stability under convolution. The latter feels stronger and more magical to me, though

**Dogma**#150

I find most math difficult to grasp on a total abstraction level and visuals nails complex ideas to the wall for me - i wish this was around when i was at school!

Really fantastic resource

**Jet**#151

Al Capone is arguably responsible for produce having “best before” dates.

Maybe not mind blowing so much as mildly entertaining…

**rick_monster**#152

This is great! My mind is duly blown that this can be explained so clearly and precisely without using equations.

**riggar**#153

I remember my father telling me when I was a boy …. ‘if you had invested a farthing in the year 0 AD at 1% compound interest - you would now have enough money to buy a lump of gold the size of the Earth for every second that had elapsed from then until now’.

**mateo**#154

Interesting tidbit I stumbled across:

At the center of the Sun, fusion power is estimated by models to be about 276.5 watts/m3. Despite its intense temperature, the peak power generating density of the core overall is similar to an active compost heap, and is lower than the power density produced by the metabolism of an adult human.

**ht73**#157

I absolutely loved this one – collecting vintage memes and running iterated image compression algorithms on new memes to achieve that vintage effect. (Lucier – “I am sitting in a room” – style). [this may be a dubious “fact”, but the ideas are wonderful…]

**otolythe**#158

I feel like this would drive richard dawkins crazy, and that makes me love it even more <3