Thursday, August 18, 2011

Hello There

Gutentag, bonjour, hello and everything! I've been away from blogging and what not - but perhaps I want this thing back alive, breathing and snoring at whatever measly number of readers I might've had eons ago. So! Without further much ado, let's get started in something interesting.

Well, I say interesting; I mean cool. While StumblingUpon random gooey things, I came across this:

I'll take no offense at the fact that StumbleUpon considers me a nerd - after all I programmed you didn't I, StumbleUpon? I'll take that silence as an evil robotic pause.

But back to the point - even though it's entirely true that I do in fact heart math, what caught my eye was the fine print. An equation that seems to approximate the popular image of the heart to such precision! I had to replicate it in mathematica, and fiddle around with it. So:

Heart demo

(You will probably need the free cdf player from the awesomeocracy of Wolfram)  And for those of you ever-curious about references, voici.

Of course there is nothing mathematically deep about this - its just some old function that happens to look like a heart. Not even a real heart - just one of those girlish heart hearts. Change the colorslider to red, and you'll see what I mean. Still - there are a couple of interesting things.

Firstly - surely someone must have thought a heart looked like this. Considering my aversion to blood and human bodies, I'd have agreed with them and moved on. Apparently the origins are debated - theories range from a vision of a catholic saint to a North African birth control herb.

Secondly - we have a rather neat mathematical representation of this? Quite incredible.

The eponymous cardioid graph (r=2(1+Cos(t))) does look similar:

Heart demo

That is a reasonably good approximation for the heart, but it turns out that a better representation is a particular level surface of 3D scalar fields that look like A = (x2 + y2 + z2 - 1)3 - x2 z3 - y2 z3 , which is in fact what I have plotted far up there. The 'heartiness' parameter selects particular level surfaces of that function - with the heart appearing at h=0. At other values of h, you get either a deformed blob, or three.

Philosophical implication time. Your favorite heart symbol was covered by mathematics - in fact, whatever inspired the symbol would have been some natural approximation of this function. Don't look so surprised: from simple fractals describing ferns to logarithmic spirals appearing on galaxies, broccoli, hurricanes and nautilus shells, mathematics is in fact, at the heart of the natural world.

To my fellow physicists, mathematics is a language (invented by physicists of course) to communicate their ideas about the natural world. This is true - but it misses a remarkable fact that hurricanes and galaxies were linked through mathematics much before we 'discovered' that. This deep level of self-similarity and pattern manifestation in nature makes mathematics nearly divine. Not to mention the infinite world of nonphysical mathematics.

More to come!

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