# euler spiral

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So, the euler spiral! It’s the big one at the top. It’s traced out by the parametric equations written under it.

I thought something interesting would happen if I replaced the cosine and the sine with the derivatives of the parameterizations of various other curves, and then this happened.

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Euler’s Identity: eiπ + 1 = 0.

Euler’s Identity is an Equation about constants π and e. Both are “Transcendental” quanti­ties; in decimal form, their digits unspool into Infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect Symmetry and closure of the Circle; it’s in Planetary Orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows.

What Euler showed is that π and e are deeply related, connected in a dimension perpendicular to the world of real things - a place measured in units of i, the square root of -1, which of course doesn’t … exist. Mathematicians call it an imaginary number. These diagrams are visual metaphors. Imagine a graph with real numbers on the horizontal axis and imaginary ones on the vertical. Exponential function, f(x) = ex, ordinarily it graphs as an upward swooping curve - the very paradigm of progress. But put i in there, Euler showed, and eix instead traces a circle around the origin - an endless wheel of Samsara intercepting Reality at –1 and +1. Add another axis for Time and it’s a helix winding into the Future; viewed from the side, that helix is an oscillating sine wave.The rest is easy: Take that function f(x) = eix, set x = π, and you get eiπ = -1. Rearrange terms and you have the famous identity: eiπ + 1 = 0.

That’s the essence of Euler’s alchemy: By ventur­ing off the real number line into this empyrean dimension, he showed that disruptive, exponential change (the land of e) reduces to infinite repeti­tion (π). These diagrams combine the five most fundamental numbers in math - 0, 1, e, i, and π - in a relation of irreducible simplicity. e and π are infinitely long decimals with seemingly nothing in common, et they fit together perfectly - not to a few places, or a hundred, or a million, but all the way to forever.

You can take this farther, too. If you write that function above in a more general but still simple form as f(x) = e(zx), where z = (a + bi), what you get is no longer a circle but a logarithmic spiral, combining rotation and growth - now both at the same time- These graceful spirals are also found everywhere in Nature, from the whorls in a nautilus shell to the sweep­ing arms of Galaxies. And they’re related, in turn, to the Golden Ratio (yet another infinite deci­mal, 1.61803 …) and the Fibonacci Sequence of Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …).

But the weirdest thing about Euler’s formula - given that it relies on imaginary numbers - is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems into more tractable algebra - like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digi­tize music, and it tames all manner of wavy things in quantum mechanics, electron­ics, and signal processing; without it, computers would not exist.

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Using an orange to illustrate an Euler spiral.

Cornu Spiral