#eulers formula

[oopsallsticks]

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[art-of-mathematics]

Euler's Identity and the 3D unit helix

Sine and cosine are the 2D projections of the unit helix.

When I first saw the concept of the unit helix some years ago, trigonometric functions made so much sense.

Unfortunately, I never heard of any of it in school. Trigonometry was taught as a very unintuitive thing, as something one could not have a connection towards, as something not really 'integratable' into one's own mind. There were no elaborations about 'why' equations are the way they are, nor how the methods are used in relation to the "why", and what these methods are in concrete - in a manner which can be plotted in one's own imagination.

The abstraction appeared alienated from the own thinking process. No inner comprehension was really possible.

Yet, seeing only this concept with no further words added, it was an "Eureka-moment" for me.

In my opinion, Euler's Identity deserves the status of the most elegant and beautiful equation.

[art-of-mathematics]

Found at Jain 108 Mathemagics on Facebook

The MATHS Of WAVES: Trigonometry: Sine and Cosine as Artistic Waves Don’t let the scary, frightening Trigonometry, that was thrust upon you in your early high-school days, haunt you anymore. There is a shift happening, through the perception of Art. Just know that these are beautiful waves describing reality at intersecting 90 degree angles. Sine and Cosine are orthogonal, 2-dimensional projections of the 3-dimensional Unit Helix, which is really the Circle rotated through Space-Time. The Helix is the Spiral, the DNA, and the Unit Cube is 1x1x1. That’s as simple as it gets. With new eyes, and new perspective, the pioneering and geometric Maths of Euler’s formula can be an ever-evolving and fascinating vibrational Language of Light. by Jain 108

[antinegationism]

The reason it's bullshit btw

is that ii only equals that number (I don't even know how to say this correctly so I'll just say) "with respect to" 0. Because in reality ii has an infinite number of values since it's secretly a function. Yeah. A function. Fuck me, right? 

I don't even know how to *think* about that. Like, I understand the proof from a technical standpoint, I can even grasp it intuitively. But when it comes down to the actual thing, I'm left with this monstrosity which is a constant raised to a constant power, but still evaluates to a function = e((π/2) 2*x*π)). WHICH ONLY RETURNS REAL NUMBERS MIND YOU.   

I don't know man, I guess it's just an outlier because there aren't really any other mathematical cases I'm aware of where the result of putting 2 very fundamental concepts together can only be evaluated if you already understand a bunch of other really deep things which took humanity thousands of years to figure out.  

[the-harmony-of-the-spheres]

Heres a derivation of Euler's formula, I know the resulting identity is all over tumblr but not many people realize the derivation is relatively simple. All that is required is a knowledge of Taylor Series Expansions and it's basic algebra from there.

I also included the fact that i^i is a real and gave its decimal approximation because its one of those things that twists the mind a little.

[cosmale-blog]