Who Said Math Isn't Beautiful?

Who said math couldn't be beautiful? Math comes in various shapes and sizes that are very attractive: triangles, squares, prisms, parabolas, waves, spirals, and even flowers! Actually, mathematicians don't call them flowers; they call them rose curves. Here take a look!

This is a rose curve that follows this equation r = cos(4θ). The reason why the curve appears like this is because of polar coordinates. Polar coordinates spice up the Cartesian plane by replacing rectangular coordinates--x and y--with θ and r. θ is the angle from the positive x-axis to any given point; on the other hand, r is the length measured from the pole (or the origin) to the given point. As you can see functions like these are cyclic which causes the curve to repeat patterns; in this case a rose petal repeats itself as θ increases from 0 to 2π radians (or 0 to 360 degrees). Pretty cool, huh?

For more generalized equations to make rose curves, follow these formulas: r = ksin(nθ) or r = kcos(nθ). The constant, n, determines how many petals the rose curve will have. If n is an odd number, the curve will have n petals. If n is an even number, the curve will have 2n petals. The other constant, k, determines how long the petals will be from the origin. Before drawing rose curves on graphing calculators, be sure that your calculator is in radian mode and using polar coordinates!

Polar coordinates can create more wild shapes, but I'll save those for future blogs.

These curves alter people's perception toward math. People see math as black and white, and they don't see its true color. Hopefully, these petals pretty-up the conventional y = mx+b. Truly, math is full of boring (most may think) equations, but it's these boring equations that make the world beautiful. Just take a second and look around you. You don't just see a computer screen, a desk, a table lamp, you also see math's product. Look outside. Our whole world is governed by mathematics: fractals, bell curves, infinite series, parabolas, inverse square laws, derivatives, integrals, geometry etc. Take the time to look at math, analyze math, and soon you will lift the veil of boredom and find a beautiful figure underneath it. 

Infinite Sums are Weird

What if I told you that 1-1+1-1+1-1+... = 1/2, and I proved it?

So lets take an infinite series, S.

S = 1-1+1-1+1-1+1-1+...

Now lets factor out a negative one from the second term to infinity...

S = 1-(1-1+1-1+1-1+...)

The grouped values are identical to the original series, S. 

S = 1-S

Now with some help from algebra, we can solve for S.

S = 1-S

2S = 1

S = 1/2

Therefore,

1-1+1-1+1-1+... = 1/2

Who knew? Well actually this is categorized as mathematical hocus pocus because this breaks math. In order to use the basic properties of arithmetic (i.e. associative property, commutative property, etc.) the series has to converge. For a series to converge the sum of the infinite series has to add up to a finite number. In this case, we are unsure what the series converges to because it can add up to 0, 1, or 1/2--which is it? If you want to see a visual proof on YouTube check out this video.

Euler, a mathematical gangster, disobeyed the rules of mathematics and found even more surprising values for infinite sums. He concluded that 1+2+3+4+5+... = -1/12. Although this sounds nonsensical, this does apply to quantum physics. Physicists go by the rule that says nothing in the universe has an infinite measure, and these insane values surprisingly appear in their experiments. Although I don't know the details or the direct application to the field, I find it amazing that there is physical proof of nonsense existing. 

Physics generally follows this order: develop an equation and experiment with it (or vice verse). Excitingly, physicists found that math is wrong--in the quantum world--and experiments are correct. Usually, math is correct and the experiments are wrong. This is mind boggling because which one should we believe? 

In our lives, we know what's nonsense and what's correct. However, our field of view is limited from our frame of reference. A human experiences the universe on a human level. If we break out of this perspective and look from a planet's perspective our view changes dramatically. We'll see--as a planet--empty space and feel like we're floating amid nothingness. We'll see our counterparts: stars, asteroids, galaxies, planets, etc. Or instead of a planet's perspective, we look through a quantum perspective. We'll see particles whizzing by, we'll feel the electrical force, and we'll possibly see other dimensions (as theorized from String Theory). What if we take this a step further? What if there is a world smaller than the quantum world or a realm larger than the universe? Is this where all the mathematical hocus pocus is spawned? 

I'm an advocate for being open-minded. I like to accumulate many theories, ideas, and suggestions and determine which is correct and wrong (based on my opinion). This is not only limited to math and science, but also to real life situations. Teachers may provide life lessons in their classrooms and so do parents in their homes. People completely ignore the teacher and follow their parents advice because it's ethically correct--some may think. I believe that one should not completely ignore suggestions due to bias or alleged nonsense; instead one should take in both and generate a more appropriate conclusion. The same goes for math and science. Who knows? Maybe there are parallel universes, time travelers, different dimensions, green aliens, flying pigs, etc.