The Batman Symbol is Mathematical

In a previous blog, I wrote about math's beauty by talking about rose curves (here's the blog post: Who Said Math Isn't Beautiful?). In contrast to math's feminine beauty, math has a second face; a face that says, "I'm a hardcore vigilante that stops evil at any costs--glorious explosions, deadly weapons, sophisticated gadgets, devious plans, etc.--because I can." 

A math teacher decided--on his or her free time--to create a piece-wise equation that generates a symbol that is widely known--the famous Batman symbol. The equation (as seen by the very first image of this post) demonstrates the harshest, ugliest, and the most complex equation--that is legitimate--that anyone has ever seen. The equation is truly a beast. The math teacher didn't state anything that said what techniques he or she used to generate the equation. I'm assuming that the teacher used the basic characteristics of conics--parabolas, ellipses, hyperbolas, etc.--and a whole lot of trial and error. You can see the basic functions in the picture: the blue and red show the simple square root equations, y= √(x); the green portions appear to be parabolas, y=x²; the blue also shows simple linear equations, y=x.

Math can do a lot of crazy things, and a lot of equations do pop out wacky solutions. This equation not only creates a curve that displays Batman's symbol, but also demonstrates a real life application. Although this equation is truly ugly, the outcome is truly fabulous. Who knew that the alphabet soup equation can be tasty eye candy? The equation shows that the horrible situation you're in can become a positive. For example, college students may take several years to graduate and work at a dead-end job but by looking at their prospects motivates them to endure the unbearable times to reach their ultimate goal. The math teacher, more than likely, took painstaking, meticulous time to create this equation, but when it was all done the teacher must have been fully satiated. Although times may be rough, just continue walking down the unfortunate lane because soon there will be a highway to paradise. 

Einstein's Thought Experiment

This morning I woke up in my Buggati, and I was wondering what if my car's speedometer reached the speed of light, 3x10^8 m/s. If my Buggati (I don't really have a Buggati) reached that electromagnetic speed, my car would transform from a Buggati to a DeLorean. However Albert Einstein thought up this scenario with trains or rockets before I did with my fictitious Bugatti.

Einstein's thought experiment derived a time-altering equation by using simple mathematics--algebra and the Pythagorean Theorem.

In Special Relativity, the speed of light is an universal constant; however, in General Relativity, a more complicated topic, light can be altered or bent in space time by gravity. The speed of light is a misnomer because there are other things--gamma rays, radio waves, X-Rays, microwaves, and Gravitons (if they exist)--that have the identical speed. Instead of the speed of light it should be appropriately called the electromagnetic speed.

For time travel to be plausible, the speed of light has to remain constant in any reference frame. For example, there is a car that is traveling at 40 m/s and there is a truck traveling behind it at 39 m/s. The truck will perceive--in its reference frame--that the car is traveling at 1 m/s. However, if the car is traveling at the speed of light, 3x10^8 m/s, and the truck is traveling at 2.8x10^8 m/s, the truck will still perceive the car traveling at 3x10^8 m/s instead of 0.2x10^8 m/s. With that being said, the equation that was derived from the video proves that time travel to the future is possible--not to the past. By traveling at the speed of light, time is infinitely slowed down--basically stopping time. To acquire this speed, however, is completely impractical; it will take an enormous amount of energy to accelerate to the speed of light, and humans--more than likely--will not withstand that force (F=ma).

But that's no fun. Let's say that time travel is possible. With time travel comes interesting paradoxes like the Grandfather Paradox...

or Stephen Hawking's experiment.

Although time travel does not appear possible by the last two videos, keep in mind these videos prove that time travel into the past is impossible. So there is still a chance that time travel into the future is possible.

Breaking Mathematics

There are several dimensions in math--the 0th, 1st, 2nd, 3rd, etc. dimensions. The most interesting one is the 0th dimension which is commonly referred to as a single point in space. A point has no dimension which means there is no measure. If you want to find the distance between two points, you will take the final point minus the initial point. For example, let's say that you want to find the distance between 6 and 2. All you have to do is this simple subtraction: 6-2 = 4. Now instead of finding the distance between two points, we want to find the distance of a single point. For example, we want to find the distance between 9 and 9. So all we have to do is 9-9 = 0. This is good that the definition of a point holds up to this example, but how does it hold up to the number line?

Between two points on a number line, there exists another point between the points. And this can go on towards infinity. So by definition there is an infinite amount of points that make up the number line. But we just proved that a point has no measure? So how does adding up infinite amount points make a measure? It's like adding zero an innumerable amount of times and miraculously appears a unit of measure.

The number line has a unique property. There is no way to determine what the next point is. For example, let's take the number four. The next number can be 4.01, but, actually, there is a number before 4.01 and that's 4.001. So in general, after each integer there is a number that has an infinite amount of zeros with the last number being 1 (i.e. 4.000000000...1). Therefore, between two points there is an infinite amount of points between those two points. Removing a single point from the number line will not affect any measure because there is no length removed, and there are an infinite amount of points that can take the missing point's place.

So drawing a point is impossible. By simply a dabbing a paper--ever so slightly--is still an exaggeration of what a point is. Also drawing graphs are impossible because they are composed of points.

This idea is part of number theory, and this topic almost broke math. Mathematicians ignored this idea and proceeded on doing math. This is a problem because all of mathematics is based on numbers and what they truly are. If this problem was not solved, all theorems, equations, and numbers, will be false. This naive notion of simply accepting the fact that numbers are numbers is horrible way to construct math.

Thanks for those who read my blogs. I want to shamelessly plug my YouTube channel that I recently created. It's just like this blog but in a video format. Here's the link: The Integration Youtube Channel. Hope you guys continue liking my blogs! Enjoy exploring math and science!

The Universe is Based on Probability

The quantum world is filled with mystery and wonder. This equation, the Heisenberg Uncertainty Principle, shows probably the most intriguing aspect of quantum physics. In colloquial language, the equation states that there is no possible way to accurately measure a particle's position and momentum at the same time. Now for the mathematics, Δx represents the particles position; Δp, the particle's momentum. The fancy-looking h, ħ, is known as plank's constant which is approximately 6.626 x 10^(-34). To pronounce ħ, say "h-bar." This equation is truly non-intuitive because from the human perspective we can accurately measure an object's velocity, momentum, position, acceleration, and can even predict it's future velocity, momentum, position, etc. by using classical mechanics; however, in the quantum world classical physics breaks down and turns into probability.

Now you may be asking, "why is it so hard to measure these quantities? Don't we have advance microscopes to measure these values?" Firstly, to view an individual particle we need to shed light on it, so it becomes visible. Photons, light particles, can cause a disturbance to the particle's position like when a pool player hits a ball; the ball will accelerate to a different location. Shedding light is equivalent to millions of photons hitting the particle we want to observe. This causes a problem when we look through a microscope. Instead let's just use one individual photon instead of millions to view the particle.When the individual photon bumps the particle, the photon acquires a new direction as well as the particle. So when the photon arrives to the microscope at a new angle, it hits the lens causing it to refract even more (refraction is when light bends). Since we perceive the particle from where the photon is coming from, the measurement is completely inaccurate because of the photon's collision and refraction.

This issue forces us to use probability to locate the individual particle.

This is a probability wave. The probability is measured on the y-axis which is labeled p; the x-axis measures the particle's location which is labeled x. The peaks, or crests, show the highest chances of the particle being at that location. While the bottom peaks, or troughs, show the lowest chances of the particle being there. By using the concept of the derivative, these likely and unlikely possibilities are easy to find (we'll talk about derivatives in a future blog).

That leads us to a question, "is our world governed by probability?" Each human is made up of these infinitesimally small particles, and these same particles have the characteristic of probability. It's like seeing a friend sitting next to you in class, but he can also be sitting across the world in China. Since that probability is extremely low in our perspective, it's almost guaranteed that he or she is sitting next to you. However, having a high probability does not eliminate the other chances. So that very small chance of your friend sitting in china is still plausible. Are we misunderstanding particles' actions? Is there something that controls the universe that deceives us, the observers, to conclude that the universe is based on probability? There has to be something we're missing... I'll leave you with one of my favorite quotes from Einstein.

"God doesn't play dice with the universe." - Albert Einstein