The Mysterious Matter: Dark Matter

According to the renowned astrophysicist, Neil deGrasse Tyson, there appears to be an unknown substance in the universe that affects gravity--dark matter. Tyson claims that physicists have known about the mysterious matter since the 1930s, and they cannot decipher what this thing is. The word itself--dark matter--can be a misnomer because the thing might not even be matter. Tyson says that it should be named "Fred or some other word that has no meaning" (I paraphrase loosely and I'm sorry if your name is Fred) because scientists truly do not what that mysterious thing that alters gravity is. 

I don't know the specific equations that astrophysicists use; however, they do calculations--probably calculating galaxies and planets' movements in space--and they need an added substance to correctly map the universe's gravity. 

Clearly, scientists do not know what dark matter is, but that's the beauty of science; scientists can hypothesize what this substance is or where it comes from. The most intriguing theory--I think-- is the one that relates to String Theory. In String Theory, there is a possibility that there is something bigger than the universe which is known as the multiverse. The multiverse is pretty self-explanatory; it's a place where multiple universes exist and possibly collide. Tyson also argues that this is a valid possibility due to human history. In history, humans have made the false assumption that earth is the largest existing body; however, humans have begun realizing that there are other planets that are sporadically placed in the sky. Humans expanded the universe from earth to its counterparts, planets. It doesn't end there; humans discovered they lived in a galaxy--for sure that's the largest thing in universe, right?. Nope. They, then, discovered there are accompanying galaxies which humans now know this is the known universe. Therefore, there is a possibility that the universe is the limited view that humans see and that there is a larger place than the universe--the mulitverse. Now imagine all these multiple universes as individual bubbles. If these bubbles collide, they create an odd shape such as mickey mouse's head. This leads astrophysicists to hypothesize that the universe is colliding with another universe which causes there to be extra gravity. In String Theory, there is a chance that gravity--or gravitons--leaves from one universe and enters the other. This explains the dark matter issue, but that is not the most interesting part of the theory. If this theory is true, that means there exists another universe and makes humans ponder on how big is this place that humans live in. It truly will make any individual feel small--literally. 

Another question can be answered from this theory is "why is the universe accelerating?" According to Tyson, the universe is accelerating due to dark matter (the universe should be slowing down, but I'll talk about that in another blog post). The scary thing is that the neighboring, unknown universe can affect our universe, and we have no control over it.

The problem with this post is that there is no way that I can make you imagine how big our universe is or even how big our solar system is. It's truly unimaginable. My similes or imagery cannot possibly explain the idea of the multiverse's size. I wouldn't even know what I would be saying if I were able to apply a simile to effectively explain the size of this place. So next time when you look up at the moonlit sky, keep in mind you're seeing an infinitesimally small fraction of the universe, and even a smaller fraction of the multiverse--if it exists--however big those things are because I don't know.  

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!

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.