The 4th Dimension

Don't you just love our universe? We can travel left and right, up and down, forwards and backwards. The X, Y, and Z axes. So many directions to go. But what if we could travel through space in a direction we've never heard of? Not on the X, Y, or Z axis, but on the... on the... let's call it the W axis. Well, welcome to the 4th dimension. If you haven't noticed you've been trapped your entire life, constricted to 3 dimensions. People have often pondered about if there were more than 3 dimensions, however, and it's very hard to imagine anything 4th-dimensional, let alone 5th, 6th, or 7th. So, in order to understand the dimension 4, it's best start at dimension 0, and work up from there. The zeroth (zeroth?) dimension is nothing but a single point. There are zero directions to travel in. It's easy to visualize, and in fact there is one at the end of this sentence. Now let's look at the first dimension. In dimension 1, you can have a line, with two end points. Next, in the 2nd dimension, you can have a square. Just like a line is defined by two points, you can think of a square as defined by two lines. Put two lines next to each other, connect their corresponding points and you have a square. Of course, you can have a lot more than just a square in 2 dimensions, but you'll see where I'm going with this square thing. Now in the 3rd dimension, you can form a cube by connecting the corresponding points of two squares:

In this picture, the two black squares are in different planes, and their corresponding points are connected by red lines. Let's look at the 4th dimension now, where it gets complicated. When you want to create the 4-dimensional version of a cube, called a hypercube, you use the same pattern as in previous dimensions. Make two cubes, and connect each of their corresponding corners with lines. Just like how two squares have to be in different planes to make a cube, two cubes have to be in different areas of 3D space. In other words, if one cube is in our universe, then another must be completely outside the known universe in the 4th dimension. Anyway, here's what a hypercube might look like, if we try to make a 2-dimensional picture of it:

Here, two black cubes are connected by red lines through 4-dimensional space, or at least, that's what I attempted to show. Obviously, this looks like a mess. You can get a much better understanding of a hypercube if you look at an animation of it rotating:

https://www.youtube.com/watch?v=cbprfcSVcyQ

Looks confusing, I know. Keep in mind that the hypercube is not actually changing shape. It is simply rotating in 4-dimensional space, causing us in 3 dimensions to perceive it as changing. You can better understand this by understanding what happens in lower dimensions. Imagine you have the frame of a cube, that is, a cube without faces, but only edges. If you were to cast a shadow of the cube onto a 2-dimensional surface, it would just look like a bunch of lines that are connected in a certain way. If the cube were to rotate around, the lines forming the shadow would move and change shape. What we're seeing with the hypercube is its 3-dimensional shadow. When the hypercube rotates in 4-dimensional space, its shadow changes. 

Now that you understand what the 4-dimensional version of a cube is like, hopefully you have a better understanding of the 4th dimension. To further your understanding of what 4 dimensions is like, here are a couple little tidbits of insight:

1 dimension has length, 2 dimensions has area, 3 dimensions has volume. That means in 4 dimensions you can measure, length, area, volume, and... hyper volume. 

Remember back to our axes. A point in 1 dimension is described by 1 number, just like on a number lines. A point in 2 dimensions is described by 2 numbers: X and Y. You're probably familiar with X and Y coordinates on a graph. In 3 dimensions - you guessed it - 3 numbers describe the location of a point: X, Y, Z. X and Y are for the first two dimensions as before, and Z describes depth. So in dimension 4, although we can't imagine very well what a fourth number would describe, 4 numbers tell you where a point is. 3 we are familiar with, plus an extra that describes "hyper depth" in the 4th dimension.

I hope you understand a lot about 4 dimensions now, but I understand that a blog post lacks some visualization that might be helpful to you, so I couldn't pass up showing you this great video explaining the 4th dimension. Maybe some time you'll graduate to the 5th dimension!

https://www.youtube.com/watch?v=rG6aIVGquOg

Oh yeah, almost forgot. You're probably wondering if this whole 4th dimension thing actually exists. We really don't know. Mathematically, there's is nothing wrong with even a billion dimensions. However, there is no way to tell (yet) if higher spatial dimensions exist beyond the 3-dimensional space of our universe. String theory, which you may have heard of before, is a desperate attempt made by scientists to find the theory of everything (movie reference not intended). There are many variations of it, but the point is that these string theories require more than 4 dimensions in order to work. Different variations of the theory propose a different number of dimensions. No one knows, however if string theory is even remotely correct. There's also the question of if time should be considered a dimension. But this was about spatial dimensions. We'll talk about time another time.

The Science of Fate

Recently in English class, this question came up: How much power do we have over our fates? This was one of the questions we had to answer regarding the classic play Antigone. We were, of course, supposed to answer this question by referencing the play and using quotes, but being the science nerd I am, I was much more interested in approaching this question scientifically. Putting Antigone aside, this is a fun, yet extremely difficult question to answer with science. First, let's establish the definition of "fate" as simply what is going to happen in the future. Somewhat more general, the question is now: How much power do we have over what happens in the future? There is another question, however, that we need to answer first: Is the future predetermined? To answer this question, we need to look at some scientific theories about the universe. First, quantum mechanics. Quantum mechanics is the most widely accepted theory by scientists about how subatomic particles interact. The reason this is important is that it involves true randomness. If the universe functions with true randomness, then the future is not predetermined, and therefore is unpredictable.

Here's an explanation of how quantum mechanics works and why it brings forward the fact that there is true randomness. If you don't care about it, just skip this chunk. Unfortunately, quantum mechanics defies some basic assumptions you may have about the world. Most importantly, it defies the the assumption that an object exists even when you're not looking at it and not detecting it. This makes it so things become very crazy when dealing with subatomic particles, such as electrons or photons, since they can't be detected unless they interact with something in a specific way. The reason quantum mechanics breaks that basic assumption can be seen if you understand wave-particle duality. Wave-particle duality just means that sometimes a particle acts like a particle, but other times it acts like a wave. More specifically, when we're not paying attention to a particle, it acts like a wave (so really it's not a particle at all), but when we try to detect it, it ends up suddenly popping up in a random position, and we detect it as if it is a particle. In physics we say that the particle "collapsed" into one position. This makes things very weird, since particles and waves don't act the same. Imagine you are sending a particle from point A to point B. There is wall at point B, and when a particle hits it, you know exactly where on the wall it hit. Simple. But here's the catch: After sending the particle from point A, it acts like a wave, since you haven't actually measured or detected it yet. When the wave arrives at point B and you try to detect it somewhere on the wall, it's as if the wave/particle chooses a random position to be in, and "collapses" into a particle at that position. The particle is more likely to appear where the wave is more intense.Let's say now that there is a wall between point A and point B with two slits in it. You know that you will get different results if the particle acts like a wave while traveling through the slits versus if it acts like a particle. Here's why: if the particle acts like a wave, then it will interfere with itself after going through the slits and create an "interference pattern" on the wall at point B. If the particle acts like a particle, then then it will simply have a 50/50 chance of going through one slit or the other, if it goes through at all. We can verify that the particle acts like a wave by shooting a bunch of particles from point A and seeing where they arrive at point B. This is a real experiment called the double slit experiment. In this experiment, if the particles act like particles, then we will simply see a clump of particles behind each slit. If the particles act like waves, however, we will see the clumps of particles representing the interference pattern of the waves (illustration at the bottom of the post). So the result? The particles form clumps representing the interference pattern. Particles indeed act like waves while we don't measure them. In addition, in order for quantum mechanics to work, the position a particle shows up in must be governed by true randomness, and it is more likely to show up where its wave was more intense, which explains why the particles in this experiment form clumps representing their waves' interference pattern. So there you go. That's why there has to be true randomness if quantum mechanics is correct. The random actions of small particles can then affect what happens on a bigger scale, making the future unpredictable, with many very different possible futures.

The fact that the future is not predetermined is only half way to answering the question of how much power we, conscious beings, have over the future. If the universe functions with randomness, then do beings with consciousness have the ability to actually make decisions? Or are we just part of the randomness of the universe? This is a hard topic that known science can not yet explain. Consciousness is an extremely hard topic to deal with. Just because we are conscious does not mean that our consciousness is directly responsible for our actions. The actions and decisions of conscious beings could simply be the laws of physics doing their work. It's almost disturbing to think that our thoughts and decisions could be nothing more than an illusion, and that all of it is just the laws of physics working. But there is still the possibility that we are thinking our own thoughts and making our own decisions. There just isn't a way to find out yet.

There is one more thing we haven't touched on, which is a theory about the universe that does say that the future is predetermined. The most promising theory that says this is Bohmian mechanics, although it is not nearly as popular as quantum mechanics (Read the part about quantum mechanics if you want to understand this part). Bohmian mechanics is an alternative to quantum mechanics that tries to explain wave-particle duality in a way that does not involve true randomness, and says that a particle always exists as a particle even when we're not paying attention to it. It says that for every particle, there is a "wavefunction" that applies a force on the particle, much like the wave of a particle in quantum mechanics. Particles in slightly different positions could potentially take very different paths because of differences in the force applied by the wavefunction. If we think back to the double slit experiment, slight differences in the particles' starting positions cause the wavefunction to push particles differently. Since the wavefunction is basically a wave, it interferes with itself after going through the slits, and will guide particles into clumps representing the interference pattern. So basically, Bohmian mechanics can accurately explain the universe in a way that doesn't require true randomness. But without randomness, everything has to be predetermined. 

...or does it. Consider this one last thought. What if everything was predetermined, except for what conscious beings could contribute to the universe. This makes things really weird, but this may be fun to think about. This idea means that when the universe was young, and conscious organisms hadn't formed yet, everything was predetermined. This means that it was predetermined that conscious organisms came to exist, but after conscious organisms formed, instantly the universe stopped being completely predetermined since the conscious beings were able to make their own decisions and choose how they affected the world around them. This would make it so the universe would become more and more unpredictable as more conscious beings developed in the universe. I shouldn't get ahead of myself, since we don't even know if conscious life exists beyond Earth. Either way, this way of thinking makes the ability of conscious beings to choose how they affect their world seem absurd. But wouldn't it be just sad if everything ever was always predetermined? That is why I like quantum mechanics better then any "deterministic" theory. The idea of true randomness seems to allow for more possibilities of what consciousness could actually be, even though nobody knows really what consciousness is or how it works. Here's one more idea I propose about consciousness. There is true randomness in the universe, except for what happens in a conscious mind. In the sane way that a particle that acts like a wave collapses to a single particle in a random position when measured, a conscious mind can decide where certain particles inside their brain will collapse to, instead of it being random, allowing for conscious beings to think their own thoughts and have quite a lot of control over the future. This idea is also a bit far-fetched, but who wants to admit that we aren't actually thinking our own thoughts, and that the laws of physics are thinking them for us?

What do you think? How much power do we, as conscious beings, have over what happens in the future?

DOUBLE SLIT EXPERIMENT ILLUSTRATION:

This shows how the wave would travel through the two slits and interfere with itself. The white lines on the right indicate where the wave is the most intense, and therefore where the particles will most likely end up (in this case, photons, since light waves are labeled in the picture).

http://milesmathis.com/double.html

First post!

Hello! Welcome to STEM STUFF! Here we'll explore basically just about anything that's interesting and has to do with science, technology engineering, or mathematics. So why not jump right into it? Recently I came across something interesting in math class. (For those who don't think anything in math class could possibly be interesting, this isn't the place for you.) In our exponential function notes, there was a seemingly harmless problem: Find the intersection of an exponential function and a linear function:

graphed with https://www.desmos.com/calculator

There was a story to go along with this. The population of a country is initially 2 million people and increasing at 4% per year (the exponential function). On the other hand, the food supply of the country can initially feed 4 million people, and it can feed an additional 0.5 million people every year (the linear function). The problem asks you to find out when the country will begin to experience a food shortage (same thing as the point of intersection). So, you have two functions: y = 2(1.04)^x for the population and y = 4 + 0.5x for the food supply. To find the intersection, simply equate the right side of both equations: 4 + 0.5x = 2(1.04)^x. Solve for x, then plug it in to one of the equations to get the y value. But there's a problem, since there is an x in the exponent as well as on the "ground". This makes solving for x not so easy. I tried doing all kinds of algebra with this, including logarithms, but to no prevail. After some Googling, I learned that this is not possible to solve algebraically. This was mind-boggling to me, since I had always assumed that you could "solve for x" in an equation no matter what. The reason this is not solvable is because there is not way to combine the two x's together. There will always be one x on the "ground" and one in the exponent, or always one in a log and one outside of a log. Unable to solve algebraically, there are two options left to find the intersection point: Guess and check, or graph the equations. Unfortunately, these two methods are horribly uninteresting I try to avoid guessing and checking at all costs (plus it's more time consuming), so the best way to solve this accurately is to put the equations in a graphing calculator and tell it to calculate the point of intersection. Instantly, we can find out that the two functions intersect at approximately (78.3, 43.2). This means the country will first experience a food shortage after about 78.3 years, once it has reached a population of 43.2 million people. Although not the most interesting solution, we've just predicted a country's food shortage. Maybe graphing calculators are the answer to world hunger!

To conclude, I want to say that I know a lot of people have no idea what I was talking about here, unless you remember all your high school math classes. In the future, expect posts that involve topics that a wider audience can actually understand (still probably not all of them), and include more interesting content, hopefully. I look forward to the next few months of STEM STUFF. See you next week!