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This image I shared above has sometimes been called the most beautiful mathematical structure in the world.
What is it?
Well, try to follow me here; this is a two-dimensional representation of a three-dimensional representation of a four-dimensional projection of an eight-dimensional object. Okay, even I could not follow it- someone else said it so I won’t take the credit. I like to keep things simple so let us get to that.
If you take an eight-dimensional object and project it in a four dimension then represent that in 3d then put that on a 2d screen like you just saw in the image above this is what you would end up with. This structure is called an e8 Lee group.
What does that mean?
It is a 248-dimensional object but it can also be thought of as an object that has eight spatial dimensions with 248 symmetries. In the 19th century, the mathematician sophist Lee created algebraic formulas to describe the shape of symmetrical objects; these are called Lee fields. Then in the late 1890s perhaps the most complex shape in our universe was described- the e8 group.
A super complex structure
This structure is so complex that the drawing you saw above took a team of eighteen mathematicians years to calculate and plot using a supercomputer.
Why is this structure important?
Well, it happens to show up in parts of string theory but also in 2007 a theoretical physicist and a surfer dude Anthony Garrett Lisi published a paper proclaiming the mathematics of this structure contained all the particles and forces in the universe. He called it the exceptionally simple theory of everything. Now, this was just a pun because there’s nothing simple about it.
How can this geometric structure be the key to the fundamental nature of the universe?
Well, what you have to understand is that all geometric structures have their underpinnings in mathematics, that is all geometry can be expressed as math. For example, a circle is 2 times pi times R, and a sphere is 4 times pi times R square. This holds for this e8 object as well; it is much more complicated mathematics but the structure can be expressed in mathematical terms and mathematics happens to be the language of the universe. What Lisi did is he found a relationship between this object and all the forces and particles we know about.
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How did he do this?
First, if you were to look at the structure in the image closely you would see that it has 248 points or intersections. Just remember that number 248 for a minute. Now let me tell you how that number is related to all the forces and particles of the universe.
The standard model
Our best understanding of the universe is represented by the standard model. According to it, there are four fundamental forces in nature- strong force, weak force electromagnetism, and gravity and there are twelve fundamental particles. In addition, each of these particles has an antiparticle of itself.
So, for example, an anti-quark or antineutrino in total makes up all the elementary particles and there are exactly 28 of them, twenty-four elementary particles and four force carrier particles. Each of these distinct elementary particles has eight quantum numbers assigned to it based on the charges each particle has. This brings the total number of particles to 220 for eight times 28. Lisi found that he could mathematically equate all of these particles to one of the points in the e8 model but the model if you recall has 248 points, not 224.
How did Lisa explain these points?
Well, he simply created 24 new theoretical particles which he thinks have not been discovered yet. So even though none of these theoretical particles has been found his work is still remarkable. Enough with the mathematical talk; I am not made for this. Kudos to the guy who explained these terms.
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