Symmetries & Group Theory

What Are Symmetries?

You may have heard of a study done where babies were shown to respond better to adults that were considered to be beautiful. Beauty, in this instance, was defined as those having a more symmetrical face. Symmetry and beauty are often intertwined; that our universe has certain patterns that allude to there being an underlying intelligence gets anyone excited. The wings on a butterfly have symmetry, for they appear to have the same pattern mirrored with each other.

When we think of things in nature having symmetry, we often think of a reflection – but in mathematics a symmetry has a more durable meaning: when an object remains invariant under any form of transformation. An object has symmetry when you can do something to it and it still retains its shape and position. When you observe a snowflake and its symmetry, we think of symmetry as a passive quality – that an object merely possesses this quality. Very much like our popular conception of beauty, that one has it if they possess symmetric features. 

But when we discuss symmetry in mathematics, it is more of an action that you perform on a shape or object rather than a characteristic that it has. In the next section you’ll learn how we can do this.

As we mentioned in earlier, any integer can be constructed through a combination of adding prime numbers together. We can do something similar with symmetries. For example a 15-sided figure could be constructed by the rotations of a pentagon and a triangle, both which are prime-sided objects. These indivisible, prime groups of symmetries are the building blocks of all other symmetries.

There are various types of groups, like infinite groups (line or circle) and finite groups (snowflake). Lines and circles are infinite groups because one could make the argument that you could permute it an infinite amount of times and still retain its identity. Finite groups like snowflakes have a bounded number of permutations that can be applied to them. There are also simple groups that are the basic components of finite groups the way integers break down into primes and molecules break down into atoms. In order to categorize all finite groups, you first need to find all simple groups, then you need to find all the ways to combine simple groups.

Three dimensional shapes can be part of the same group if they share the same symmetries. You can picture an octahedron in a cube and a cube being in an octahedron by having their vertices at the focal intersection point of their face. In other words, you can fit one side within another; in some visual way sharing that property means having them be within the same group. You can start to see just how much we abstract out groups and hopefully this gives you queues when looking at different objects to see if they share any symmetries.

How beautiful is it that as we learn more about numbers, we realize that their physical applications in shapes and objects possess similar properties. Math and science tends to do that — understanding one domain leads to discoveries in other domains.

Group Theory

Group Theory is a subset of a branch of mathematics known as abstract algebra. Abstract algebra is the study of mathematical objects and structures. In the late 1800s, mathematicians noticed that similar mathematical tools were being applied from one field to another. Rather than reinventing the wheel, the branch of abstract algebra was created to try and generalize this tooling to be applied across these various domains, some like algebra, geometry, number theory, and topology.

At a high level, a group consists of sets and sets consist of any combination of things: numbers, functions, shapes, objects, matrices, etc. What separates groups from sets is that a group always has an identity element. Sets are usually just a collection of elements while a group are a set of elements with an operation. Group theory aims to show that certain sets are part of the same group, so long as they adhere to certain elements and properties. Once their grouping is shown to be true, we develop a unique type of math that can bridge the gap between our various  math domains, but also extend out to quantum physics where group theory has assisted in novel breakthroughs.

Let’s start with an example, let’s take an equilateral triangle where each angle is 60 degrees. Imagine you have a cardboard cutout of this equilateral triangle on hand, how can you demonstrate the various symmetries this triangle has? (Remember earlier we defined symmetry as transforming the shape or object in such a way that it retains its shape and position). Well, you’d probably start to rotate it once and find that it looks the same; maybe rotate it again once more. You may flip it in various ways as well. This is the correct approach, but we can apply some math to our methods by associating each transformation with a variable. 

Above you’ll see on the top left corner a triangle with each vertex labeled with a number – this is to help us visualize the transformations we’re applying. Doing nothing to our shape is actually a unique type of symmetry called the identity property. To the right we have R1 where we rotate it 120 degrees. R2 is rotating it another 120 degrees to now position it at a full 240 degrees. If we rotate it once more, we are back to our same position. Now, we can flip our triangle along the y-axis to get F1 and flipping it along each diagonal x-axis gets us F2 and F3. We’ve exhausted all our transformations for this triangle and are now left with our set, or said differently, our composition of transformations: S = {I, R1, R2, F1, F2, F3}.

Different groups can have a binary operation applied to it, which is saying “how can we further transform these elements within this set?” A binary operation is a fancy way of saying we’re going to apply some symbol as we continue to transform these elements, sometimes it’s an addition sign, other times it’s a multiplication sign.

Before we begin combining elements within a set, these compositions need to adhere to the following axioms:

1. Closure → If I combine any 2 elements in a set, the result will always be another element in a set. (∀ x, y ∈ S, x*y ∈ S)
2. Associativity → Basically ordering of things doesn't matter, like when you’re multiplying elements and some are enclosed in parentheses. (∀ x, y, z  ∈ S, x*(y*z) = (x*y)*z)
3. Identity → There must exist an element that leaves any element in the set unchanged.            (∃ e ∈ S, ∀ x ∈ S,x*e = e*x =x)
4. Inverse → Like how 1 of x is a multiplicative inverse of (Like how x * 1/x = 1/x * x = 1) (∀ x ∈ S,∃x-1∈ S,x*x-1=x-1*x=e)

Once our sets follow these axioms, we can then confidently use deductive logic to build our groups. Remember how we mentioned we could use an addition or multiplication binary operation? Let’s elaborate on that, technically you’re just applying arithmetic to these different elements. We can use addition, but we don’t use subtraction because subtraction is really just the addition of a negative number – so for simplicity sake we stick with the addition operand. Multiplication applies as well and you’ll see later they’re usually reserved for certain types of sets (remember, we can use this for numbers, matrices, shapes, objects, etc). Division in this case wouldn’t work because we can easily violate the property of closure, which states that it must result in an element in another set. If you have 5 as one element and 3 as another, division can make it so that you end up with ⅗ or 5/3, which is a non-integer that we wouldn’t have in a set. But we do have an alternative where we can use multiplication in its place, since 4 divided by 8 is really 4 x ⅛.

Binary operations can also not be commutative, i.e. (a * b  ≄ b * a) and with division we can limit ourselves from expressing the relationship between elements within different sets – so this was omitted at the inception of group theory. 

With our set on hand, we have an array of the first transformation we applied to our equilateral triangle as follows → S = {I, R1, R2, F1, F2, F3}. Now, we need to determine if this set abides by the four axioms we have listed above. We can do this with something known as a Cayley table (shown below) that allows us to pit the same set against itself in a manner similar to a multiplication table. With this, we can determine if this group is abelian, which means that the order in which the binary operation is applied to these two elements of transformation does not matter. If we find that it in fact does matter, it can still be part of a group, but known as non-abelian.

First, does this set have closure? You’ll see above that every element is represented once within each row, which satisfies the element of closure.

  Does this set have associativity? While it would be an exhaustive list for us to represent that each permutation within this table adheres to: a * (b * c) = (a * b) * c, for the sake of simplicity, we can assume that this is the case here.

Next, does this set maintain identity? It does because we can see that there is an identity element, I, that is present within each row.

Finally, does this set contain an inverse? If you look at where we have “I” listed above in the table, this is known as the identity skeleton. For a handful of these values, we find that the identity element is achieved when pairing the same values for some of these: F1 & F1, F2 & F2, and F3 & F3. For R1 & R2, the pairing of R1 & R2 achieves the identity element twice, adhering to x * x-1 = x-1 * x = e. So, we can confirm after adhering to all four elements, that each set of compositions and its binary operation comprise a group.

Regardless of which type of operation we use, there is always going to be some way to find its identity element. When we have an additive operation, you get the identity by adding 0. When we have a multiplicative operation, you get the identity by multiplying it by 1. Each of these leaves the number unchanged.

We explained earlier why subtraction is really addition of the inverse of that number, and division is the multiplication of the inverse of that number. The inverse of an additive operation is the negative of the number (so the inverse of 5 is negative 5) and the inverse of a multiplication operation is a fraction (the inverse of 5 is ⅕).

At the heart of group theory is taking various sets and determining if they are part of the same group – we showed that we have tools to use to determine if an element in one set maps to another. Not every distinct element in one set needs to map to another distinct element in another set in a one to one fashion to be part of a group, but if it does, then this is a bijective relationship known as isomorphism.

In our example, an equilateral triangle has 3 sides and 3 vortices, which would mean it has 3! arrangements, or 6 arrangements. It also has 6 configurations as well which matches what we have listed. However, a square, with 4 sides and 4 vortices, has 4! possible arrangements, or 24, but only with 8 configurations. You can also apply sets of symmetries in a sequential fashion, known as compositions, which are their own types of symmetries. Our Cayley table helps us unearth these additional configurations using our binary operation.

We just defined the symmetry group of a triangle, but this exercise can also be extended to polygons and anything, really. I mentioned before that this can be done with a set of matrices, too. While it might look complex and cluttered, abstracting each element as a variable makes it less intimidating and applying the methodology above allows one to determine the characteristics of the group. Remember, symmetry is what we do to an entity. So what can we do with a clock? Well, there’s plenty of symmetry within a clock and its movement of its hands from 12 to 12 in a circular fashion and there is a unique type of math that applies to this called modular arithmetic.

If you’re on the 7 mark of a clock and I ask you to add 1, then you move the hand until you arrive at 8. Simple enough, but what do you tell me when I ask you to add 5 to 8? The answer in this case is 1. What if I ask you to subtract? Remember, we would add a negative number in the binary operation for symmetries. Subtracting would just be moving the hand of the clock back. If you started at the 12 mark and I asked you to add 12, you would end up in the same spot, which sounds a lot like when we did nothing to our triangle and preserved its identity element. Well, each axiom in comparing sets can be done with clocks  and you’ll find that it also uses addition as its binary operation. Its inverse can also be determined by turning the hand of the clock backwards and the associative and closure elements can be gathered as well.

Noether’s Theorem

Throughout this book we touched upon the back and forth relationship between math hinting at a novel scientific discovery and science hinting at the need for a new branch of mathematics. One vital account helped fill the gap in one of the most important theories: Einstein’s field equations for general relativity.

While the theory was promising when published in 1915, it had violated an important fundamental principle in physics: the conservation of energy. Using the framework of general relativity, you’d get odd results when trying to write an equation for the conservation of energy; the math didn’t add up.

Luckily during this period, group theory was in full swing and one important part in this branch of abstract algebra that could be the key to solving this, is the use of invariants: properties of functions or groups of functions which remain unaltered when the function is transformed. Emmy Noether, a mathematician who was an expert in invariants, was sought after by the mathematicians Felix Klein and David Hilbert to see if she can unravel the inconsistencies in the framework of general relativity.

She found one in 1918; her theory known as Noether’s Theorem states: “For every symmetry, there is a corresponding conservation law.” More specifically, “if the Lagrangian of a system has a continuous symmetry, then there exists an associated quantity which is conserved by the system, and vice versa.” Essentially she found a bridge between symmetries within the law of nature and a conservation of quantity.

Simply put, she showed that these laws of physics applied in one coordinate in space should apply in another as well. Energy can neither be created nor destroyed, we learned this in physics class, but does that energy remain conserved regardless of where it is in space? The short answer is yes. 

Noether states this is because there is an underlying symmetry which is evident within these different conservation laws. We saw the identity property was applied when an object remains unchanged when transformed, if an object is able to retain its energy elsewhere within space, isn’t that implying the same thing? Wouldn’t this insinuate that there is a symmetry this object adheres to which can be applied? This goes for the laws of motion, momentum, angular momentum, etc. With the mathematical proof she provided, any law of nature adhering to the Lagrangian of a system, or the least action principle, can now be mapped to a symmetry. What you can do in one part of space you should be able to do in another. Because of this, space has something called a translational symmetry. One thing we learned from Albert Einstein is that space and time are really just different aspects of the same thing: spacetime. Therefore, time has the same translational symmetry as space. If a system is symmetric under time translations, then it must conserve energy if the Lagrangian of the system does not explicitly depend on time (as it may with velocity).

Because time is not an absolute quantity as it is in the Newtonian world, it warps and stretches with space. So, time translation symmetry would apply in cases where spacetime is flat (or asymptotically flat). Outside of that, energy doesn’t need to be conserved which resolves the issue Einstein faced with his field equations.

In our earlier example of symmetries, we rotated our triangle 120 degrees to reach a symmetry. Only by rotating the triangle in 120 degree increments do we achieve symmetry, so it is known to have discrete rotational symmetry. A circle on the other hand can be rotated by any amount and always maintain its symmetry, so it’s known to have continuous rotational symmetry. Noether was implying that the conserved energy that is found in nature also has a continuous rotational symmetry.

An electrical charge is also conserved which implies there is an underlying symmetry that is at play. This is what the field of Quantum Electrodynamics tackles – Noether’s work has brought symmetries into the world of physicists and the translation of symmetries and conservations of quantities is also being used to prove the existence of gravitational waves. The famous physicist Richard Feynman spent much of his career working on Quantum Electrodynamics (or QED for short) and became popular for the simplicity in which he was able to explain these rather difficult concepts, using visuals dubbed “Feynman diagrams”.

Evariste Galois

In 1832, Evariste Galois made his contributions to group theory and abstract algebra through a hastily written farewell letter the night before he was due for a duel that ultimately took his life at the age of 21. In it, he wrote a method for determining when to use radicals in an equation as well as how to handle quintic equations (those that contain a variable raised to the 5th power).

As we covered earlier, various civilizations found solutions and formulas for solving quadratic equations. They extended the search to higher powers, solving cubic and quartic equations. The question now was whether there’s a way to do this for quintic equations, or those that are raised to the fifth power. When do we know to use radicals to solve equations to the Nth dimension?

Often when solving an equation like x2 - x - 2 = 0, we break this formula down into (x - 2)(x + 1) = 0 which can be broken down even further into x - 2 = 0 and x + 1 = 0. We took this equation and beat it down with a hammer until it cannot be broken down any further, we call these irreducible.

With the example above, the quadratic equation had two solutions and for cubic solutions you would have three. It follows that the number of solutions for your equation is equivalent to the number of dimensions it contains (as stated in the Fundamental Theorem of Algebra by Gauss in 1815). These are known as polynomial functions to the Nth degree.

What’s unique about irrational solutions is that they come in multiples and possess a symmetry amongst each other. The number of solutions within a given equation possess a particular type of symmetry called a permutation. A permutation is a switching of a sequence of an object or a cluster of some sort. For instance if we have 10 apples and I swap the last two, I’ve created a new permutation. To exhaust all the different ways in which I can swap all 10 apples in a unique combination would be its full permutation. You often find this by quantifying the factorial of the number of items/digits in the set. In this case for the apples it would be 10 factorial, which is 3,628,800 different permutations.

A permutation is a type of group, because really a group is a type of action that expresses the symmetry. We’re used to thinking of symmetry in the visual sense, where you can rotate and flip an object to retain its identity – but it will be easier if we refer to symmetry here as a verb, a thing you do to a group to retain its identity. In our case, the permutations within a given equation is known as the Galois group. If you remember from earlier, you should be able to take two elements within a group and produce another element, which makes sense for why the number of permutations is equivalent to the factorial of the number of elements within the group (it relates to the number of combinations one cycles through as they do the permuting.

There are quadrilateral, trilateral, and problems where the x to the highest power is five or more. Solutions to those problems come in the form of roots, where you have a square root, cube root, etc. In order to solve those, you have a function that has been created to find the x. In the case of those to the power of 2, we have the quadratic function: x = -b b2 - 4ac2a. Solutions for polynomial functions for up to the 3rd power were used by Babylonians, formalized in a geometric form by Omar Khayyam, and then standardized in numerical form by Scipione del Ferro. Ludovica Ferrari discovered a function for polynomial functions to the power of 4, but solutions to those of powers 5 and higher were difficult to find. They remained this way for a while, until Evariste Galois came along and expressed his ideas through his final letter before his death.

What Evariste Galois did that was so innovative, was rather than creating a function that would explicitly find the ‘x’s in the solution, he used symmetry and group theory to determine what properties a quintilateral function would have.

Galois’ approach was to group the allowable permutations of the operations that are applied to the sets within a group, which are known as Galois groups. Just as in the Goldbach Conjecture, we have every integer that can be broken down to its base groupings of prime numbers which cannot be broken down any further. And so, we do the same with the operations in hand to the group. A flip is one, rotation of 90 degrees is another, rotation of 120 degrees is another, etc, and the number of times you must do a single operation to reach this point is known as its order. For example, a square can have a 90 degree rotation which can be done 4 times before reaching its identity, so it has an order of 4. A group that is generated by a single operation is known as a cyclic group and its size is the order of the operation in which we started with. When the order of an operation of a group is a prime number, it is known as a prime cyclic group.

Just like with prime numbers, a prime cyclic group cannot be broken down further, implying that it has only one cyclic group of that size. But unlike in real numbers, not all cyclic groups can be broken down into prime cyclic groups and this observation surfaced the key Galois was looking for in solving the quintilateral functions (and those of degrees higher). What Galois uncovered was that those which can be broken down into prime cyclic groups can be solved using roots and those which cannot be broken down into prime cyclic groups cannot be solved using roots.

Unlike in real number sets where prime numbers are the basic building blocks to generate new numbers from, in the world of Galois groups, the smallest group size that acted as the basic building blocks of symmetry outside the prime cyclic groups, was the even integer 60, which means that for us to determine the number of permutations available between five objects, there are 60 permutations. It’s hard to imagine how an even number can be a basic building block considering it is always divisible by 2 – our gut reaction might be to question why this cannot be broken down into two size 30 groups. 

But here’s why: the number of permutations that can be applied must be either even, or odd – it can never be both. Transposing two objects can be like switching seats between two people. To get them back in their original spot, you need to switch them again, which is even-numbered. When you’re dealing with permutations of five objects or more, the group of all even-numbered permutations is considered simple, hence an atom of symmetry and cannot be deconstructed further into the prime cyclic groups.

This reasoning was used to describe why the solution for quintic functions cannot use roots of any sort (because it cannot be broken down any further) and led Galois to state, in an elegant and unconventional way, that there are no functions one could devise to solve for polynomial functions with powers of 5 or greater.

There is an entire branch of mathematics dedicated to the way in which sets of elements and their mathematical relations can be permuted, known as combinatorics. It is often used in computer science when analyzing the complexity behind certain algorithms. Abstracting anything to groups and using the underlying mathematics to group them by their combinations is a powerful tool that opens up doors to new discoveries as you’ll find with the Umbral Moonshine Conjecture, which proposes there are these 24 mathematical objects known as moonshines, which involve the dimensions of a symmetry group on the one hand, and the coefficients of a special function on the other. The Monstrous Moonshine is currently the largest and most-complex symmetrical object to theoretically exist. It’s discovery took the coordination of many mathematicians and several years.

Groups can also have subgroups, and we traditional represent a group as G and a subgroup (or subset) as H. To show that H is a subgroup of G, we write H < G, which is pronounced “H is a proper subgroup (or subset) of G.” G can also be a subset of itself, which is very similar to our discussion around identity elements, and we can represent that as G G. The “less than or equal to” implies that it can be equal to it as well, but less than in general means it’s a subset of the value to the right.

Sub-groups can be used to inform the greater group as a whole. You may find that a group has 24 permutations and a sub-group has 8 – any subgroup should be able to cleanly divide into the number of permutations of a larger group as stated in Lagrange’s Theorem. It would be written as follows |8| |24|. This tells us that if we are to look for the subgroups of a group with orders of 24, we would first include the identity element: |1| & |24|. We would also include its discord |3| and |8|. But 24 can also have the following divisors: |2|, |12|, |4| and |6|, does this mean that there MUST be subgroups with those orders?

No, Lagrange’s Theorem states that just because it is divisible by the total number of permutations, it doesn’t always have to exist, except in the instance that one of the divisors is prime. Yet again, prime numbers make their way as a fundamental building block. A concept called cosets was created to prove Lagrange’s Theorem.

Often what you’ll find in the history of mathematics is that new terrain in mathematics are discovered that seem far away from the traditional body of mathematics. In due time, concepts are created to bridge that seemingly far away terrain with the existing body of mathematics, only to find that they are intrinsically connected. Algebra and geometry are a great example, ushering in the field of trigonometry – with Galois groups, we find that polynomial functions are dictated by the grouping of how one can solve them.

Anytime we deal with functions of a higher degree, we comprehend their many dimensions through the use of fields. As you’ll find later, the structure of fields and their mathematical construct give us a newer lens to view the intricacies of this mathematical construct. Galois’s theory showcased the relationship between the structure of fields and the structure of these groups – with these we see how the roots of the polynomial relate to one another (which exist in what is called the splitting field of f(x)). From the Fundamental Theorem of Galois, applying the symmetry of the Galois groups to the splitting fields unveils the structure of the group and the field.

With Galois groups, Evariste Galois was able to bring algebraic equations into the realm of group theory. Now you’ll learn of Sophus Lie who was able to bring differential equations into the realm of group theory.

Lie Groups

Building on Group Theory was a 19th Century Norwegian mathematician, Sophus Lie. Just as Galois sought to explain polynomial equations in the language of symmetry, Lie sought to do the same but for differential equations, which are equations that involve rates of change. As we saw earlier with Galois’s algebraic functions, we had a finite number of results as dictated by the value of the power inherent in the function. Unlike polynomial functions, differential equations can have an infinite number of results. For example if we had a differential equation describing a vibrating string, you could try and narrow it down to where someone had pressed upon a string, at a specific point in time.

Lie wanted to consider all of these infinite possibilities in the form of groups and since these are continuous functions, how one answer morphs into another. The concept of something morphing into another will be touched upon later when we cover topology, but both topology and symmetry rely heavily on this idea of group theory. The thing with rates of change that fluctuate is that they always do so in a gradual fashion and so Lie focuses on something called continuous transformations.

This notion of continuous transformations lies at the heart of Lie Groups. Lie groups are a finite, continuous group which means that within a group there are a finite number of degrees of freedom where continuous transformations can occur. There is something known as Zeno’s Paradox where  someone progresses forward in a race but to finish the race, they must get halfway there. To get to that mark, they need to get halfway there as well (quarter of a way), and so on and so forth. By this logic, there will be an infinite number of halfway-progressions and the athlete will never end their race. The reductio ad absurdum paradox here is that time is seen as a sequence of events rather than the true fluidity in which it functions.

Just as Zeno saw a fluid sense of time as really being a myriad of sequential instances in time, we can use the Planck length to act as the broken down bits of time -- the units we’ll use in these differential equations. A Planck length is the distance light travels in a vacuum to cover a Planck time, anything smaller than this and quantum physics breaks down. Lie uses multidimensional geometry to work out the specifics of these differential equations.

If we take a CD and put our finger through the hole, we can spin the disc. Each time you rotate the disc you are enacting a type of symmetry and those groups of rotations, which as you can imagine continuously morph amongst each other, are part of a single Lie group -- a one-dimensional Lie group.

Because the flat disc was rotating around our finger, it had one degree of freedom, but if we placed that CD on a piece of paper and moved the disc around it, we’re able to operate within three degrees of freedom (one degree for rotation and two degrees for the two-dimensional surface the piece of paper has). It’s weird to think of that CD moving around on a piece of paper as three-dimensional since the piece of paper is two-dimensional -- the curvature adds that third dimension, a three-dimensional curved space, which is different from the three-dimensions of Euclidean space (length, width and height). If we were to replace the CD with an apple and have the apple inhabit our three dimensions of Euclidean space -- then we’re left with six degrees of freedom (three degrees for velocity and another three for its spin). Because that apple can theoretically have six degrees of freedom in a three-dimensional Euclidean space, we can think about what effect this might have on the fabric of our universe. What does this mean for atoms and electrons -- or the smallest quanta of energy (string theory)?

You’ve probably seen the image above and even learned of the different energy levels of an atom. For an electron to move up or down an energy level, it doesn’t do so in a continuous manner but it does so instantly, something we call a quantum jump. This “ecstatic” and sudden quantum jump normally wouldn’t work with Lie groups since Lie groups deal with continuous change, but because electrons also have a wave-like nature outside the atom, its orbit isn’t exactly like the way Earth orbits the sun -- instead it has a more continuous nature. 

Because an atom is spherical and exists within a Euclidean third dimension, we can piggyback off the ideas of six degrees of freedom we learned about earlier with the apple. Because no two electrons can have the same energy state and same orbit in an atom, we find that electrons that exist in the same orbit have opposite spins. There is a harmonious symmetry between the two.

The Lie groups extend to the basic forces of our universe: electromagnetism, weak force, strong force and they are known as gauge groups. The gauge group for electromagnetism has one degree of freedom. The weak force which is responsible for holding a neutron steady has three degrees of freedom. The strong force which holds together the nucleus of the atom has eight degrees of freedom. Scientists around the world are searching for the holy grail of laws that unify all of these forces together and many are doing this by finding the one Lie group that can encompass all these gauge groups to unveil a deeper symmetry amongst the elementary particles (everyone is always searching for the fundamental theory of everything – could this be a good candidate?) 

Lie groups were then ordered into a period table of Lie groups, by a high school teacher named Wilhelm Killing in 1888. They were sorted into 7 families from A to G.

The number corresponding to the alphabetic family pertains to its rank and this implies the number of dimensions it can operate in. The alphabetic family ranks from simplest to more complex from A onwards with A being simplest. You’ll notice some of the alphabetic families end at dimensions 2,4, and 8. If one were to create an object within that family in dimensions beyond those numbers, the object operates within infinite dimensions. Pertaining to numbers in the alphabetic family defaults to the rank in the family prior (E4 is really D4). Below you’ll find a more complex periodic table that has been developed over years of additional discoveries.

When Killing arranged the Lie Groups into families, they often dealt with geometric structures that were curved. This makes sense, since differential geometry deals with rates of change, rates of change is the essence of calculus. In calculus, you’ll always notice that there are curves on the graphs because often when things are taken to a higher power, they represent fluctuations (unlike taking something to the first power where it remains linear). But, there are of course linear rates of change if the object that is said to be moving does so in a continuous fashion. Lie Groups did well with curves for that reason but not in cases that were finite. 

The essence of determining the rate of change is deriving the rate of change in each instance with an infinite number of slices and then integrating the parts together. The curved family groups needed to be broken down in a similar fashion to their flat approximations so they can be applied in finite cases more generally. This is an imperfect method, however. Using real numbers, abstracting continuous change through a tangent line works, but when you use finite arithmetic (a series of numbers with a constant interval) the results become incredibly fragmented and pointless. This method wouldn’t work, no. But this did spawn the perfect question: how can we distill the family groups in a way that can be more generally applied, even in cases of finite arithmetic?

In our prior essay we spoke of Nicolas Bourbaki, a pseudonym used by French mathematicians in the post-war era to re-imagine mathematics in a more abstract and logical manner. Using these methods, new branches of mathematics can spawn easier. One of those mathematicians in the exclusive group was Claude Chevalley, who leveraged the momentum of this abstract and logical approach in rebuilding the edifices of mathematics, to construct the finite versions of the Lie groups.

A revolution was born as mathematicians learned of how Chevalley was able to do this. More and more discoveries were made that found finite versions of new family groups. Jacques Tits, Robert Steinberg, Michio Suzuki, and Rimhak Ree all independently discovered additions that would then be added to the newer periodic table of finite simple groups (as shown above). Most of these results were discovered algebraically (including Sophus Lie & Evariste Galois), but Jacques Tits was the only one who arrived at his discovery through geometry.

Crystals

The title of this book is manifest and the idea is that in mathematics, specifically Euclidean geometry, when we transcend our construct into higher dimensions, it manifests itself. When we transcended to a higher dimension via our coordinates, we manifested another coordinate to the right in order to represent it. When we transcended via another dimension in Euclidean geometry, we added another polar axis to our space. Another way to manifest is through crystallization.

Most minerals occur naturally as crystals. With the right environment, the internal structure of the atoms manifest themselves through a process where they build upon themselves in a recursive manner until the geometrically appeasing form takes shape. The cube shape of a grain of salt or the visually symmetric snowflake mirror their internal atomic structure. The variables that generate unique combinations of crystals are their heat and chemical compounds, each variation adjusting the algorithm that produces the unique crystals we see.

If we look at our Platonic shapes, we can understand their symmetry now that we have that tool in our tool kit. For example, we used an equilateral triangle before, but what does this equilateral triangle look like when it transcends to a higher dimension, say the third? We end up with a shape called a tetrahedron. We know what this tetrahedron will look like, for it too must contain symmetry. We do this using a rubric that adheres to the vertices, edges, and faces of the shape/object in question. Triangles are part of the A2 family group and its tetrahedron is part of the A3 family group. The way in which this shape sort of grows when it enters this new dimension is a natural rule that we also find in real-life crystals, but these are known as multi-crystals.

Because we know the rules for how the shape must “evolve” when entering a higher dimension, we can extend this out into higher dimensions as well as across different family groups which have a similar set of rules. A square is a B2, a cube is a B3, and a tesseract is a B4.

Visualizing these higher-dimensional objects is difficult, but with the rulesets we have around the shape’s faces, vertices and edges, we can use math to help us uncover their properties, as well as discover new symmetrical objects.

Feit-Thompson Theorem. Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of odd order.

The Monstrous Moonshine

Because we can break shapes down to numbers and coordinates, we can use math and computers to see shapes that our eyes and mind would find difficult to comprehend. There exists a shape known as 'M' for "Monster" because it is an object that has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries (that’s more than the number of all the atoms in a thousand Earths!) If you're wondering why we have difficulty imagining it, it's because we're used to living, experiencing, and thinking in a world of 3-dimensional space. Even the fourth dimension is difficult and meditations on mandalas are used to train our minds to see these grand structures in another dimension. But the Monster is an object that exists, at a minimum, in a 196,883-dimensional space.

How did we come to this point? Well previously we discussed how objects can “manifest” into another dimension, dictating the shape with which it is made. The concept of symmetry atoms, those which are the basic building blocks of symmetries, help us discover newer symmetrical objects in different dimensions. There are two important concepts you’ll want to get familiar with that are used in discovering new symmetry atoms: groups of permutations and an involution centralizer. 

The involution centraliser is a method in which you can use the concept of mirroring to determine if there are any other symmetry atoms inherent within the group. By observing each individual symmetry atom, you can find the absence of a new symmetry atom using this method to either prove that it wasn’t meant to be there, or prove its hidden existence. A decades-long project called the great Classification project had a group of mathematicians applying these methods in their quest for discovering more symmetry atoms.

You can establish the existence of a symmetry atom through the use of a character table, which is a square array of numbers that describe the group – any missing numbers and your step-by-step discovery process becomes more refined and focused as you piece everything together. Having one number helps you determine what the other is until you’ve solved it. With the groups of permutations method, there is a central tenet known as ‘transitivity’.  The groups of permutations on a set of objects are transitive if it could send any object to any other the same way that you could switch seats with someone. If you could send any pair of people sitting in their own seats to switch with another pair of people, then this is considered 2-fold transitive and as you can imagine you can extend this to N-fold transitive. N must be less than 6 for at 6 and beyond, this is no longer possible as proven through an exhaustive technique. Both of these concepts would converge by several mathematicians until the race to find the Monster would come to an end.

The Universe, Grouped

There’s a famous question in philosophy, asking whether the future is predetermined or has free will. When you look at everything in the universe from the microscopic to the macrocosmic, you can’t help but feel like everything has a mind and will of its own. Even as individuals, we feel we express strong individuality and are different from others, but fields like machine learning show that with mathematical concepts like matrix factorization, we’re really just a reducible being by our actions and tastes.

Group theory adds fuel to this fire – if we can group anything together and find a common, fundamental truth between those entities, what does that say about the universe as whole? With group theory we came to the realization that there are different types of infinities, some bigger than others. What else can we come to discover about the nature of reality through group theory?