Stepping Into The Grand Mathematical Play: A Dance With Isomorphisms
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Stepping Into The Grand Mathematical Play: A Dance With Isomorphisms

When taken in the right mindset, a walk through the familiar can reveal undreamed of mysteries, reaching all the way to the interconnectedness of reality.

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Stepping Into The Grand Mathematical Play: A Dance With Isomorphisms
Saad Slaoui

In the grand play of Mathematics, the protagonists are solely defined by what they can do. It is a play centered around the appreciation of interconnected motion; a play in which the characters are themselves transparent, presenting a select few of their properties whenever deemed necessary, yet seldom unraveling the entirety of their manifold self.

Within the Eastern spiritual school of Jainism, the Anekantavada doctrine - often translated as “many-sidedness” - refers to the fundamental limitation of a human being’s ability to perceive the infinitely many facets of any given object. According to the Jains, because of a thick layer of karmic-dirt we have accumulated over our many lives, our inner self can only see a partial rendering of reality. Only through dedicated practice can a seeker of truth slowly progress towards omniscience and free him or herself from the constraints of the human condition.

In a sense, this worldview could hardly be more fitting in the attempt to describe the evolution a Mathematician goes through in his or her unending journey to attain “Mathematical omniscience.” We all start as neophytes, blind to the underlying reasons of seemingly dull computations, yet some of us decide to push the veil further and to sacrifice our time in exchange for brief moments of vivid clarity, inextricably linked to the uncovering of a new facet in a recently discovered mathematical object, or perhaps even in an old friend.

The motivation article stemmed from the reminiscence of an old saying a Buddhist monk once told me (and proceeded to hammer into my mind by the sheer strength of undying repetition): “Learning by Doing.” You can only truly understand the things you experience. In what follows, I will build upon familiar concepts which will lead us to a central idea in Mathematics. This idea, in turn, will hopefully allow you to see a Mathematical object you’ve known for most of your life in a new and rather magnificent light.

Consider the following drawings:

Each of these objects is made of four points A, B, C and D, linked to one another in alphabetic order in a sort of cyclic structure: A -> B -> C -> D -> A -> … Intuitively, you can see that the four drawings really aren’t all that different. You can mentally move the dots and line of any one of them to get any of the others. Stepping back into the theater analogy for a moment, we could observe that each of these objects does the same thing: it links the same pieces in the same order. It really doesn’t matter how the drawing is arranged on a piece of paper. As Mathematical objects, these drawings are essentially identical. In other words, they are the same up to “isomorphism.” Formally, we say two objects are “isomorphic” whenever we can find a correspondence between the components of each object which preserves their structure.

Finding an isomorphism between two objects is in a sense telling us more about each of these objects, as we can then “merge” our partial understanding of each of the two objects into a deeper, more nuanced appreciation. One of my favorite examples stems from the simplest of structures: think of a square in the plane, four sides of equal length meeting at right angles. Try to visualize that square as embedded in a mold which perfectly contains its four edges. That is, whenever you “lift” the square out of its mold, you are able to see a subtle trench where the square used to lie.

In a natural generalization, let us define the “symmetries” of a geometrical object as the set of all spatial displacements of that object such that at the end of the displacement, the object is “back into its mold.” For example, we could take our square, rotate it by half a turn clockwise, and get the square back into the mold. Then, we would consider the motion “rotation by half a turn clockwise” as a valid symmetry of the square. Now, close your eyes and try to imagine all possible symmetries that can be obtained from the motions of a square. There are eight such symmetries, including the one presented above and the “identity” symmetry which leaves the square in place. When you think you have them all, take a look at the drawing below.

As you might have guessed, there are 4 possible “rotation” symmetries, corresponding to turning the square clockwise by 0, 90, 180 or 270 degrees - notice that rotations by d degrees clockwise are the same as rotations by (360 - d) degrees counterclockwise. In addition to these, there are four “reflection” symmetries, which can be obtained by either reflecting the square along each of its four axes - vertical, horizontal and both diagonals - or choosing an axis to reflect upon and then follow the reflection any of the four rotations.

Now for the fun part: having just determined all possible symmetries of the square, it is very easy to determine the symmetries of any polygon with n equal sides, with n being any natural number greater than 2: just as the square, a polygon with 4 equal sides, has a set of 4 rotations fully determined by the rotation r = 90 = 360/4 degrees - namely the set {0, r, 2r, 3r}, which can be denoted {1 = r^0,r^1,r^2,r^3} -, so can the polygon of n sides be seen to have a set of n rotations, fully determined by the "generating" rotation r = 360/n degrees and given explicitly by the set {1, r, r^2, ..., r^n-1} . Finally, if we let s be any reflection of the square, the 4 remaining symmetries of the square can be written {s, sr, sr^2, sr^3}, and similarly the n remaining symmetries of any n-gon are {s, sr, sr^2, ..., sr^n-1} .

Having gained control over the symmetries of any geometrical structure with n-equal sides, let us push our exploration one step further and incorporate a plot twist that never fails to make the play radically more exciting: let n go to infinity. Imagine a “polygon” with infinitely many equal sides, extending endlessly from both edges of any given vertex. In that sense, we could define any one of those vertices as the “center” of that infinitely long polygon, as there are equally many vertices on each side of the vertex. Fortunately, the symmetries we had found for a finite n still hold for this infinite case. We can still “rotate” the polygon by any integer coefficient, with the convention that a single rotation r corresponds to moving each vertex to where its neighbor used to lie in the mold analogy. For example, r^5 would correspond to moving our chosen center 5 vertices to the right; by virtue of a rigid motion, this movement will be reproduced by every other vertex and the overall shape will fit back into its mold. Moreover, by taking the line cutting through perpendicularly throughout chosen center, we can define a valid axis of symmetry along which to perform all the reflection symmetries.

We have now covered enough ground to appreciate the isomorphism I am about to introduce. Take our polygon with infinitely many sides, and choose a center. Label it “0.” Now, mentally grasp the chains of linked vertices extending from each side of the center, and slowly flatten them to extend in a straight line going through the center, just as you would cut a circle and extend it into a straight line. Finally, label the vertices on the right of the center as “1”, “2”, “3”, and so on, in order of departure from the “0” vertex. Similarly, label the vertices on the left of “0” as “-1”, “-2”, “-3”, ..., with respect to their distance from the center.

Notice that none of the above transformations changed the structure of our original polygon in a significant, non-reversible way. We did not tear anything apart, nor did we change any connection. In that sense, we have established an isomorphism between the polygon with infinitely many equal sides, having symmetries under rotation and reflection, into what is perhaps the very first Mathematical object elementary school children are introduced to: the number line, with associated symmetries being a “sliding-over” of the line by a given number, and a reflection of the line across the perpendicular line through 0, sending every number to its negative counterpart, possibly followed by a sliding-over. These two objects have the exact same structure, and we can freely shift from one facet to the other to better appreciate their shared properties.

If you found the above isomorphism interesting, expect the following concept to be breathtaking. In conclusion of this attempt to share the pleasures of Mathematical insights, I am about to finally discuss the formula that served as the cover picture for this article. In order to do so, let me mention one last nuance as to the ways one can define an isomorphism. Above, we have repeatedly taken the intuitive route in determining why objects can be said to be isomorphic to each other, by mentally twisting and relabeling a structure to obtain the other one. In more formal Mathematical language, this process consists in finding a “bijective” function between the two objects which also preserves their respective structure. Conversely, the existence of such a function between two objects is sufficient to establishing that they are isomorphic to one another.

The reason the following isomorphism is so fascinating is that without first realizing that there is a such a structure preserving, bijective function between the two objects, it is very hard to realize that the objects are isomorphic. In other words, we could be using the two structures on a day-to-day basis and never really appreciate their fundamental similarities. The objects I am talking about are the following: ⟨R, +⟩, the entire real line with a structure defined by addition, and ⟨R+, ∗⟩, the positive half of the real line with a structure defined by multiplication. By structure, I mean that the specified operation can be used to combine any two elements of the corresponding object - in both cases, elements are simply real numbers - to produce a third one.

These two objects are isomorphic. The bijective function, in this case, is the exponential function, defined over the entire real line. The reason this function is so special is because it has the following property:

e^(a+b) = e^a * e^b , for all a,b in R

That is, it preserves the structure of its input within its range. Because it perfectly matches every element of the real line with a unique pair in the positive real line, it is an isomorphism between the two objects.

The ultimate realization lies not in the definition of the isomorphism, but in the investigation of its consequences. What we have shown so far is that we can relabel every single real number (both positive and negative) into a unique, strictly positive real number. We are effectively mapping two copies of infinity, (−∞, 0] and (0, +∞), into a single one. The question then becomes: where does each infinity go? And this is where things get really stunning. The first step is to realize that for us to create an isomorphism, we must relabel each element of one object into an element of the other that has the exact same properties. Specifically, the identity element 0 in the entire real line, which doesn’t affect any other element under addition, is the only one with such a property, and so is the identity element 1 in the positive reals under multiplication. Then, there is no choice but to map 0 into 1.

Well then, the entire structure necessarily follows in a surprising way: on the right of 0 in the additive structure, every element will be mapped somewhere on the right of 1 in the multiplicative structure. The positive reals under addition, this infinite sequence of uncountably many positive numbers, is mapped into the entire positive line on the right of 1 under multiplication. This seems reasonable, and one could think of “sliding over” the positive line by one unit to find a correspondence. The thing is, we haven’t yet mapped any of the negative numbers, and all but the unit segment in the positive reals has already been used by the positive infinity half of the real line! What this tells us is that the entirety of the negative real line under addition is somehow being injected into the tiny [0, 1] subset under multiplication. The hallucinating vastness of the infinitely large negative reals is being reproduced, atom by atom, within the laughably small unit segment of the positive reals, with a perfect translation of the additive structure in multiplicative language.

What started as a rather neutral observation turned out to be an absolutely riveting tale about the correspondence between the infinitely large and the infinitely small. The mind-boggling complexity of the infinitely large negative real line is also to be found in the structure of the tiny unit segment, under a different structure but with the same overarching pattern. Just as some of the deepest intricacies of outer space, such as the possible existence of infinitely many universes, spatially co-existing with our own, constantly expanding into nothingness, seem to be mirrored by the shattering suggestions of quantum physics, including hints towards infinitely many co-existing universes, temporally co-existing with our own, constantly dividing themselves as a result of the probabilistic behavior of matter at the atomic scale, so do the fundamental structures of Mathematical thought leave us baffled by deep, elementary correspondences. The extent to which these physical and mathematical correspondences are one and the same is left for further investigation.

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