Algebraic Symmetry and Lessons from Origami

Algebraic Symmetry A short AI summary:

Algebraic symmetry describes situations where a mathematical object remains unchanged under certain transformations, and these transformations form a group because they can be combined and reversed. For example, consider an equilateral triangle: rotating it by (120deg), (240deg), or doing nothing leaves the triangle looking the same. These rotations form a group called the cyclic group (C_3). In algebra, we say the triangle has symmetry because the set of operations (the group elements) preserves its structure. This idea extends beyond geometry — in equations and polynomials, groups describe how variables or solutions can be rearranged without changing the underlying relationships.

Symmetry

After reading above passage, you may ask, so what? I am struggling to find my left shoe. Or I am trying to write this damn thesis on God knows what. So, why should I learn about Algebraic Symmetry. If you have that question, my answer is Don't. (You may still carry on mathematics if you are "Born" to do that.) But the reason is Algebraic symmetry is hypothetical.

Let's look at some of the examples from daily life. If you see, there is nothing you can imagine think of doing without knowing basic understanding of mathematics. After all, who ever thought when we share a plate with sibling, we were looking for algebraic symmetry of a piece of Brownie?

Examples of Algebraic symmetry from day-to-day life.

1️⃣ Sharing money equally

If two friends share Rs. 100, we can write it as[x + y = 100] Swapping who is (x) and who is (y) doesn’t change the total. The equation is symmetric — fairness and equal exchange use this idea.

2️⃣ Balancing a weighing scale

A scale is balanced when both sides are equal: [a = b] Adding or removing the same weight from both sides keeps it balanced: [a+c = b+c]This “do the same thing to both sides” rule is an algebraic symmetry used daily in measurements and cooking.

3️⃣ Arranging seats or teams

Suppose two teams A and B play a match. The result “A vs B” is the same as “B vs A” in many schedules. Swapping labels without changing the outcome reflects symmetry, similar to symmetric variables in algebra.

4️⃣ Patterns in art and design

Floor tiles or rangoli patterns often repeat in a balanced way. Behind the scenes, designers use repeating rules like: [f(x,y) = f(y,x)] which means swapping directions keeps the pattern unchanged — an algebraic symmetry idea applied visually.

5️⃣ Passwords and encryption apps

Many everyday apps use cryptography built from algebraic groups (like cyclic symmetry). The math ensures that encoding and decoding follow structured transformations that preserve certain properties.

Whenever you can swap, rotate, or change parts of a rule without changing the result, you’re seeing algebraic symmetry — whether it’s sharing money, balancing objects, or designing patterns. In life even without mathematics we have been doing these things, day in and day out. But as you can see there is a problem. Call me obsessive, but I cannot see any symmetric pattern which exist in the world. It seems algebraic symmetry is just an "Approximation of the reality with a critical error of assumption" not exactly the reality itself.

Lessons from Origami.

Origami Tulips

I used to do Origami. Not like an expert but like a kid. If you do not know, in Origami, you are supposed to fold the paper, mostly in symmetric manner. However, I had hard time getting perfect fold. Likely I was clumsy at folding papers. But I saw my friends doing worse. When it comes to the edge, there would be a tiny miss alignment that I could not help noticing. It didn't bother me all these years. Now when I read about bending square papers and talking about algebraic symmetry in a mathematics book. My mind goes, that is wrong. You can mathematically show that there is symmetry. But all that mathematics is just approximation of reality. In real world there are no symmetrical Folds on a square paper. It is because there are no perfect squares in real world. They are just Platonic forms in human minds. A perfect square is Unreplaceable imaginary dream.

I know, a lot of people would disagree with that. But the argument holds. The symmetry that we see is just an apparent one. Everything built upon this simple math is just another form of imagination. (From addition to subtraction) Here is another twist. If you include time as a factor, we can argue the paper, in the past is not the paper in the present. And there is no way of bending the paper through time. Every paper including we seems to be in a continuous motion through time without achieving any symmetry. I am not sure what you are going to do with this piece of information. Maybe some quantum Physics? :)

"Algebraic Symmetry is just an approximation of reality. Reality does not have true symmetry at all"