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Algebra. Geometric Decomposition of Algebraic identity

Audience:

From (a−b)² to x³+y³+z³−3xyz: A Geometric Exploration

I have been exploring geometric interpretations of algebraic identities taught in school mathematics. My aim is to see whether a suitable algebraic decomposition can make an identity easier to visualize and understand.

I first created pictorial geometric representations of the familiar identities

(a+b)^2=a^2+2ab+b^2

and

(a-b)^2=a^2-2ab+b^2

Instead of treating $2ab$ as one term, I split it as $ab+ab$ :

(a-b)^2=a^2-ab+b^2-ab

Although algebraically equivalent, this decomposition gives a different way of viewing the geometry. This idea helped me explore $(a-b)^3$ .

I experimented with two different representations of $(a-b)^3$ : one using physical carrom coins, and another using FreeCAD to create a 3D visualization.

YouTube demonstrations:

$(a+b)^2$ — https://www.youtube.com/watch?v=G-mybx9ZI-I

$(a-b)^2$ — https://www.youtube.com/watch?v=lIpBRiWCxWk

$(a-b)^3$ using carrom coins — https://www.youtube.com/watch?v=On84n_m7o4A

$(a-b)^3$ using FreeCAD — https://www.youtube.com/watch?v=dtQB1yiPwTY

These explorations eventually led me to the more challenging identity

x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)

The difficult term was $3xyz$ . Following the same idea, I split it as

3xyz=xyz+xyz+xyz

and grouped the terms as

(x^3-xyz)+(y^3-xyz)+(z^3-xyz)

Similarly, I considered

(x^2-yz)+(y^2-zx)+(z^2-xy)

This revealed an interesting sign pattern. For positive $x,y,z$ with $x\geq y\geq z$ ,

x^2-yz\geq0

and

z^2-xy\leq0

while $y^2-zx$ can be positive or negative depending on the values of $x,y,z$ .

Let

p=x^2-yz,\quad q=y^2-zx,\quad r=z^2-xy

Then

xp+yq+zr=x^3+y^3+z^3-3xyz

and

p+q+r=x^2+y^2+z^2-xy-yz-zx

Therefore,

\frac{xp+yq+zr}{p+q+r}=x+y+z

This resembles a weighted average, although the weights need not all be positive. This gave me another way to think about the factorisation and its geometric structure.

I created an interactive application in which students can enter different values of $x,y,z$ and explore the identity.

Interactive application: https://sravanam-keerthanam-smaranam.github.io/Algebraic-Identity-Visulization/x3_y3_z3.html

GitHub source: https://github.com/sravanam-keerthanam-smaranam/Algebraic-Identity-Visulization

AI assistance: I developed and refined the mathematical thought process through iterative discussion with ChatGPT. After finalizing the mathematical approach, I used ChatGPT and Claude AI through iterative prompting to generate the complete HTML, CSS and JavaScript implementation. I reviewed, tested and repeatedly directed modifications to tailor the generated code to my requirements.

My central idea is that geometric interpretation need not begin with geometry. It can begin by finding the right algebraic decomposition. Splitting $2ab$ into $ab+ab$ and $3xyz$ into three $xyz$ terms changed the way I could see these identities. I hope these visualizations encourage students to experiment with algebra rather than merely memorise identities.

Analytics

3.95 Overall score*

37 Rank

4 Votes

4 Comments

Comments

3.9

I may just bee stupid. But you list me at step “Positive vs Negative, Side by Side”. I can calculate it mathematically. But the visual cue makes no sense to me. How am I supose to picture that instead of just calculating it?

The entry description promises to explain algebraic equalities with geometric interpretation that is claimed to give intuition. Instead of geometric, though, I think it’s more of some kind of a “graphical” illustration. I didn’t find that illustration helps intuition in a meaningful way. Besides, the overall design of the demo make it very far it from being an explainer, because it’s not mentioned anywhere at all why we should care about this topic.

Notes:

On state 3/15, how squares and rectangles are paired and why is not explained.
It shows n/15 states at the top, but I always reached 11 states at the end

4.6

A little convoluted, a little dense. The big important takeaway or insightful step didn’t stand out.

2.2

I really like the idea, but the execution could be improved. I like the idea of using a physical metaphor to make an identity feel more approachable. I think the animations are smooth and pleasant, although they do look very AI-generated.

What worked well? What could be improved? Suggest one actionable improvement

I think the animations worked well, but the visualization does not really show why the identity is true. I think your short youtube video of a^2 + b^2 and a^2 - b^2 with the butterflies work much better.

Each positive square next to its matching negative rectangle x² − yz y² − zx z² − xy

It is not immediately clear to me why x^2 is “matching” with yz.

With x=y=z=3, both sides are 0⁄0 — undefined, but the algebra still holds I don’t think this is accurate - the algebra does not hold in the case where the denominator is 0

Discover the Identity

I don’t want to be too harsh, but to me this visualization really felt more like “verifying” than “discovering.”

My main piece of feedback is that the visualization shows that the identity is true for the numbers I plug in, but not why it is true in general. I felt as if we were handed a formula at the beginning, and I wasn’t sure why I was given it. The rest of the “interactive” consists of just pressing next to see that it holds true for the examples that I have plugged in. But plugging in examples doesn’t prove that this identity is true, it just does arithmetic for a specific case (or a specific set of cases).

Instead of opening with the finished formula, I would recommend that you start with the geometry and let the identity emerge. Build the volumes first, guide the learner as to how the pieces combine, and then reveal the equation as the punchline. I think 3blue1brown does this very well in his video “Derivative formulas through geometry | Chapter 3, Essence of calculus”, specifically starting at 3:51:

https://www.youtube.com/watch?v=S0_qX4VJhMQ

Finally, and this may be a very western-centric criticism, but I wasn’t familiar with what a carrom coin was before this interactive. I’m not sure if this is just a me problem though.