The Magic of Binomial Expansions: A 3D Experience in Structure & Patterns
Binomial expansion is one of my favorite topics to explore visually. Did you know you can engage with visually exploring & understanding binomial expansion with zero prior Algebra understanding? In this post, we will be using 3d visualizations to observe the binomial expansion. Don't worry if you're not sure what you're looking at, the point is to challenge yourself to think outside the box & explore.
Let's start by taking a look at the binomial expansion below. Take a moment to study the binomial expansion below, & jot down what you notice & what you wonder about. As you explore, consider the following questions:
What does the RED line represent? What does the BLUE line represent?
What does ‘a’ represent? What does ‘b’ represent?
What patterns or structure do you notice involving the RED & BLUE lines? How are they spaced & organized in the image?
Now that you’ve explored our first binomial expansion, let’s step up the complexity & see how the expansion changes as we raise (a+b) to a higher power of 2. Take a moment to study the evolution of the binomial expansion below, & jot down what you notice & what you wonder about. As you explore, consider the following questions:
What does the RED box represent? What does the Blue box represent? How do the RED & BLUE boxes relate to the RED & BLUE lines from earlier?
What does the GREEN box represent?
What patterns or structure do you notice involving the RED, BLUE, & GREEN boxes? How are they shaped, spaced, & organized in the image?
Where did the GREEN box come from? Where do you see ‘a’ & ‘b’ in the GREEN boxes? Why is there a ‘2’ with the GREEN boxes?
Why did our previous binomial expansion not have a GREEN box?
Finally, let’s step up the complexity & see how the expansion changes as we raise (a+b) to a higher power of 3. Take a moment to study the evolution of the binomial expansion below, & jot down what you notice & what you wonder about. As you explore, consider the following questions:
What new shapes do you see?
What new colors do you see?
How do you see ‘a’ and ‘b’ interacting visually?
What pattern or structures do you see in these new colors & shapes?
Congratulations! You’ve successfully learned a bit about binomial expansions using 3d visualizations! By using these visualizations, we were able to see the patterns & structure of the expansion in a way that I hope was fun to comprehend & interact with. Remember, the aim of this journey was not to find the perfect answers, but to foster curiosity & the desire to discover; so don't be afraid to think differently & challenge yourself. Until next time, keep exploring & keep learning!
BONUS: Interested in a mind-bending challenge? The binomial expansion of (a+b) to the 4th power is below. Why does it look the way it does? Can you see the structure & patterns? Why or why not do you think?
