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Using Fiboacci-Like Equations to Compute Pi

Audience:

Tags: complex-numberslinear-recurrences

This originated as a lesson designed to motivate complex numbers for a precalculus class. I show examples of linear recurrences (generalizations of the Fibonacci sequence), starting with simple population equations. Playing with parameter values leads to instances that create periodic behavior and sine waves, and then we see a specific set of recurrences that create sine waves with half periods very close to 10pi, 100pi, 1000pi. In order to explain this mystery, we build up to the theory of how to solve these recurrences. In finding our sine wave examples have complex roots, we then learn about taking powers of a complex number and how that explains the period of our solutions. Finally, we see how to visualize these exact solutions via a machine which adds the outputs of two 3d complex spirals. Most scenes invite you to play with the same visualization I am showing on Desmos or Geogebra. Here are the links: https://www.desmos.com/calculator/4tj7pu6eon (Scene 1) https://www.desmos.com/calculator/f9353a1d60 (Scene 2) https://www.desmos.com/calculator/pldzawaqok (Scene 3) https://www.desmos.com/calculator/fclt20icqx (Scene 5) https://www.geogebra.org/m/qvxyrxqb (Scene 6) https://www.desmos.com/3d/hwdxpabhib (Scene 7)

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6 Overall score*

77 Rank

11 Votes

4 Comments

Comments

8.2

Fantastic motivation to explore complex numbers! Also a great slow ramp-up in difficulty, and the DIY visualizations made this very interactive. Although this was long, it didn’t feel like an hour at all. Nice job!

Overall main thoughts:

The video clearly took a HUGE amount of work and dedication. The topic of how second-order linear recursions end up bringing in complex numbers in a fundamental way, and the link to estimating pi, is very beautiful.

But the video is incredibly fast! I don’t think it would be humanly possible for someone to follow all the details of what you’re saying in real time for the whole hour; of course, for very short videos, it can be okay to expect the viewer to need to regularly pause and occasionally repeat - but not many people are going to be dedicated enough to do that continuously for a whole one-hour video. To my mind, a maths exposition video can serve the purpose of informally introducing the viewer to a beautiful idea, or serve the purpose of teaching a topic to the extent that the viewer can then solve exam-style questions on the topic, or both. In this video, however, there is too much going-through-details for a viewer just wanting to see the main beautiful ideas, and these details are presented far too quickly for any student to be able afterwards to know how to solve problems about complex numbers or problems about linear recurrences. Ironically, if the video were much slower and thus much longer, the task of watching the whole video would have been less taxing and more enjoyable.

Motivation:

The idea of showing how complex numbers are central to behaviours just involving real numbers is very nice.

The exact logic of the motivation presented in the introduction [namely,

differential equations in physics reveal the centrality of complex numbers to the real world;
but that topic requires a more advanced concept than what you want to assume for your audience;
so you will look at a similar topic that reveals the same kind of idea without the need for that more advanced concept]

was a bit difficult to follow the first time round, again because things are presented quite quickly. After coming back to the introduction again at the end, it was clearer.

Clarity:

Ideas are introduced quickly and then assumed for the rest of the video. Key mathematical concepts are sometimes introduced in the spoken words without a corresponding equation or diagram being displayed at the same time. (Again, in a relatively short video with sufficiently slow explanation, that could have been okay.) For the “pause and play”, a slower introduction to how to use the linked tools would have been helpful. The connection between the first “pause and play” and the content presently immediately before it was a bit confusing. For the last “pause and play”, it took me a bit of time to see how to use some of the necessary Desmos functioning.

Novelty:

This is probably the first video I’ve seen that seeks to explain to a not-very-advanced audience how complex numbers appear in a fundamental role to the solutions of fairly simple real-world problems. The idea of using discrete time-steps rather than calculus is particularly nice in this regard. The idea of linking the topic to the estimation of pi is also nice and novel; and I particularly like the way that the question of why pi appears is raised before complex numbers (which ultimately provide the answer) have been introduced.

Memorability:

There are some nice moments in this video that have the capacity to be memorable; but the large amount of fast-paced detail, which somewhat disorients the viewer in regards to the big-picture flow, may partially obscure those moments from memory.

3.2

Motivation is really poorly presented. Then it is so long… Sorry but I would not use that with students.

Wow, great use of the tools. I remember doing a summation that gets you to Pi some years back. I think I missed a step as to “why” this works like this, the link between fibonacci and pi? This might be a bit too advanced for pre calculus. Why did you make a Desmos for scenes 1,2,3,5,6,7? I would expect a fibonacci based one for scene 1,2,3,5,8,13. :)