Doodles: Sources of Tiny Mathematical Discoveries

Eyes drooping, mind numbing, back trying to stay straight, dreaming of a warm, warm bed during this AM. Head perched on ink-stained hand, ink pen dangled precariously in the other as I draw and listen to the classroom discussion of synthesis essays. Dr. Kelso, if you’re reading this, I’m sorry. I love your class, but for some reason, today, the humanities side within me is sleeping blissfully in her warm, warm bed and math just can’t stay asleep in her cold, cold bed. So, math picks up the pen and draws squares with sides size one in the space next to the passage we’re discussing — I apologize to the Alabama clergyman we’re discussing, too, but squares and lines seem more inviting.

My pen scrapes across the paper as I draw light lines within a crude square, making tinier and tinier squares from halves — the half becomes two fourths, one fourth becomes two eighths, and so on… A juvenile drawing at best, but the mathematician within me later sees the connection between my doodle and what Mr. Ambrose has been teaching me.

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Can I ask you something? Why are we always in love with halves? Because there are always two peas in a pod? Two arms, legs, hands, eyes, every other body part, excluding the organs? Why don’t give that much passion to the trifectas, the three musketeers? Or the four seasons?  Because I think the other numbers need more love… sorry, two. Before I know it, my previously sleep-addled mind had been lit up by these rudimentary lines and squares before prompting me to draw thirds and fourths.

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And after a few calculations finding the area of each rectangle, I recognize the pattern that’s present.

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The sequence of the areas is (1/2), (1/4), (1/8), (1/16), (1/32)… which is just one divided by the powers of 2, since you’re multiplying each remaining space by (1/2).
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The sequence of these areas is (1/3), (2/9), (4/27), (8/81), and so on… this is just (3 – 1)^n/(3)^n, n being the number of the rectangle you’re on. For example, the second rectangle is (3-1)^2/(3)^2, which is 2/9.

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The same goes here! The sequence of areas is (1/4), (3/16), (9/64), (27/256), and so one. This is (4 – 1)^n/(4)^n, n being the number of the rectangle you’re on.

Now, once you add each term of in each sequence together, you get a value that is approximately one. That’s clear from the diagrams above; all of these rectangles put together still make up the area of 1, which is the area of the whole square. Need more proof? Plug it into a program!

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Theorem (or is it?)*: for a square with sides of length 1, one can form a rectangle with area (1/n), and then form another rectangle with area (n-1)/(n^2), and another with area (n-1)^2/(n^3), and so on. Adding all of these areas together will result in a series that converges to one.

Doodles, as much as they exist to satisfy my childish obsessions with hearts and flowers, are my source of seeking mathematical conclusions on my own. Taking a look into all the drawings scrabbled on looseleaf and history sheets clumped at the bottom of my backpack from past tenth grade — eyes, retro hearts — it’s clear that math hadn’t quite infiltrated my mind yet. However, my curiosity and passion for math has grew, so my doodles have grew as well — Pascal’s triangles out of shapes and numbers other than 1, surfaces, tangent lines to double helixes, the squares I illustrated above. Art is for the artist as math is for the mathematician, but now art is for the mathematician, too.

*Proof is coming soon… geometric series = fun

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