Daniel, a local homeschooler, blogs for Equalis

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Maria Droujkova

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Aug 19, 2010, 10:18:02 AM8/19/10

to CHS, wakehsco...@yahoogroups.com, natur...@googlegroups.com, scratchint...@googlegroups.com

Equalis is a new social hub for mathematicians. Daniel Chiquito, a local homeschooled teen, is one of the few invited bloggers there. I first met Daniel at our Physics and Modeling unClass last year, where he quickly became a Teaching Assistant and an overall software guru. His first post on the blog is up: http://www.equalis.com/members/blog_view.asp?id=565749&post=107447

I am copying it here entire, because it's funny and it features one of my favorite proofs ever. Well done, Daniel!

You can comment at Daniel's forum on the site, linked in the post.

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Editorial Staff Note: On Wednesdays we take a welcomed break from professional-grade mathematics blog topics and focus on highlighting rising stars in mathematics. Our 2Infinity blog series features children in middle and high schools that exhibit a gift and passion for math-centric topics. Their enthusiasm for the subject matter is contagious and we believe it will bring you back to the days that you first were inspired to pursue the field of mathematics. Enjoy.

About 2 months ago, my dad got sick of the age and grunge of our bathroom and decided to give it a total makeover. Unfortunately for him (and his children), he decided that the wall needed to be tiled. So he made the bathroom a family endeavor and (naturally) asked me to pick a pattern. He had already picked two types of tile to be used; one was twelve inches square, the other was six inches square. Before you read on, try to find a pattern yourself.

After a bit of guessing with diagonal cuts, I remembered a pattern I had seen in an Art of Problem Solving book. Turns out that, with a few extra lines, our bathroom wall turns into a proof of the Pythagorean Theorem! See if you can figure it out without clicking the link!

These "Proofs without words” are generally defined as a picture or diagram without any explanations that effectively proves a theorem. There are several wordless proofs of the Pythagorean Theorem, the oldest of which was found in an ancient Chinese manuscript dating from around 200BCE (mine is attributed to Annairizi of Arabia, circa 900CE).

Proofs without words have probably survived all these years because of their elegance and accessibility. After all, there is no formal training required; it's obvious that, for instance, you can just reassemble the smaller squares into a bigger one. This ease of understanding, while not as rigorous as a full-blown proof, is certainly more rewarding.

I found some interesting proofs in this comment thread (pretty dense, some of them) and at the USAMTS site. There is also a compendium of proofs by Roger B. Nelsen (I haven't looked at it, but it looks good). If you have any proofs to share or points to discuss, I'll look forward to seeing you at my forum!

P.S. The bathroom turned out great! My dad is happy because it's done, my mom is happy because she has her own bathroom back, and I'm happy because the tiling is mathematical!