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August 21, 2007
proporzione divina
Bless Mike of Epenthesis, from whom all naked nerdiness flows. You would think that he would have solved it after a few hours after posting, but no, he had to solve it in MERE SECONDS. Teachers love/hate that kid with their hand already raised as you pose the question.....
Let's call the original unknown side X. Creative, I know. That means the shorter side of the blue rectangle is X-1, while its longest side is 1. If shapes are similar, then the ratios are proportional. Example is short/long on one rectangle = short/long on the other rectangle.
So 1/x=(x-1)/1.
Numerous ways to go forward, but let's multiple everything times X to get rid of the left denominator. We now have 1=x^2-x, and we can subtract one from both sides, showing the equation 0=x^2-x-1. My kids have a Pavlovian response to seeing this, saying "Ooh, Mistah! Factor that quadratic!" This brings tears to my eyes and treats to their brains, of course, as we plug into the Quadratic Formula (x = (-b +/- sq.r.(b^2 - 4ac))/2a) the values a=1, b=-1, c=-1, and yields the answer
What's that, gasping children? Do we recognize that approximation? Remember my example of comparing your smallest two knuckles to the third knuckle (add together, they equal the third), then the 2nd and 3rd knuckles to your palm (add together, they equal the palm). Why is this special? Where else do we see this sequence? No, not golden showers, Johnny, stop thinking dirty all the time. Thank you, the Golden Ratio! I'm so proud of all of you!
Posted by G at August 21, 2007 11:25 AM
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Comments
Just the naked nerdiness, then?
Never mind. I'll take it.
Posted by: Mike B. at August 21, 2007 12:01 PM
Aaaahhhhh1!!!!!
This is why I switched from computer science to English/philosophy in college. After having gotten a D in calculus during my midterms, I said there goes that A average.
Posted by: Stash at August 24, 2007 05:08 PM
The problem with the golden ratio is that it is claimed to exist on the basis of seriously approximate measurements. Really, (1+√5)/2 is a very precise number. Just as π ≠ 22/7, Φ ≠ 1.7 ≠ 1.5. I really doubt that average children's knuckles are in that proportion, though they may look close enough for comfort.
Posted by: Joe Clark at August 24, 2007 06:21 PM
My head is spinning.
Posted by: Todd HellsKitchen at August 28, 2007 01:23 PM