# Category Archives: Mathematics

# The Symmetry of Bach

# The Divine Proportion

Sometime in the mid-80’s, I decided to build a redwood deck in the back of our Mountain View home, just down the street from where Google is today (off Stierlin Road). I had been studying the Golden Mean and was inspired to build a spiral-shaped deck. As I struggled through the mathematics, I realized it was much harder than I had thought. We needed 3.5″ integrals of linear dimensions on a spiral-curved plane. Wicked hard math. So I called my friend Larry (MIT, Ph.D. Math, Magna Cum Laude). The next day he brought me two pages of calculations which concluded in a formula that translated polar coordinates to linear dimensions relative to the location of boards. Wow. Another friend, Martin, built the structure from the most perfect redwood sticks we had ever seen. Not one knot in the entire deck (ironically, the entire house was built of redwood – back when it was about the same price as Douglas Fir). We have photos of that deck somewhere around here, but couldn’t find them. Actually, I hadn’t thought about the spiral deck until someone sent me this stunning video animation which captures the mathematical essence of universally perfect spirals: the golden rectangle – the Fibonacci Series – the Divine Proportion. One part of the animation looks EXACTLY like the deck we designed. Imagine a matrix of exquisite redwood planks forming a spiral that looks just like this (about 6 meters across). Anyway, here’s the visualization from Spanish filmmaker Cristobal Vila. It is under 4 minutes, and very much worth your

# Certainty

Last night we watched a movie called “21” – about a group of MIT math students who made a fortune in Las Vegas by “counting cards” at the Blackjack tables. The movie was so-so, but one scene reminded me of an old mathematical nemesis — the Monty Hall problem. In the scene, a math professor (Kevin Spacey) tells the student there is a car behind one of the three chalkboards, 1, 2, or 3. He asks the student to guess which chalkboard hides the car. Student picks board #1. At this point, professor reveals that behind door #3 there is no car. So now we know that the car is behind either board #1 or board #2. Professor then asks student if he wants to change his guess. Student says “yes” and changes his guess to board #2, telling professor “I have a 66% chance behind board #2, but only a 33% chance behind board #1.” This is unintuitive to me. When one choice becomes eliminated, my intuition tells me that the probability of the car behind either remaining door is 50:50, regardless if I had previously made a choice, or not. So I created a test with my son. We did 40 trials. He guessed 1 of the 3 options, then randomly said “keep” or “change” without even knowing the choice I had eliminated. Indeed, when he changed his original choice, he was right roughly 2/3 of the time. When he did not change his choice, he was right only 1/3 of the time. (for those still puzzled, a good analysis can be found here). Does this reveal a larger metaphor? More times in life than I care to admit, I have found myself holding on to a form of certainty that was later found to be totally unfounded. Even when presented with unassailable evidence, we often refuse to acknowledge the clarity set before us – favoring deeply ingrained “religious” certainties. And some of you are still saying “no, there are two remaining doors – it’s plainly obvious to anyone that the odds are