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Snippets about: Math

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Mandelbrot and The Geometry Of Nature's Roughness

Traditional Euclidean geometry deals with smooth, ideal shapes like lines, triangles, circles etc. However, the shapes seen in nature (mountains, coastlines, trees etc.) are rough and jagged.

Mandelbrot discovered fractal geometry to describe this roughness mathematically. Fractal shapes exhibit self-similarity - they look "the same" at different scales. Many natural and man-made phenomena exhibit fractal properties. Financial markets also display fractal behaviors, with similar patterns at different time scales

Section: 3, Chapter: 16

Book: The Black Swan

Author: Nassim Nicholas Taleb

The 37% Rule: When To Stop Looking And Commit

The 37% Rule provides guidance on the optimal time to stop searching and commit to a particular choice, whether you're looking for an apartment, hiring an employee, or finding a spouse. In short, look at your first 37% of options to establish a baseline, then commit to anything after that point which beats the best you've seen so far.

To be precise, the optimal proportion to look at before switching to "leap mode" is 1/e, or about 37%. So if you're searching for an apartment and have 30 days to do it, spend the first 11 days (37% of 30) exploring options, then on day 12 pick the next place that tops your current best.

This algorithm offers the best chance of finding the single best option, though it will still fail a majority of the time. But it shows the power of establishing a "good enough" baseline before jumping on something that exceeds it.

Section: 1, Chapter: 1

Book: Algorithms to Live By

Author: Brian Christian

Why Silver Medalists Are A Lie

the 1800s, Lewis Carroll (of Alice in Wonderland fame) pointed out a flaw in single elimination tournaments, like those used in lawn tennis at the time. The problem is that the tournament format can only definitively determine 1st place. The person who loses in the finals could theoretically be the 2nd best player, but they could also be the 3rd, 4th, or worse. That's because they only lost to the eventual champion - if they had faced off against others, they may have lost those matchups.

As Carroll calculated, if player skills are evenly distributed, the odds that the 2nd and 3rd best players face off in the semi-finals is about 50/50. So awarding a silver medal to the finalist creates a lie half the time. The same is true of bronze medals determined by a 3rd place game. Single elimination formats simply don't provide enough information to rank anyone but the overall winner.

Section: 1, Chapter: 1

Book: Algorithms to Live By

Author: Brian Christian

Optimism In The Face Of Uncertainty

A key insight from the multi-armed bandit problem is the power of "optimism in the face of uncertainty." That is, when choosing between options where some information is known and some unknown, optimism is the mathematically correct approach.

Suppose you walk into a casino and see two slot machines. The first, you're told, pays out 20% of the time. The second machine's payoff rate is unknown. Which should you choose?

Rationally, you should try the mystery machine. That's because it COULD pay out at >20%, in which case it's the better choice. But you'll only find out if you try it. Mathematically, the expected value of an unknown option is higher than a known suboptimal one.

So in life, when facing uncertainty, choose optimistically - assume the best of a new person, place, or experience. Optimism maximizes your chance of finding something great. Pessimism can lead to overlooking hidden gems.

Section: 1, Chapter: 2

Book: Algorithms to Live By

Author: Brian Christian

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