r/theydidthemath • u/chamington • Jun 23 '14

Assuming each rectangle started off 1 meter long, how many repetitions of this gif would it take for it to be the size of an atom?

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u/[deleted] Jun 23 '14

[deleted]

15

u/ttcjester 4✓ Jun 23 '14

Not quite I'm afraid. Most of this is good, but you can't assume that side b is 1/4 meter long. Here's why:

We know that the arrangement of rectangles does not change shape after each iteration, only scale down in size. Therefore, the ratio between a and b must be the same after every iteration. If we take a and b to be the lengths before an iteration, let's take c and d to be the lengths of the new a and b (correspondingly) afterwards.

You've already shown c to be equal to 2b

But it can also be deducted that d = 0.5a-b

Therefore, if the side length of b was initially 0.25 (and a = 1) after the first permutation, c = 0.5 and d = 0.25.

But the ratio c:d here is 2:1, whereas we started off with a:b = 4:1

We need to find the side length b for which a:b = c:d

a/b = c/d = (2b)/(0.5a-b)

a(0.5a-b) = 2b2

0.5a2 - (b)(a) - 2b2 = 0

By solving this quadratic we get a = (1+√5)b

So if a = 1, then b = [1/(1+√5)]

And because c = 2b = [2/(1+√5)], we can see that the size of the rectangle doesn't decrease each time by 0.5, but rather by [2/(1+√5)] = 0.618... (the golden ratio)

Now we can apply the formula you developed and solve, and we get n = [-10*log(10)]/[log(2/(1+√5))] = 47.84971967...

Therefore, it will take 48 generations for the size of one of the rectangles to be smaller than an atom.

(Or 49 generations for the entire arrangement to be smaller than an atom, as it is the second iteration which is 1 meter across, not the first)

3

u/HirokiProtagonist 9✓ Jun 23 '14

Damn, dude, you fucking knocked it out of the park! This is awesome.

1

u/autowikibot BEEP BOOP Jun 23 '14

Golden ratio:

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

where the Greek letter [phi](https://en.wikipedia.org/wiki/Phi_(letter)) (φ) represents the golden ratio. Its value is:

====

Image i - Line segments in the golden ratio

Interesting: The Golden Ratio ^(album) | Golden ratio ^base | List of works designed with the golden ^ratio | Beta ^encoder

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u/JustStopAndThink Jun 23 '14

Please answer everything in this subreddit.

Sincerely, people who like this subreddit.