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Line Segment - Thought Experiment
BAndrew Offline
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#1
Line Segment - Thought Experiment

Imagine a line segment of 1 unit(s) long (meters,feet,whatever you want).

You remove 1/3 of the segment located in the middle and you are left with 1/3 in the left and 1/3 in the right. Do this again for each of the two pieces of the segment left. Now you should have 4 smaller segments. Continue this process infinitely many times (I mean imagine it). The process is illustrated bellow:

[Image: lmzsb2s]

The thing is. You obviously have removed infinetely many points (points are considered 0 dimensional), but how many points are left? Moreover, how much length of the original segment is left in the end?

•I have found the answer to the universe and everything, but this sign is too small to contain it.

[Image: k2g44ae]



04-23-2014, 08:11 PM
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Froge Offline
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#2
RE: Line Segment - Thought Experiment

1 - (1/3) - 2(1/9) - 4(1/27) - 8(1/81)....
= 1 - (2^0)(1/3) - (2^1)(1/3^2) - (2^2)(1/3^3) - (2^3)(1/3^4) - ....
= 1 - Sigma (n=0 -> inf) 2^n / (3^n+1)
= 1 - Sigma (n=0 -> inf) (1/3) (2/3)^n

Consider just the Sigma for a moment. The formula for summation of infinite geometric series is Sum = u1 / (1 - r) where u1 is the initial term and r is the ratio between each terms.

In this series, u1 = 1/3 and r = 2/3, so Sum = (1/3) / (1 - 2/3) = (1/3) / (1/3) = 1

Substitution back into the equation, the the answer is 1 - 1 = 0

None of the segment is left.

[Image: p229xcq]
(This post was last modified: 04-23-2014, 08:25 PM by Froge.)
04-23-2014, 08:22 PM
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BAndrew Offline
Senior Member

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Joined: Mar 2010
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#3
RE: Line Segment - Thought Experiment

@Froge

WOW! I actually wasn't expecting that response, but you are absolutely right! This is the so called cantor set.

Cantor Set

•I have found the answer to the universe and everything, but this sign is too small to contain it.

[Image: k2g44ae]



04-23-2014, 08:30 PM
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Froge Offline
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#4
RE: Line Segment - Thought Experiment

free reps pls

[Image: p229xcq]
04-23-2014, 08:35 PM
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Statyk Offline
Schrödinger's Mod

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#5
RE: Line Segment - Thought Experiment

I was skimming the Cantor Set page and this guy's name just killed me: "Henry John Stephen Smith"

It's like... How generic can your name get? lol

(no offense, I just find that funny)
04-24-2014, 04:52 AM
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