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A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

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Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#31

Post by tetha »

zackattack wrote: post #27
Oompa wrote: post #14

Earlier quoted context omitted.

Since the triangle is broken up into 4 equal sections and one is highlighted.

What are the 4 equal sections?

There are four equal (modulo rotation) parts: three share a corner with the original triangle and a fourth in the middle does not share a corner with the original triangle. The one in the middle is highlighted grey as 'the third', the two triangle parts sharing the lower left and lower right corner of the original triangle are ignored and the triangle sharing the top corner of the original triangle is segmented further

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#32

Post by jordyhoyt »

LogicHoleFlaw wrote: post #29
gabeybaby wrote: post #24

Xeno's Paradox

Gah, Zeno's Paradox. A professor tried to stump the class with that one in an introductory philosophy course I took. I then proceeded to introduce him the fundamental principles of calculus with respect to limits. I think I threw in some snark about how this was the difference between mathematicians and philosophers - mathematicians actually find solutions! Then scientists ensure they apply to reality, and engineers…

Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox.

Further, very interesting reading: http://www.mathpages.com/rr/s3-07/3-07.htm

Math just gave us a way of coming up with the obvious answer, it does not describe the nature of the universe, which is what the philosophy was attempting.

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#33

Post by ftse »

michael_nielsen wrote: post #26

A very cute algebraic way to see this is to do arithmetic in base 4. In base 4 the series on the left is: 0.1 + 0.01 + 0.001 + ... = 0.111... (recurring) Multiplying the right hand side by 3 gives 0.3333... = 1, and so the original series must have just been 1/3.

Cute. To illustrate the sum of limits, however, I prefer a graphic with sum -> 1. Take an empty circle then add a chunk of shaded semicircle, then add one quarter, then add one eighth... you get the idea.

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#34

Post by anthonyb »

malkia wrote: post #5
ramidarigaz wrote: post #4

Earlier quoted context omitted.

I'm not so sure. It seems less immediately obvious.

I think the original example is more obvious.

There's also a merger of the two on Wikipedia:

http://en.wikipedia.org/wiki/File:Geometric_series_14_triang...

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#35

Post by 321abc »

jordyhoyt wrote: post #32
LogicHoleFlaw wrote: post #29

Earlier quoted context omitted.

Gah, Zeno's Paradox. A professor tried to stump the class with that one in an introductory philosophy course I took. I then proceeded to introduce him the fundamental principles of calculus with respect to limits. I think I threw in some snark about how this was the difference between mathematicians and philosophers - mathematicians actually find solutions! Then scientists ensure they apply to reality, and engineers…

Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox. Fur…

Also, just because a bunch of puny humans manage to convince each other that the world must be a certain way doesn't mean it actually is.

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#36

Post by dfranke »

LogicHoleFlaw wrote: post #29
gabeybaby wrote: post #24

Xeno's Paradox

Gah, Zeno's Paradox. A professor tried to stump the class with that one in an introductory philosophy course I took. I then proceeded to introduce him the fundamental principles of calculus with respect to limits. I think I threw in some snark about how this was the difference between mathematicians and philosophers - mathematicians actually find solutions! Then scientists ensure they apply to reality, and engineers…

The problem with philosophers is not that they can't find solutions, it's that they're incapable of rejecting bad ones. There are plenty of good ideas in philosophy, but you can't pick them out from the flood stupid ones without thinking everything through from square one on your own. Hence the reason that millenia later we're still here debating Zeno, while physicists no longer have to bother giving much thought to 100-year-old rejected theories.

A university dean approaches the chair of the Physics department and tells him, "Look, we really need to talk your budget. Every year it's particle accelerator this and supercomputer that. You're bleeding the college dry. Why can't you be more like the Math department? All they ever ask us for is pencils and chalkboards and wastebaskets. Or better yet, why can't you be like the Philosophy department? They don't even ask for wastebaskets!"

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#39

Post by jibiki »

tenzero wrote: post #38

While pedantic, the above "expression" is NOT = 1/3 1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n The "result" converges towards 1/3. You can get as close to 1/3 as you like, but the result will never quite equal 1/3. Cheers Dion.

I am on the fence about whether this is a good troll or a bad troll. It certainly exploits the "someone on the internet is WRONG" ethos of HN, but I don't think it does so in a particularly amusing way. I'm going to say that it's a rather boring troll.

(I do find myself compelled to say that the mathematical convention is that an infinite sum is defined to be equal to the limit of the partial sums, if it exists.)

Re: A visualization of why 1/4 + 1/16 + 1/64 + 1/256 + ... = 1/3

#40

Post by lallysingh »

Oompa wrote: post #3

I like this one better. http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without...

Indeed, the reason for the exact relation sigma (1/4^n) == 1/3 becomes more obvious. Split an area into fourths, use up one fourth, and another fourth of that fourth, and the whole set becomes an infinite series of triples, with the middle (shaded) one filled in.
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