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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

#21

Post by I_got_fifty »

LogicHoleFlaw wrote: post #8
jimmybot wrote: post #6

Earlier quoted context omitted.

In the original, the (1/4)^n is more obvious to me, while in the second, the 1/3 part of it is more obvious. But the coloring scheme in the two are different too. The first uses three colors, the second two colors. What if the light gray in the first was white instead? I think then the 1/3 might pop out better. Wait a second, does everyone even see the same thing? Although it doesn't matter which color you pick to re…

The white, black, and gray can all represent the series - they are equal in area! The triangular representation does convey the idea of "1/3ness" more naturally to me, but the white/gray/white scheme seems to throw off the comparison. The equilateral triangle divided into four smaller such triangles, and the square divided into four smaller squares both have advantages as representations. Hmm. Do any other simple geo…

If the triangle would get the same colour-scheme, then it would be equally obvious. Of course it would be better if it was animated.

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

#23

Post by sp332 »

baddox wrote: post #9
jimmybot wrote: post #6

Earlier quoted context omitted.

In the original, the (1/4)^n is more obvious to me, while in the second, the 1/3 part of it is more obvious. But the coloring scheme in the two are different too. The first uses three colors, the second two colors. What if the light gray in the first was white instead? I think then the 1/3 might pop out better. Wait a second, does everyone even see the same thing? Although it doesn't matter which color you pick to re…

In the original, I can't see the one third at all. I originally saw the grey as being the items being summed, but after a second look I think the white may be this (not that it matters, the black could be it as well). It took me a while to realize that the point of the colors is to show that there are 3 of each size square.

Try looking at the different "layers" of squares. Start with the three big ones at top-left, bottom-left, and bottom-right. See how exactly one of the three is white (or black or gray, just pick one it doesn't matter). Now you see that one-third of this L-shaped piece is white, but what about the remaining quarter? Well, it's just the same. Look the the gray, black, and white square in the same configuration at the left & bottom of that quarter. Obviously one-third of that "L" is also white. Continue until you are convinced the one-third of every remaining part of the square is white.

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

#25

Post by pkulak »

Oompa wrote: post #3

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

I'm not sure how you show that the second triangle is half the volume of the first with that picture. With the Wikipedia proof, you really needed only the picture.

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

#26

Post by michael_nielsen »

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.

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

#27

Post by zackattack »

Oompa wrote: post #14
zackattack wrote: post #12

Earlier quoted context omitted.

This one seems immediately obvious of the thirdness, as other posters have pointed out. But I don't understand how each row ALSO represents 1/4^n?

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

What are the 4 equal sections?

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

#28

Post by jakkals »

zackattack wrote: post #13
jakkals wrote: post #11

I like this! Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4 To generalize: 1/x + 1/(x x) + 1/(x x*x) + ... = 1(x+1) 'Proved' by looking at a picture :-)

You mean, = 1/(x-1) ;) Can someone please post a non-visual proof of why this is the case? In the meantime, I am working on figuring out my own.

Rats!

Yes, I did indeed mean 1/(x-1). Thanks for the correction.

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

#29

Post by LogicHoleFlaw »

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 make them useful.

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

#30

Post by jonsen »

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.

Very nice indeed, but you can do it in binary, where the series is

  0.01 + 0.0001 + 0.000001 + ... = 0.010101010101...
Multiplying the right hand side by 2 gives

  0.101010101010...

    0.010101010101...
  + 0.101010101010...
  -------------------
    0.111111111111... = 1 = 11*0.010101010101...
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