Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

en.wikipedia.org

Post reply on HN
Print view View original

1–10 of 50 posts

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

#6

Post by jimmybot »

ramidarigaz wrote: post #4
Oompa wrote: post #3

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

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

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 represent the sum of the geometric series, I defaulted to the black squares representing the series. In the second, I assumed the gray represented the squares of the series. How about others?

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

#7

Post by rottencupcakes »

jimmybot wrote: post #6
ramidarigaz wrote: post #4

Earlier quoted context omitted.

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

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…

No one color represents the series. They all do.

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

#8

Post by LogicHoleFlaw »

jimmybot wrote: post #6
ramidarigaz wrote: post #4

Earlier quoted context omitted.

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

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 geometric shapes easily divide into self-similar shapes?

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

#9

Post by baddox »

jimmybot wrote: post #6
ramidarigaz wrote: post #4

Earlier quoted context omitted.

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

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.
Post reply on HN