9 ms·

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

200 points by Flemlord 17y ago

ram1024 17y ago
preposterous
Oompa 17y ago
I like this one better. http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without-words/ http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without...
- ramidarigaz 17y ago
  I'm not so sure. It seems less immediately obvious.
  - malkia 17y ago
    I think the original example is more obvious.
    - anthonyb 17y ago
      There's also a merger of the two on Wikipedia: http://en.wikipedia.org/wiki/File:Geometric_series_14_triangle.png http://en.wikipedia.org/wiki/File:Geometric_series_14_triang...
  - jimmybot 17y ago
    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?
    - rottencupcakes 17y ago
      No one color represents the series. They all do.
      - jimmybot 17y ago
        Yeah, so it would be equivalent to swap out any of the colors, which is why I stopped for a moment to think about why I automatically assumed black represented the series. It does make some sense visually, being the only of the three to occupy the diagonal, though of course mathematically, any of the three are equivalent. But the whole point was the visualization!
    - LogicHoleFlaw 17y ago
      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?
      - I_got_fifty 17y ago
        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.
    - baddox 17y ago
      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.
      - sp332 17y ago
        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.
- zackattack 17y ago
  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?
  - Oompa 17y ago
    Since the triangle is broken up into 4 equal sections and one is highlighted.
    - zackattack 17y ago
      What are the 4 equal sections?
      - tetha 17y ago
        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
- pkulak 17y ago
  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.
- rottencupcakes 17y ago
  I like these trapezoids: http://math.rice.edu/~lanius/Lessons/Series/infinite.htm http://math.rice.edu/~lanius/Lessons/Series/infinite.htm
- lallysingh 17y ago
  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.
dustmop 17y ago
If the grey were made white instead I would have gotten this immediately.
- Xichekolas 17y ago
  The point of having three colors is to show that there are three boxes of each size. As such, if you pick any of the colors to represent the series, it's plain to see that all the boxes of that color will make up 1/3 of the total area.
  - TheSOB88 17y ago
    Too bad, that doesn't change the makeup of the average person's mind.
jakkals 17y ago
I like this! Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4 To generalize: 1/x + 1/(xx) + 1/(xx*x) + ... = 1(x+1) 'Proved' by looking at a picture :-)
- zackattack 17y ago
  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.
  - swapspace 17y ago
    Infinite GP: a/(1-r)
    - zackattack 17y ago
      Ah, true. http://en.wikipedia.org/wiki/Geometric_series#Formula http://en.wikipedia.org/wiki/Geometric_series#Formula
      - I_got_fifty 17y ago
        This wouldn't happen on Digg.
  - jonsen 17y ago
    You can get the proof idea from the picture: 1/4 is 1/3 of 3/4 1/4 of 1/4 is 1/3 of 3/4 of 1/4 etc. In math: 1/4 = 1/3*3/4 1/4^2 = 1/3*3/4*1/4 1/4^3 = 1/3*3/4*1/4^2 etc. Summing equations: (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 * (1 + 1/4^1 + 1/4^2 + ...) <=> (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 + 1/3 * 3/4 * (1/4^1 + 1/4^2 + ...) <=> x = 1/3 * 3/4 + 1/3 * 3/4 * x <=> x - 1/4 x = 1/4 <=> 3/4 x = 1/4 <=> x = 4/3 * 1/4 <=> x = 1/3
  - jakkals 17y ago
    Rats! Yes, I did indeed mean 1/(x-1). Thanks for the correction.
gabeybaby 17y ago
Xeno's Paradox
- LogicHoleFlaw 17y ago
  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.
  - jordyhoyt 17y ago
    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 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 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.
    - 321abc 17y ago
      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.
      - joeyo 17y ago
        That depends on what your meaning of the word 'is' is.
    - ewjordan 17y ago
michael_nielsen 17y ago
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.
- ftse 17y ago
  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.
  - eru 17y ago
    With this method you won't be able to tell converging sequences from diverging ones. Suppose I used your circle and put in the first hundred terms of the harmonic sequence (divided by 10 or so, to fit in the unit area).
- jonsen 17y ago
  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...
  - michael_nielsen 17y ago
    The advantage of my proof is that it generalizes to give the sum of 1/n + 1/n.n + ... Just replace 4 everywhere by n, and 3 by n-1: 1/n + 1/n.n + ... = 0.1111... (in base n) Multiply by (n-1) to get: 0.(n-1)(n-1)(n-1)... = 1 So the original sum must have been 1/(n-1). This is not so easy in binary.
tenzero 17y ago
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.
- jibiki 17y ago
  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.)
  - tenzero 17y ago
    Well perhaps I delivered the statement poorly. My intent is not to troll, I detest Trolls. The convention I have seen is to use the symbol of an arrow such as -> to denote the concept of approaching. However I did state I was being pedantic. D.
    - jibiki 17y ago
      Considering when you created your account, and the subject of your post, "troll" seemed like the most likely explanation. Sorry about that. (It's sort of a typical troll to make a technically incorrect statement and then watch people get all agitated as they correct you. In fact, it's one of my favorite trolls, right behind making a sarcastic statement which you know will be taken literally by half of the audience, a la "A Modest Proposal".) Anyways, assuming that your comment was in earnest, the arrow is typically used for functions (or sequences). E.g., 1/n -> 0 as n->infinity You could write: 1/4 + 1/16 + ... + 1/4^n -> 1/3 But you would write 1/4 + 1/16 + ... = 1/3 You wouldn't write (or at least I've never seen it) 1/4 + 1/16 + ... -> 1/3 It's not really a mathematical issue, just a definitional one: the left hand side is considered a real number, not a sequence of real numbers (or function :N->R, or whatever).
- 17y ago
pbhjpbhj 17y ago
hacker news needs to allow mathml!

200 points by Flemlord 17y ago

preposterous

Oompa 17y ago

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

ramidarigaz 17y ago
I'm not so sure. It seems less immediately obvious.
- malkia 17y ago
  I think the original example is more obvious.
  - anthonyb 17y ago
    There's also a merger of the two on Wikipedia: http://en.wikipedia.org/wiki/File:Geometric_series_14_triangle.png http://en.wikipedia.org/wiki/File:Geometric_series_14_triang...
- jimmybot 17y ago
  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?
  - rottencupcakes 17y ago
    No one color represents the series. They all do.
    - jimmybot 17y ago
      Yeah, so it would be equivalent to swap out any of the colors, which is why I stopped for a moment to think about why I automatically assumed black represented the series. It does make some sense visually, being the only of the three to occupy the diagonal, though of course mathematically, any of the three are equivalent. But the whole point was the visualization!
  - LogicHoleFlaw 17y ago
    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?
    - I_got_fifty 17y ago
      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.
  - baddox 17y ago
    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.
    - sp332 17y ago
      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.
zackattack 17y ago
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?
- Oompa 17y ago
  Since the triangle is broken up into 4 equal sections and one is highlighted.
  - zackattack 17y ago
    What are the 4 equal sections?
    - tetha 17y ago
      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
pkulak 17y ago
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.
rottencupcakes 17y ago
I like these trapezoids: http://math.rice.edu/~lanius/Lessons/Series/infinite.htm http://math.rice.edu/~lanius/Lessons/Series/infinite.htm
lallysingh 17y ago
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.

dustmop 17y ago

If the grey were made white instead I would have gotten this immediately.

Xichekolas 17y ago
The point of having three colors is to show that there are three boxes of each size. As such, if you pick any of the colors to represent the series, it's plain to see that all the boxes of that color will make up 1/3 of the total area.
- TheSOB88 17y ago
  Too bad, that doesn't change the makeup of the average person's mind.

jakkals 17y ago

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

zackattack 17y ago
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.
- swapspace 17y ago
  Infinite GP: a/(1-r)
  - zackattack 17y ago
    Ah, true. http://en.wikipedia.org/wiki/Geometric_series#Formula http://en.wikipedia.org/wiki/Geometric_series#Formula
    - I_got_fifty 17y ago
      This wouldn't happen on Digg.
- jonsen 17y ago
  You can get the proof idea from the picture: 1/4 is 1/3 of 3/4 1/4 of 1/4 is 1/3 of 3/4 of 1/4 etc. In math: 1/4 = 1/3*3/4 1/4^2 = 1/3*3/4*1/4 1/4^3 = 1/3*3/4*1/4^2 etc. Summing equations: (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 * (1 + 1/4^1 + 1/4^2 + ...) <=> (1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 + 1/3 * 3/4 * (1/4^1 + 1/4^2 + ...) <=> x = 1/3 * 3/4 + 1/3 * 3/4 * x <=> x - 1/4 x = 1/4 <=> 3/4 x = 1/4 <=> x = 4/3 * 1/4 <=> x = 1/3
- jakkals 17y ago
  Rats! Yes, I did indeed mean 1/(x-1). Thanks for the correction.

gabeybaby 17y ago

Xeno's Paradox

LogicHoleFlaw 17y ago
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.
- jordyhoyt 17y ago
  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 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 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.
  - 321abc 17y ago
    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.
    - joeyo 17y ago
      That depends on what your meaning of the word 'is' is.
  - ewjordan 17y ago

michael_nielsen 17y ago

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.

ftse 17y ago
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.
- eru 17y ago
  With this method you won't be able to tell converging sequences from diverging ones. Suppose I used your circle and put in the first hundred terms of the harmonic sequence (divided by 10 or so, to fit in the unit area).
jonsen 17y ago
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...
- michael_nielsen 17y ago
  The advantage of my proof is that it generalizes to give the sum of 1/n + 1/n.n + ... Just replace 4 everywhere by n, and 3 by n-1: 1/n + 1/n.n + ... = 0.1111... (in base n) Multiply by (n-1) to get: 0.(n-1)(n-1)(n-1)... = 1 So the original sum must have been 1/(n-1). This is not so easy in binary.

tenzero 17y ago

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.

jibiki 17y ago
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.)
- tenzero 17y ago
  Well perhaps I delivered the statement poorly. My intent is not to troll, I detest Trolls. The convention I have seen is to use the symbol of an arrow such as -> to denote the concept of approaching. However I did state I was being pedantic. D.
  - jibiki 17y ago
    Considering when you created your account, and the subject of your post, "troll" seemed like the most likely explanation. Sorry about that. (It's sort of a typical troll to make a technically incorrect statement and then watch people get all agitated as they correct you. In fact, it's one of my favorite trolls, right behind making a sarcastic statement which you know will be taken literally by half of the audience, a la "A Modest Proposal".) Anyways, assuming that your comment was in earnest, the arrow is typically used for functions (or sequences). E.g., 1/n -> 0 as n->infinity You could write: 1/4 + 1/16 + ... + 1/4^n -> 1/3 But you would write 1/4 + 1/16 + ... = 1/3 You wouldn't write (or at least I've never seen it) 1/4 + 1/16 + ... -> 1/3 It's not really a mathematical issue, just a definitional one: the left hand side is considered a real number, not a sequence of real numbers (or function :N->R, or whatever).
17y ago

pbhjpbhj 17y ago

hacker news needs to allow mathml!