4 ms·

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...

by Oompa 17y ago

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.

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...

by 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.