4 ms·

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 t

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.

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.