Summing Powers of 1/6 via a Square Dissection

Let's see a surprising geometric series dissection inside a unit square giving us the sum of the geometric series with first term 1/6 and ratio 1/6. Before we start, we note one key point. If we have two squares side by side and shade one of them, then we can cut the shaded triangle from a midpoint of a side to an opposite vertex and rotate the resulting triangle into a second square. In a diagram like this, the triangle, which we call special, has an area equal to one square. Now, start with a unit square. That is, consider a square with side length one and thus area one. First, divide each side into six equal lengths thus creating a 6 by 6 grid in the square. Each smaller square in this grid is 1/36 the area of the original square, and since the square has area one, each sub square here has an area of 1/36. Now, if we shade the yellow trapezoidal region, we've shaded an area of 1/6 as this region has an area equal to six of the sub squares in the grid. This follows as there are four fully shaded squares and two of the special triangles in adjacent squares. Next, the blue shaded triangle is a special triangle and thus has an area equal to one of the sub squares, and so we've shaded an extra 1/36. Now, we can use self-similarity to repeat this process on the sub square in the grid sitting on top of the shaded trapezoidal region. We cut this sub square into 36 squares, shaded trapezoid equivalent to six sub squares enclosing an area of 1/6 cubed, and a triangle equivalent to one sub square enclosing an area of 1/6 to the fourth. Then repeat again in the new sub square on top of the newly shaded trapezoid, shading a trapezoid with area 1/6 to the fifth, and a triangle sixth. And just keep repeating this process. Each time, cut the sub square on top of the newest trapezoid into 36 smaller squares and shade a new trapezoid region and a new triangle region, thus shading areas corresponding to the next two powers of 1/6. If we do this indefinitely, in the limit, the shaded area matches the area of a triangle in the square that looks like this. This triangle also arises as one of four triangles in the central square created by connecting the vertices of the square to opposite side midpoints like this. And here is the amazing part. By cutting the four triangles along the shown lines and rotating the smaller cut triangles 180 degrees, we see that each of these regions is exactly 1/5 of the area of the square, so each region has an area equal to 1/5. But that means that the bottom red triangle has an area of 1/5, and that means that the infinite sum of the positive powers of 1/6 is equal to 1/5. If you don't like that argument, we can also let the shaded region corresponding to the infinite sum of the positive powers of 1/6 be represented by S. And then we see that S is equal to the yellow trapezoid region enclosing an area of 1/6 plus the blue triangular 1/36 plus a region that is an exact scaled copy of the original shaded region with a scaling factor of 1/36. So S is equal to 1/6 + 1/6 squared * S. Then using a little algebra, we see that 35/36 * S is equal to 7/36, so that S is exactly equal to 7/35 or 1/5. Again, we conclude that the infinite sum of the positive powers of 1/6 is exactly 1/5. If you like this proof, check out my channel for geometric series dissections in a unit square for ratios 1/2, 1/3, 1/4, 1/5, and 1/9. Can you find a geometric series dissection in a unit square for the series with ratio 1/8?