After I saw Matt Henderson’s visual demonstration of how the geometric series 1/4 + 1/16 + 1/64 . . . adds to 1/3, I thought about how to generalize this for any geometric series. This led to a way to visualize any real number from its decimal representation. As far as I know it’s original.
Start with a circle. Let’s say that its area represents one unit. Imagine an infinite sequence of smaller concentric circles inside it. Each of these has 1/10 the area of the next larger circle. The spaces between the circles form bands or rings.
Nine radial lines divide each band into nine equal parts. Each segment in the outer band has one-tenth the area of the whole circle. For the band just inside this one, each segment has .01 the area of the whole circle, and then .001, and so on as we move inward.
We could also move outward. Draw a new circle around the original circle such that its area is ten units. Each of the nine segments of this new outer band has an area of 1.
To represent π, we could color 3 of these outer segments (each with area 1), then 1 of the .1 area segments in the next inner band. After that, we color 4 of the .01 are segments, and so on. As we go, we build 3.14159 . . . .
With this approach, we can represent any real number, whether rational or irrational. Although the number of segments available for coloring is a countable infinity, the number of numbers that can be represented is uncountable, as Cantor showed.
I asked Matt to see what he could do with this idea. Here is his hypnotic animation of π as we zoom in to smaller and smaller scales:
Update: I should also point out that this kind of visualization could help explain the properties of repeating decimals. For example, 0.333… would correspond to three colored segments in each band, forming a wedge equal to 1/3 of the total.
If two digits repeat, say with .010101…, then we need to use circles whose areas shrink by hundredths instead of tenths. Each band would be 99 times larger than the circle it surrounds. We would use 99 radial lines to divide each band into 99 very thin pieces. For .010101…, we would color one segment of each band, creating a wedge equal to 1/99 of the whole.