What happens when you divide the space between 0 and 1 into finer and finer slices using the fractions 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5…? Some interesting things. In the images below, the white line on the left with a white dot below it represents 0 and the white line on the right with a white dot below it represents 1. The single line in the middle of the first image represents 1/2 between 0 and 1, then the lines in the second image represent 1/4, 1/3, 1/2, 2/3, 3/4 between 0 and 1. And so on:
1/2 between 0 and 1 (click for larger)
1/4, 1/3, 1/2, 2/3, 3/4 between 0 and 1
a/b <= 7/8 between 0 and 1
a/b <= 15/16 between 0 and 1
a/b <= 31/32 between 0 and 1
a/b <= 63/64 between 0 and 1
a/b <= 127/128 between 0 and 1
a/b <= 255/256 between 0 and 1
Animation of a/b <= 255/256 between 0 and 1 (created at ezGif)
The lines are dividing space into finer and finer slices, but the lines aren’t distributed evenly. Some gaps close quickly, some slowly. Here’s some color for the slicing (when the colored circles below the lines are solid, they mark a/b <= 7/8; when they're hollow, they represent 1/9 <= a/b <= 11/12). The lines are now adjusted in height to the fraction a/b they represent, so they create a wedge rising from left to right:
1/2 between 0 and 1
a/b <= 1/3 between 0 and 1
a/b <= 2/3 between 0 and 1
a/b <= 1/4 between 0 and 1
a/b <= 3/4 between 0 and 1
a/b <= 4/5 between 0 and 1
a/b <= 5/6 between 0 and 1
a/b <= 6/7 between 0 and 1
[…]
↓
[…]
a/b <= 11/12 between 0 and 1
a/b <= 19/20 between 0 and 1
a/b <= 29/30 between 0 and 1
a/b <= 39/40 between 0 and 1
a/b <= 49/50 between 0 and 1
a/b <= 99/100 between 0 and 1
Animation of a/b <= 99/100 between 0 and 1 (created at ezGif)
Here’s how the colored lines representing a/b <= 11/12 get swamped as b gets larger and larger:
a/b <= 11/12 between 0 and 1
a/b <= 15/16 between 0 and 1
a/b <= 31/32 between 0 and 1
a/b <= 63/64 between 0 and 1
a/b <= 127/128 between 0 and 1
a/b <= 199/200 between 0 and 1
a/b <= 255/256 between 0 and 1
Animation of a/b <= 255/256 between 0 and 1 (created at ezGif)
Post-Performative Post-Scriptum
In fact, thrice isn’t slice as twice, because the line marking the slice of 1/2 gets swamped after the lines marking 1/3 and 2/3, which get swamped after 1/4 and 3/4, and so on.
