The tiles in some particular bathroom form the following pattern:
Up to a certain height, the tiles form a typical up/down/left/right grid, and above that height, the tiles form a grid at a 45 degree angle to the original grid.
At the line where the two intersect are the long diagonals of the 45 degree rotated tiles. The tiles above and below the line are the same tiles, so the
diagonals are √2 times as long as the sides of the lower tiles.
Notice that they don't line up perfectly: 2 of the longer diagonals is just short of the length of 3 of the sides, and 5 of the longer diagonals is just
slightly longer than the length of 7 of the sides. This gives us the following approximations:
The interactive below allows you to explore how we can use this sort of tiling to come up with rational approximations to all sorts of numbers:
Rational Approximations
Instructions
On the number line below, multiples of the selected number (α) are shown above a standard number line.
Click and drag or use the left and right arrow buttons to move the number line back and forth until the center blue line is over a spot where a multiple of α nearly lines up with a whole number on the number line.
This will give you a good rational approximation for α. A comparison of this approximation with the true value is shown above.
Further information about this approximation is shown below.
You can zoom in and out on the number line using the scroll wheel, or by adjusting the minimum x value and maximum x value shown in the window (xmin and xmax).
Increasing the info tolerance value will cause information about approximations to pop up more frequently (i.e. for worse approximations).
It may be helpful for finding good approximations to jump from some point on the number line to some multiple of it. The multiplier feature will help you do that.
Interactive
α = √n n:
α = n√m n:
m:
α = logm(n)
n:
m:
α =
α =
Click and drag or use the buttons below to scroll left/right, use the scroll wheel to zoom.
xmin:
xmax:
info tolerance: (increase this number to get information about more (worse) rational approximations)
multiplier: