I uncovered something very interesting while working out a practical problem to build a circular patio with square blocks. I am so FLOORED by what I found I'm wondering how this works, and how I can use it to simplify things in the future. I've attached supporting files, an excel file and a PNG of the patio.
To plan for the patio, I'm drawing out a couple options in Adobe Illustrator. In this version, my square blocks are staying square, with pie-shaped gaps.
Here are the starting stats: Initial diameter circle: 36 inches Block size: 6 inches
At 36", I needed 18.85 blocks, but I'm not cutting the blocks, so I rounded to 19 and recalculated the circumference. I now needed a 36.29" diameter.
I manually continued on this way for a while, creating a new ring in Illustrator, measuring it roughly, rounding up, recalculating, rinse and repeat.
After a while, out of curiosity (and out of desire to figure out how many rings I'd need to create a 16' ring), I created a column to calculate the difference between each successive diameter.
I was SHOCKED to find I didn't need to average anything; the differences were exact to 9 decimal places! And then I noticed that the number of blocks in each ring was increasing at the same rate too, 7 new blocks for each ring.
How is this so precise and linear?! What principle governs this, and how could I use it to change my parameters, like block width?
My guess is this has something to do with the circle being reduced to a polygon shape by the blocks, and polygon diameters are more straightforward and precise than circles. Am I in the ballpark? What's the exact answer?