Anyway, I woke up during the night with a teaser floating around my head and I remembered most of it this morning and tried to play with it. I got so far and then got stuck.
"The Potter’s Wheel.
In the annals of history, before the wheel came the potter’s wheel, and depending on what one was constructed from meant how to transport it from here to there. Certainly having it on its edge became the norm, rolling it along, and when some smart bugger decided that moving two of them along could be better done if an axle linked them, he did so…and wallah! Something really amazing eventuated after that when someone put a box on the axle and created a cart.
I’m going to say that his two potter’s wheels were perfectly round and one was twice as large as the other, so that one could be considered as 1 unit and the other 2 units in radius. They both had a little nick in their edges. The axle linking them was perfectly straight and the wheels were solid in their position and perfectly upright. One day, they were rolled out after several days of rain and a couple of days of good sunshine, and the wheels went upon drying mud.
It was noticed that at a certain spot, both wheels and their nicks were in perfect alignment, and further along the small wheel showed its nick in the mud while twice as far along the large wheel showed its nick in the mud. Being curious, the potter got his ruler out of his bag, the one used to measure the radii of the wheels from, and found that there was a perfect association between both wheels and their circumferences…so much so that he decided he would investigate such a thing, and set about delineating his measure down to the smallest even parts that could fit upon the ruler.
In estimating the circumferences of the wheels, he found that they were between 3 1/8 and 3 1/7 larger than their radii. Ah, but he could better delineate his measure and did so, and did so again and again until he had a fine measure upon his ruler that offered up a hair’s breadth as a distance between one line and another on his ruler. He had very good eyesight.
Upon measuring again, he found that he had a better estimation of the circumference from the larger wheel against the smaller wheel, twice as good actually, and decided he would create two more potter’s wheels to see if he could better define this wonder of math that was appearing before him…he made one a half-unit radius, and the other 5 units radius, and created a small wallow and wheeled his potters wheels through it so that their edges were straight up and they had nicks and all four lines could be measured against a ruler that could tell one hair’s breadth from another. And he found that there was a simple formula that expressed what he was seeing. The measure of the wheel’s edge in relation to its half-width upon a face was: 2 x r x (628/200 + 1/628)…and he was satisfied that that was the best he was going to do under the circumstances.
In playing with the numbers the potter was using and seeing how the larger wheel related to the smaller wheel, he found that there was a nice surprize in such a number. 628 = 500 + 128
By adjusting the multiplication of such a number to go inward and outward according to the size of the wheel and determining the relationship between scale, he could have twice the figure showing an association to radius and circumference as he could with half the figure, and he also found something quite unique with this 500 + 128 identity. He tried something…
250 + 64; 125 + 32; 62 ½ + 16; 31 ¼ + 8; 15 5/8 + 4; 6 13/16 + 2; 3 13/32 + 1; 1 45/64 + ½.
He went the other way: 1000 + 256; 2000 + 512; 4000 + 1024; 8000 + 2048; 16,000 + 4096; 32,000 + 8192; 64,000 + 16,384; 128,000 + 32,768.
He pondered the ramifications…having the original 500 + 128 went 8 steps in either direction where 256 and 64 showed to be instrumental numbers. On the way down, 64 on one side became 64ths on the other; on the way up, 256 on one side became 256 x 500 and 128 on both sides. He pondered the ramifications of the 628 and the 64 first, and found 9 13/16 as the ratio between them, which was quite odd considering the 3 and 13/32 association to 1. He pondered that the ratio between greater and lesser numbers like 500 to 128 was strange. He had pondered what he saw with this piece of pi and observed 3.1415925…1 / .1415925 = 7.0625…1/.0625 = 16 therefore 0.1415925 = a seventh and a sixteenth of 1. Now he saw this same sixteenth in conjunction with nine tenths plus 3 of 500 to 128.
There was a difference between 500/128 and 64/16 of a tenth and 2/3s of 1."
...I got stuck because my head got fried in trying to express such things as 'a seventh of 1 plus a sixteenth of 1' relating to what 0.1415925 is. If I can grasp that, I'll be bale to fix up where the 500 and 128 were heading to.