Posted by John M. Nichols on February 14, 2007 at 20:01:43 from 165.91.46.46 user Mcneacail.
In Reply to: Re: Dick question posted by John M. Nichols on February 10, 2007 at 23:07:07:
I am reading a book called Nahin, Paul, J.(2006) Dr. Euler's Fabulous Formula - Cures Many Mathematical Ills, PUP: Princeton and Oxford
Nahin (2006, 43) describes how to create n-gons with Fermat (F) primes. It is an interesting read and to summarize the main point
Assuming that all sides of a n-gon are equal and as far as we are concerned are unit length.
. a point is a unigon or as we call it properly a point. Zero sides, one point, all sides are orthonormal or parallel.
---------- a line is a duogon, we can not argue that the sides are orthonormal which is interesting as the sides are parallel. This is the paradox of a definition, the limit of a point defies logic.
But the true n-gons in Nahin's book are argued to start at three. A point and a line can trivially be constructed with a pencil and a ruler.
a triangle is a trigon, but here it has the special name of a equilateral triangle.
a square is quadgon, where all the sides are at 90 degrees
etc.
Some of these gons can be constructed using plain geometry basically described as + - multipication and square roots. Euclid did this 2000 years ago upto but not including 17. It is easy to construct even n-gons from lower even gons, but the odd sided gons are harder. No-one before Gauss (age 19) had shown mathematically it was possible to construct a heptadecagons (17 sided).
It has been subsequently shown that there are a limited number of such n-gons that can be constructed using a ruler and a compass.
Gauss gave a formula for determining those n-gons that could be constructed with a ruler and a compass. n=2(power k)FiFjFk.... where Fi are Fermat numbers. Of interest here are the first few:
F0 = 3 sided (2 raised to power 1) + 1 etc....
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
unfortunately F5 is not a prime and so we miss the next n-gon
Richelot constructed the 257-gon in 1832 (24 years old)
Klein derived the point data for the 65537 sided polygon in 1894 (took 10 years)
The fifth such nn-gon is 3*5*17*257*65537 = 4,294967295 sides
Nahin states "It (nn-gon) would be indistinguishable from a circle."
My first issue
1. A circle by definition has an infinite number of points,
2. The nn-gon has a finitely countable set of points that I could count at a rate of one per second in 136 years. The last time I looked that is not a infinite time.
3. The area of all gons must be less than the inscibing circle, so it can be distinguished, it may not be visible tot he human eye, but it can be distinguished.
My second issue:
4. A pure mathematician should not make a general observation about what they perceive as being the same. Nahin is an EE.
My third issue
let us define the angle subtended by one unit of the nn-gon side as a Nahin.
Then the Nahin would be defined as a metre long on a radial length of 683.565275576 kilometres. There would be 3314.017497 Nahins per second of angle. or 1 Nahin equals 3.017485e-4 second.
I grant Nahin that it would be difficult to measure, but they are getting down to mm accuracy on GPS using differential measurements, so it is not impossible.
So that is the long story. It actually raises more questions than I answer.
1. Are there x-gons between the 3-gon and the 4-gon, and if so are there an infinite number of them?
2. Can I calculate x-gons? Where 3 < x < 4.
Interesting.
Sorry it took so long, but all my otehr computers do not have my TARS account password and so I had to wait till I got over my ills and came to work.
It is very cold here, like Nancy in the pond weather.
JMN