Geometrical construction and cumulative error

[Philip Davies]

I have had quite a lot of practice in compass & straightedge geometry, sacred and profane. I do not think, however, that I have ever completed a construction without making at least one error that requires erasing and emending.
Here’s a case in point ( excuse pun): I want to make a simple display board and have laid out a hexagonal template. All right, the compasses are not best quality, but you may be able to make out at least one spot where the centres of the daisy wheels do not coincide.
Question is: have you ever seen a crop circle with erasures? Many are very complicated: but done in moonlight, and without mistakes!
Is this a can of worms?

[mklotz]

As I'm sure you know, dividing a circle into six equal parts is particularly easy. Now, are you aware that dividing a circle into seven parts with only compass and straightedge is impossible but five parts is doable?

The more general question of which divisions are possible and which are not is a fascinating bit of mathematics...

Gauss proved 200 years ago the possibility of dividing a circle into N equal parts (N=2,3,...) ONLY when N has the form:

N = 2^k*p1*p2*....pm, where k=0,1,2,3,....,p1, p2, ..., pm are "m" DISTINCT prime numbers (m=0,1,2,...), each of the form:

2^(2^q) + 1 (so called Fermat primes, q=0,1,2,...)
.
Hence it's possible for N = 2, 4, 8, 16,...

With N=3, we have 3 = 2^(2^0) + 1, so you can divide a circle into 3, 6, 12, 24, .... parts;

With N = 5 =2^(2^1) + 1, so you can divide a circle into 5, 10, 20, 40, .... parts.

Seven is, of course, prime but it is not a Fermat prime hence a division into seven parts is not possible.

Probably more than you wanted to know.

[Philip Davies]

Yes, I know.

[jgjgjg]

Typically crop circles are seen from the air, and the medium is not as precisely visible as pencil/paper. Perhaps we would think more to the story if, as previously pointed out, there was a crop circle evenly divided into seven parts.

I have less experience than you with compass/straightedge, though I have on occasion had to use them on a carpentry project. It's a lost science. The last time was during a post flood reconstruction project, and I used the technique to cut through some otherwise complicated tomfoolery. The other (professional!!) carpenters looked at me like I was Merlin suddenly risen from the under the hill.

[Philip Davies]

Jgjgjg, that’s bad about those carpenters! My interest in geometry arose from my training in carpentry and joinery. But there are very few in this country who can match the facility in geometry that the French, South Korean trained carpenters have. These are the two nations (there may be others) who excel in the international builders’ competitions. But, if crop circles are larger, surely any mistakes would be magnified?