[quote=davieddy;99361]This problem suggests that choosing a multiple of 36 for the number
of degrees in a circle was inspired. When does it date from?[/quote]After I googled "Origin of 360 degrees in a circle", the first two hits,

[URL]http://mathforum.org/library/drmath/sets/select/dm_circle360.html[/URL]

and

[URL]http://www.wonderquest.com/circle.htm[/URL]

eventually both lead to the apparently authoritative book [U]The Exact Sciences in [/U]
[U]Antiquity[/U] by Otto Neugebauer.

Not having Neugebauer's reference at hand, I combined April Halladay's answer at WonderQuest with the eventual answer at MathForum, plus a little of my own, to get (in my wording):

In Mesopotamia the Sumerians had, by 2400 BC, a calendar of 12 months of 30 days each. Apparently they valued the arithmetic niceties of the number 360 more than they were irritated by the five-day yearly discrepancy. They also invented the 360-degree circle, but not subdivisions of degrees, which came later.

About 1500 BC, Egyptians invented the 24-hour day, but with variable-length hours. Roughly the same time in Mesopotamia, Babylonians invented base-60 arithmetic. Later, Greeks made the hours equal and constant.

About 300-100 BC, Babylonians subdivided both the degree and the hour into 60 minutes of 60 seconds each.

[QUOTE=davieddy;100057]I still wait with interest, Phil[/QUOTE]

I created the graphic with a program called Geometer's Sketchpad which saves its work in the form of a program-specific graphics file. I am able to copy the displayed graphic and paste it into a Word document, but I do not know how to convert it into a format (jpg or gif ?) that I can upload to the forum. Suggestions are welcomed.

[COLOR=White].[/COLOR]

That's just what I anticipated!

Thanks for posting that, Mike. I'll try to explain what I found interesting. It was provoked by David's observation of reflection. Suppose we label the vertices consecutively as V[sub]1[/sub], V[sub]2[/sub], V[sub]3[/sub], ... V[sub]18[/sub]. Then what we notice is that the following line segments are all concurrent: V[sub]1[/sub]V[sub]7[/sub], V[sub]2[/sub]V[sub]9[/sub], V[sub]3[/sub]V[sub]12[/sub], and the reflections of the first two segments across the last: V[sub]4[/sub]V[sub]15[/sub] and V[sub]5[/sub]V[sub]17[/sub]. My guess is that there is some underlying symmetry that explains why these lines are concurrent, but I haven't done much research on it. Could something related to Pascal's theorem be at work here?

Are there other regular polygons with similar
concurrencies of diagonals?

David

A diagram without the diameters might look
more spectacular?

Perhaps, but it is the fact that the intersections of the other diagonals fall directly on those diameters that is of interest.

[quote=philmoore;100126]Perhaps, but it is the fact that the intersections of the other diagonals fall directly on those diameters that is of interest.[/quote]
Two lines intersecting each other is not interesting
which is why removing the diameters (and the trivial symmetric
triple intersections) would emphasize what we are trying to illuminate.

But it is the fact that three lines intersect at one point that is interesting, and one of those lines is the diameter.

I'm sure I have a reply to this. Just give me time:)