Billion dollar ideas

[SeaWasp]

The problem of constructing an angle equal to the one third of the given one has been pondered since the times of antiquity. Probably to make the notion of 'geometric construction' more exciting the Ancient Greeks have restricted the allowed operations to using a straightedge and a compass. It's thus specifically forbidden to use a ruler for the sake of measurement. Three famous construction problems lingered until early 19th century when it was shown that it's impossible to solve them with the help of only a straightedge and a compass. The three problems are: to trisect a given angle, to double a cube, and to square a circle. However, one illicit solution that has been found in the works of Archimedes.

http://www.geom.uiuc.edu/docs/forum/angtri/

http://www.uwgb.edu/dutchs/PSEUDOSC/trisect.HTM

and this may be of interest...

http://www.cut-the-knot.org/pythagoras/archi.shtml

[terry5732]

The largest gold mining operation in Colorado extracts one ounce per twenty tons of burden at a good profit.

[ken_behrendt]

SeaWasp noted:

Probably to make the notion of 'geometric construction' more exciting the Ancient Greeks have restricted the allowed operations to using a straightedge and a compass. It's thus specifically forbidden to use a ruler for the sake of measurement.

I think Archimedes had the right solution and that it's okay to cheat a little by using one's compass to mark off divisions on one's straight edge and thereby turn it into a ruler!

With the advent of modern scientific calculators, I'm surprised that anybody is still interested in these ancient geometry problems. Yes, they can help stimulate one's sense of geometric relationships, but today, with the push of a few buttons, they can all be solved with an precision to a dozen or more "significant" figures. I think modern math students do not even want to be bothered with using Newton's (or is it Leibnitz'?) calculus to calculate such things as areas, volumes, and centers of mass anymore. They just want to get it done as quickly as possible with a program and then get onto something more relevant to their lives...like playing video games!


ken