Pi-day today, so a nice occasion to write something about circles. π is the ratio between the circles' perimeter (περίμετρος in Greek) and diameter.

In the previous post, I explored some angular tricks using circles. The circle is a key geometrical object when dealing with angles. For example, Archimedes' construction to trisect an angle uses a circle and a straight line with a marked unit. But for clarity's sake, I prefer to draw a set of circles which show how a straight line can mark odd multiples of a unit angle α on intersecting circles, see Figure 1. In that way, it is easier to see how Archimedes' trisection circle relates to the multiplication and division of angles by odd integers, see alternative Figure 2.

This makes me think that Archimedes' angle trisection is in fact the inverse of "Archimedes' angle tripling".

## Monday, March 14, 2011

### Archimedes angle trisection or tripling?

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Arjen,

ReplyDeleteAwesome stuff.. I especially like the idea from the last post that the two arcs cut out on any circle will always have a constant difference. These two posts could make a really good exploration in a HS geometry class.. Thanks

Pat

Hi Pat,

ReplyDeleteIt would be nice indeed to ask HS students, or even children, to explore such patterns. I am sure some would discover new ones faster than we would. My 9-year old daughter sometimes draws geometric figures next to me. I try to teach her some recursive drawings with circles or triangles. It's amazing how fast she catches it up.

Arjen

New method about general angle division and angle trisection on a link

ReplyDeletehttp://www.mathramz.com/math/node/1726

Discovered By

Eng/ Ahmed elsamin

(Arab Republic of Egypt)