Monday, March 14, 2011

Archimedes angle trisection or tripling?

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".

3 comments:

  1. Arjen,
    Awesome 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

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  2. Hi Pat,

    It 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

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

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

    Discovered By
    Eng/ Ahmed elsamin
    (Arab Republic of Egypt)

    ReplyDelete