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        riddles >> medium >> Geometric construction proposition
        
(Message started by: Christine on Feb 21^st, 2008, 4:09pm)

Title: Geometric construction proposition
Post by Christine on Feb 21^st, 2008, 4:09pm

Has anyone read the book "Mathematics and the Imagination" by Edward Kasner and James Newman ?

This is a great book for an introduction to Mathematics which includes in it for the first time, googol and googolplex, as well as some other concepts such as the "hyper or ultra radical" and "tubines" and "cycles". One other is the construction of what they call a "parhexagon" . Given six random points in a plane, connect these points with straight lines to form a closed figure. Construct the six centroids of three points that are connected consecutively, then connect the six centroids in order. The result is a hexagon that has opposite sides equal and parellel. If 2n random points in a plane where n is an integer and lines are drawn to make a closed figure between these points, and the centroid of n points connected consecutively are drawn and the centroids are connected, then a parallelo-2n- gon is made with the same properties. This is the main part of the proposition. I've noticed that if the compass has a fixed radius large enough to complete the constructions, then exactly 4*n^2 construction etches are needed to construct the ||-2n-gon given the 2n points from n=1 to n=4. Is this just a coincidence or does it continue when n>=5?
Supposing that 100 etches are needed for a ||-10-gon, how do you go about figuring how many etches would be necessary to construct the centroid for 5 points and can it be constructed? What is the formula for finding the number of etchings for the centroid for n points if this proposition is correct?

Title: Re: Geometric construction proposition
Post by BenVitale on Feb 23^rd, 2008, 11:38am

I think you could get the centroid of five points by constructing centroid of four of the points then mark off a distance 1/5 of the way from the remaining point on the line connecting it with the centroid of the four, but it would take many many etchings in total.
I don't know whether or not there's a formula for the minimum number of markings to construct the centroid of n points.
I'm sure someone on this site can answer this question.