Mascheroni Algorithms

 Lorenzo Mascheroni (May 13, 1750 – July 14, 1800) was an Italian mathematician. He was born near Bergamo, Lombardy. At first mainly interested in the humanities (poetry and Greek language), he eventually became professor of mathematics at Pavia. In his Geometria del Compasso (Pavia, 1797), he proved that any geometrical construction which can be done with compass and straightedge, can also be done with compasses alone. However, the priority for this result (now known as the Mohr-Mascheroni theorem) belongs to the Dane Georg Mohr, who had previously published a proof in 1672.

 Mascheroni construction for the Pentagon 

 1) Draw circle A. Find B, a random point on circle A. 2) Draw a circle at B with radius BA. Circle B intersects circle A at C. 3) Draw a circle at C with radius CA. Circle C intersects circle A at D. 4) Draw a circle at D with radius DA. Circle D intersects circle A at E. 5) Draw a circle at B with radius BD and draw a circle at E with radius EC. They intersect at F. 6) Draw a circle at A with radius AF. This new circle intersects circle B at G. 7) Draw a circle at G with radius GB. Circle G intersects circle A at H. 8) Draw a circle at H with radius HG. Circle H intersects circle A at points I and J. 9) Draw two circles, one each at I and J, each with radii ID and JC. They intersect at K. 10) Draw a circle at B, this time with radius BK. This circle intersects circle A at points L and M. B,L,M are points of the pentagon. Through them the other two can be found.

 Mascheroni construction for the Hexagon

 

Simply draw a circle and choose a point on that circle (yellow). Then draw a circle centered at that point of identical radius. Where that circle intersects the original circle, draw two more circles of equal radius. Continue doing this until you get 6 points on the circle. Those are the 6 points of the hexagon.

I keep asking myself (civil engineer, 84) about the joy for mathematics that today youth might have in the future due to the absorbing power of cybernetics (computers and similar).