Harmony in Mathematics
Table of Contents
Harmony in Perspective
A quite different approach to the problem of constructing the Harmonic Mean geometrically arises from the early work of Desargues in establishing the field of Projective Geometry. In a work published in 1639, Desargues sets forth the foundations of the theory of what he calls the four harmonic points, not as it is done today using co-ordinate geometry techniques, but using projective methods.
To divide a line segment AB harmonically involves finding a pair of points C and D which divide the interval internally and externally in the same ratio, the internal ratio CB : AB being equal to the external ratio BD : AD on the extended line.
The Theory of Harmonicity states that if the external point of division of a line segment is given, then the internal point can be constructed using only intersections of straight lines - a purely projective technique.
Click anywhere on the first image showing a TI-Nspire construction to open an interactive version which you can control using HTML5 from Geometry Expressions, or watch the video of the GXWeb construction.
To divide the interval AB harmonically:
- An arbitrary triangle is drawn on the base AC.
- An arbitrary line drawn from the external point D cuts the triangle into a smaller triangle on a quadrilateral.
- The corners of the quadrilateral thus formed are joined to give a point of intersection of these diagonals.
- The vertex of the large triangle O and this point of intersection of the diagonals determine a line which cuts AB at the point H.
If we let AB = a and BC = b then BH gives the Harmonic Mean of a and b. Thus, to construct geometrically the Harmonic Mean of 6 and 12, rule an interval 12 cm long, and mark on it a point 6 cm from one end. Performing the above construction gives the length BH of 8 cm.
For comments & suggestions, please e-mail Steve Arnold.
