100 Famous Problems of Elementary Mathematics and How to Solve (some of) Them with Geometry Expressions and Mathematica

Philip Todd

Saltire Software

Abstract

The combination of Geometry Expressions and Mathematica is a powerful platform for the automation of certain aspects of mathematical modeling and problem solution.  Geometry Expressions lets you sketch geometric figures and extract symbolic expressions from the figures.  These expressions may be copied into Mathematica for analysis, and the results of the analysis pasted back into the geometry system.  We illustrate this process with a set of problems taken from Heinrich Dorrie's book "100 Famous problems of Elementary Mathematics, their History and Solution" [1].

Among the problems to be discussed will be Regiomontanus' Optimization problem.  This poses the question:

From what point on the earth's surface will a vertically suspended bar look the longest.

A Geometry Expressions model of this problem consists of a geometric sketch along with some symbolic constraints.  The circle representing the earth is given a radius R, the bar is given length L and distance R+h from the center of the earth.  The viewing point is located at angle x from the line joining the bar to the center of the earth.  Given this geometric input, Geometry Expressions is able to generate a symbolic output for the viewing angle.

This may simply be copied into Mathematica where standard calculus techniques may be applied to the expressions:

Solve[D[-arctan(((L·R·sin(x))/((R+h-R·cos(x))·(-(L+R+h)+R·cos(x))-(R^2)·((sin(x))^2)))),x]==0,x]

The solution may be copied back into Geometry Expressions in place of the variable x.

Geometry Expressions gives the user the ability to inspect the solution, and to traverse the problem space by interactively altering the numeric values attached to problem parameters.  Important insights may be gained in this way, for example in Regiomontanus' problem, one observes that the circumcircle of the triangle formed by the ends of the bar and the viewing point is tangential to the earth.  This insight can lead to a geometric proof of the result.

A variety of examples will be presented, including Euler's Tetrahedron Problem, Cardan's Spur Wheel Problem, the Poncelet-Brianchon Hyperbola Problem, Archimedes' Squaring of the Parabola, The Curvature of Conic Sections.   The common theme will be the power of the combination of geometric modelling in Geometry Expressions, mathematical analysis in Mathematica, and solution exploration in Geometry Expressions.

[1] H. Dorrie (1965), “100 Great Problems of Elementary Mathematics”, Dover, New York,