Optimization Michael Lahanas

Optimierungsprobleme im Antiken Griechenland

Bees were endowed with a certain geometrical forethought .... There being, then, three figures which of themselves can fill up the space round a point, viz. the triangle, the square and the hexagon, the bees have wisely selected for their structure that which contains the most angles, suspecting indeed that it could hold more honey than either of the other two. Pappus of Alexandria

Geometric Optimization

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As Geometry plays a important role in Greek Ancient Science it is not very surprising that optimization problems have been considered and solved.

In Euclid's book III of the Elements there is a discussion of the greatest and least straight lines that can be drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from editions of his works) the parallelogram of greatest area with a given perimeter.

Apollonius of Perga (Απολλώνιος ο Περγαίος) investigated the greatest and least distances of a point from the perimeter of a conic section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular hyperbola.

Some remarkable theorems on maximum areas are attributed to Zenodorus (Ζηνόδωρος ο Γεωμέτρης), and preserved by Pappus of Alexandria (Πάππος ο Αλεξανδρεύς) and Theon of Alexandria (Θέων ο Αλεξανδρεύς):

• Of polygons of N sides with a given perimeter the regular polygon encloses the greatest area.

• Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.

• The circle encloses a greater area than any polygon of the same perimeter.

• The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.

• Of segments of a circle of given perimeter, the semicircle encloses the greatest area.

• The sphere is the surface of given area which encloses the greatest volume.

Maximization

Archimedes solves the problem to find the maximum of f(x) = x² (x-a) as part of the problem to divide a sphere in two parts so that the volume ratio of these parts is equal to a given ratio m/n. The solution was lost but Eutocius who wrote comments about the work of Archimedes was able to reconstruct the solution which involves the use of conic sections.(as intersection points of a hyperbola and a parabel)

Physics