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Pappus of Alexandria

Pappus was born in approximately 920 in Alexandria, Egypt. He was the last of the great Greek geometers and one of his major theorems is considered to be the basis of modern projective geometry (“Pappus”). Pappus flourished in the fourth century, writing his key work, the Mathematical Collection, as a guide to Greek geometry (“Biography”). In this work, Pappus discusses theorems and constructions of over thirty mathematicians including Euclid, Archimedes and Ptolemy (“Biography”), providing alternatives of proofs and generalizing theorems.

The Collection is a handbook to all of Greek geometry and is now almost the sole source of history of that science (Thomas 564). The separate books of the Collection were divided by Pappus into numbered sections. In the fourth section, Pappus discusses an extension on the Pythagorean Theorem (Thomas 575) now known as Pappus Area (Williams).

Pappus drew parallelograms on two sides of a triangle, extended the external parallels to intersection, connected the included vertex of the triangle and the intersection point, used the direction and length of that segment to construct a parallelogram adjacent to the third side of the triangle, and proved that the sum of the areas of the first two parallelograms is equal to the area of the third parallelogram (Williams, Thomas 578-9).

Section five of book five of the Collection discusses regular solids with equal surfaces and their varying sizes (Heath 395). Pappus’s conjecture was that the solid with the most faces is the greatest (Heath 396). He proved this using the pyramid, the cube, the octahedron, the dodecahedron, and the icosahedron of equal surfaces. Pappus noted that some of the other major Greek geometers had already worked out the proof of this conjecture using the analytical method, but that he would give a method of his own by synthetical deduction (Heath 395).

Using 56 propositions about the perpendiculars from the center of a circumscribing sphere to a face of the solids, Pappus proved that if the dodecahedron and the icosahedron were inscribed in the same sphere, the same small circle in the sphere would circumscribe both the pentagon of the dodecahedron and the triangle of the icosahedron (Heath 396). He went on to show that the cube is greater than the pyramid, the octahedron is greater than the cube, and so on (Heath 396).

One of Pappus’s biggest contributions to geometry is Pappus’s Theorem, which states, “If the vertices of a hexagon lie alternately on two lines, then the meets of opposite sides are collinear” (“Pappus”). When put another way, “If A, B and C are three distinct points on one line and if A’, B’ and C’ are three different distinct points on a second line, then the intersections of AC’ and CA’, AB’ and BA’, and BC’ and CB’ are collinear” (Smart 26), Pappus’s Theorem spawns the Geometry of Pappus.

This is a finite geometry consisting of exactly nine points and nine lines. The pairs of points making up the intersecting lines are interchangeable (Bogomolny 2). Also, Pappus’s Theorem is self-dual (Bogomolny 2), meaning that if the words “point” and “line” were interchanged in the theorem, it would still hold true. Thanks to the duality principle, any theorem proved for Pappus’s geometry is also true for the dual geometry.

According to Pappus, the purpose of the Collection was to explain the propositions established using geometrical methods by the ancient Greek mathematicians in a shorter and easier to understand from, and to introduce some useful theorems he himself discovered (Heath 429). Thanks to Pappus and his Collection, the world better understands theorems, propositions, and conjectures made by geometers such as Euclid, Archimedes and Ptolemy. And Pappus’s Theorem and the resulting geometry helped to bring the idea of duality to life so it could be applied to axioms from other geometries.

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