Pythagoras and the Pythagoreans - Department of Mathematics

Pythagoras and the Pythagoreans 27
6.4 More Pythagorean Geometry
Contributions 22 by the Pythagoreans include
• Various theorems about triangles, parallel lines, polygons, circles,
spheres and regular polyhedra. In fact, the sentence in Proclus
about the discovery of the irrationals also attributes to Pythagoras
the discovery of the five regular solids (called then the ‘cosmic
figures’). These solids, the tetrahedron (4 sides, triangles), cube (6
sides, squares, octahedron (8 sides, triangles), dodecahedron (12
sides, pentagons), and icosahedron (20 sides, hexagons) were possibly
known to Pythagoras, but it is unlikely he or the Pythagoreans
could give rigorous constructions of them. The first four were associated
with the four elements, earth, fire, air, and water, and
because of this they may not have been aware of the icosahedron.
Usually, the name Theaetetus is associated with them as the mathematician
who proved there are only five, and moreover, who gave
rigorous constructions.
Tetrahedron Cube Octahedron
Dodecahedron Icosahedron
• Work on a class of problems in the applications of areas. (e.g. to
construct a polygon of given area and similar to another polygon.)
• The geometric solutions of quadratics. For example, given a line
segment, construct on part of it or on the line segment extended a
parallelogram equal to a given rectilinear figure in area and falling
22 These facts generally assume a knowledge of the Pythagorean Theorem, as we know it.
The level of rigor has not yet achieved what it would become by the time of Euclid