The Garden of Archimedes  A Museum for Mathematics

Pythagoras and his Theorem

Regular solids

	Among the discoveries attributed to Pythagoras is that of the cosmic figures, meaning the regular solids. To understand what this is about we first have to say a few words about polygons. A polygon is a plane figure delimited by a broken line, the segments of which are the sides of the polygon. The points in which the sides touch are called vertexes of the polygon. The angles that the sides form in each vertex are the angles of the polygon. Each polygon has as many sides as it has vertexes and angles - it takes its name from this number. For example a polygon with three sides (therefore with three sides and three vertexes) is called a triangle, the one with four is called a quadrilateral, with five a pentagon (from the Greek pente, five and gonia, angle) and hexagon, octagon, and so on. Note the difference between the triangle, the quadrilateral, and other polygons.
	A polygon is called "regular" if its angles and sides are has all equal. To draw one of them, one can divide a circumference into the wanted number of parts, and join the points of division. In this manner a pentagon can be obtained by dividing the circumference in five equal parts, like in the figure. A property of the regular polygons we will deal with is that their angles increase as the number of the sides increases. The angles of an equilateral triangle are 60 degrees, those of a square 90 degrees, a regular pentagon has 108 degree angles, the regular hexagon has 120 degree angles and so on.
	Now let us move on to polyhedrons, solids with equal regular polygons as faces, such that the same number of faces gather in each vertex. The faces are attached along the edges, which in turn gather in the vertexes, points around which are placed three or more faces. Unlike regular polygons, which can have any number of sides, there are only five regular solids: three with triangular faces, one with square faces and one with pentagonal faces. To understand the reason for this pattern, let us take a regular polygon, and see what happens around one of its vertexes, leaving only the faces around this vertex and taking away all the others. If we now cut again along one of its edges, the remaining surface can be unfolded on a plane. What we see is a certain number (three or more) of equal regular polygons, that all touch in the vertex, plus a certain angle that is created when the polyhedron is unfolded to be placed on the plane. The sum of the angles that are around the vertex is less than 360 degrees, and since around the vertex there are at least three polygons, these must have angles of less than a third of 360 degrees, which is less than 120 degrees. But, as we saw, there are only three regular polygons with angles smaller than 120 degrees - the triangle, the square and the pentagon. The hexagon already has 120 degree angles, and hence three hexagons fill all the space around the vertex and don't leave any angle to close the polyhedron.
	Therefore only regular solids with faces of three, four or five sides are possible. In the last two cases there can only be three faces around a vertex (four squares fill all the space around the vertex, four pentagons make more than 360 degrees). The respective solids are the cube and the dodecahedron, the first with six square faces, the second with twelve pentagonal faces. In the case of triangular faces, each having a 60 degrees angle, around a vertex there can be three, four or five faces but not six, which would fill all the available space. The corresponding solids are the tetrahedron which has four faces in total, the octahedron, and the icosahedron with twenty.