Archive for November, 2010

Polyhedrons

November 28, 2010

Solid geometry has always fascinated me and has fired up my imaginative power. I was introduced to Solid Geometry by my elder brother (L V Sundaram) when I was barely 10 years old. He had a habit of reading and explaining to himself as a part of his study process. When I was around as a young boy, he would do the explaining to me, not caring how much of it I understood. But he was rather surprised that I understood a lot of what he told me especially in geometry. As I learnt more of plane geometry, I developed a habit of extending the results of the theorems to the third dimension.

Starting from triangle, quadrilateral and pentagon, there are infinite number of polygons. Polygons with all sides and all internal angles equal are called regular polygons and there are infinite numbers of regular polygons with 3 to infinite numbers of sides. In the limit the regular polygon becomes a circle.

Extending to the third dimension, there are infinite numbers of polyhedrons, starting from tetrahedron.

Two non-parallel straight lines in a plane make an angle. Three straight lines in a plane which are non parallel to each other make a triangle, which is a polygon of least order – 3 sides and 3 angles.

Similarly, two non-parallel planes make an angle all along the common edge. Three planes which are non parallel to each other make a solid angle at a unique point of intersection of all the three planes. This solid angle is referred to as trihedral angle, as they are made by three planes. The solid angle at the vertex of a typical polyhedron is known as a polyhedral angle, made by three or more planes. But three intersecting planes still do not make a closed polyhedron. A fourth non-parallel plan is needed to get a polyhedron of least order – 4 faces, 6 edges and 4 vertices; this is known as a Tetrahedron or sometimes as triangular pyramid. We may also think of a polygonal pyramid with the base as a polygon of n-sides. For a regular polygon of n-sides, this polygonal pyramid will become a circular cone in the limit.

Regular polyhedrons are those whose faces are all regular polygons, congruent to each other, whose polyhedral angles are all equal and which has the same number of faces meet at each vertex. An interesting fact is, there are only five regular polyhedrons: the Tetrahedron (four triangular faces), the Cube (six square faces), the Octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the Dodecahedron (12 pentagonal faces), and the Icosahedron (20 triangular faces). (as below) Polyhedrons are classified and named according to the number and type of faces. A polyhedron with four sides is a tetrahedron, but is also called a Triangular pyramid. The six-sided cube is also called a hexahedron. A polyhedron with six rectangles as sides also has many names—a rectangular parallelepiped, rectangular prism, or box. However there are infinite numbers of semi-regular and non-regular polyhedrons which closely approximate to a hollow sphere in the limit. Imagine cutting off the corners of a cube to obtain a polyhedron formed of triangles and squares, for example. Other common polyhedrons are best described as the same as one of previously named that has part of it cut off, or truncated, by a plane.

Euler characteristic

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

For a simply connected polyhedron, χ = 2.

For regular polyhedrons as below

Cube ;     χ = 8-12+6 = 2

Tetrahedron     χ = 4 -6 +4 = 2

Octahedron     χ = 6 – 12 + 8 = 2

Dodecahedron χ = 20 – 30 + 12 = 2

And so on ….

Duality For every polyhedron there exists a dual polyhedron having:

• faces in place of the original’s vertices and vice versa,
• the same number of edges
• the same Euler characteristic and orientability.

Dodecahedron and Icosahedron are the dual of each other. Same is true with Cube and Octahedron. The dual of a Tetrahedron is another smaller tetrahedron.

The five regular polyhedra were discovered by the ancient Greeks. The Pythagoreans knew of the tetrahedron, the cube, and the dodecahedron; the mathematician Theaetetus added the octahedron and the icosahedron. These shapes are also called the Platonic solids, after the ancient Greek philosopher Plato; Plato, who greatly respected Theaetetus’ work, speculated that these five solids were the shapes of the five fundamental components of the physical universe.

Are we sure there are only five regular polyhedrons? By definition the faces of regular polyhedrons should all be regular polygons. Here are the possibilities:

• Triangles. The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet at a vertex. If there were 6 or more, their angles would add up to 360 degrees or more and hence not admissible. Consider the possibilities:
• 3 triangles meet at each vertex. This gives rise to a Tetrahedron.
• 4 triangles meet at each vertex. This gives rise to an Octahedron.
• 5 triangles meet at each vertex. This gives rise to an Icosahedron
• Squares. Since the interior angle of a square is 90 degrees, at most three squares can meet at a vertex. This is indeed possible and it gives rise to a hexahedron or cube.
• Pentagons. As in the case of cubes, the only possibility is that three pentagons meet at a vertex. This gives rise to a Dodecahedron.
• Hexagons or regular polygons with more than six sides cannot form the faces of a regular polyhedron since their interior angles are at least 120 degrees.

Hence convex polyhedral angles of regular polyhedrons can only be of three types: Trihedral (Tetra-, Hexa- and Dodeca-hedrons), Tetrahedral (Octahedron) and Pentahedral (Icosahedon). Hence we can say there can be only five regular polyhedrons as above.

The following are the patterns that may be cut off from paper or a card board and may be used to make 3-dimensional models of the regular polyhedrons.

My next blog on the subject will concentrate only on Tetrahedrons.