Spiral

Spirals are classified by the mathematical relationship between the length r of the radius vector, and the vector angle q, which is made with the positive x axis. Some of the most common include the spiral of Archimedes, the logarithmic spiral, parabolic spiral, and the hyperbolic spiral.

The simplest of all spirals was discovered by the ancient Greek mathematician Archimedes of Syracuse (287-212 B.C.). The spiral of Archimedes conforms to the equation r = a θ, where r and θ represent the polar coordinates of the point plotted as the length of the radius a, uniformly changes. In this case, r is proportional to θ.

The logarithmic, or equiangular spiral was first suggested by Rene Descartes (1596-1650) in 1638. Another mathematician, Jakob Bernoulli (1654-1705), who made important contributions to the subject of probability, is also credited with describing significant aspects of this spiral. A logarithmic spiral is defined by the equation r = e^{a θ}, where e is the natural logarithmic constant, r and θ represent the polar coordinates, and a is the length of the changing radius. These spirals are similar to a circle because they cross their radii at a constant angle. However, unlike a circle, the angle at which its points cross its radii is not a right angle. Also, these spirals are different from a circle in that the length of the radii increases, while in a circle, the length of the radius is constant. Examples of the logarithmic spiral are found throughout nature. The shell of a Nautilus and the seed patterns of sunflower seeds are both in the shape of a logarithmic spiral.

A parabolic spiral can be represented by the mathematical equation r² = a^{2 θ}. This spiral discovered by Bonaventura Cavalieri (1598-1647) creates a curve commonly known as a parabola. Another spiral, the hyperbolic spiral, conforms to the equation r = a/ θ.

Another type of curve similar to a spiral is a helix. A helix is like a spiral in that it is a curve made by rotating around a point at an ever-increasing distance. However, unlike the two dimensional plane curves of a spiral, a helix is a three dimensional space curve which lies on the surface of a cylinder. Its points are such that it makes a constant angle with the cross sections of the cylinder. An example of this curve is the threads of a bolt.