Spiral

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Spiral

A spiral is a curve formed by a point revolving around a fixed axis at an ever-increasing distance. It can be defined by a mathematical function which relates the distance of a point from its origin to the angle at which it is rotated. Some common spirals include the spiral of Archimedes and the hyperbolic spiral. Another type of spiral, called a logarithmic spiral, is found in many instances in nature.

Characteristics of a spiral

A spiral is a function which relates the distance of a point from the origin to its angle with the positive

KEY TERMS

Logarithmic spiral —A type of curve defined by the relationship r = e^a q. It is a shape commonly found in nature.

Origin —The beginning point of a spiral. Also known as the nucleus.

Spiral of Archimedes —A type of curve defined by the relationship r = aq. This was the first spiral discovered.

Tail —The part of a spiral that winds away from the origin.

x axis. The equation for a spiral is typically given in terms of its polar coordinates. The polar coordinate system is another way in which points on a graph can be located. In the rectangular coordinate system, each point is defined by its x and y distance from the origin. For example, the point (4,3) would be located 4 units over on the x axis, and 3 units up on the y axis. Unlike the rectangular coordinate system, the polar coordinate system uses the distance and angle from the origin of a point to define its location. The common notation for this system is (r,θ)where r represents the length of a ray drawn from the origin to the point, and θ represents the angle which this ray makes with the x axis. This ray is often known as a vector.

Like all other geometric shapes, a spiral has certain characteristics which help define it. The center, or starting point, of a spiral is known as its origin or nucleus. The line winding away from the nucleus is called the tail. Most spirals are also infinite, that is they do not have a finite ending point.

Types of spirals

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 BC). The spiral of Archimedes conforms to the equation r = aθ, 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 = a²^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.

See also Logarithms.