A fundamental part of the natural world and art itself, a pattern is a unit (or group of units) that is recurring in a collection or a whole piece of art. Traditionally artists have used patterns as part of the decorative process, either as a method of composition, decoration or simply as part of the overall artistic theme. However, in recent years many people have come to see patterns not merely as elements to be visualized, but as works of art themselves. Some people have even taken patterns and turned them into sculptures, other people have used patterns in photographic canvas paintings, and still, others have created entirely fictional works of patterns that are themselves very beautiful and worth studying.
In the area of applied mathematics, patterns are used extensively in the study of discrete mathematics, such as arithmetic and geometry. Discrete math deals largely with dealing with sets of objects which are finite in nature. In fact, all the different types of real numbers, from the number of pebbles on a beach to the largest number calculated by a calculator, are the result of discrete math. The study of real numbers involves discrete patterns and the repetition of certain numbers over again. It is this aspect of real math that lends it to the study of patterns.
The subject of patterns can be studied in two main areas of mathematics. The first area of mathematics which deals primarily with patterns is pathfinding. This particular area of an object is searched for by identifying the shortest path between two points on that object. Pathfinding can be applied to a number of different areas, including geometry, calculus, optics, and sonar imaging, and mathematics itself.
The second area of mathematics which greatly benefits from patterns is finance. Financial mathematics deals largely with the calculation of various mathematical expressions and graphing symbols such as lines, bars, and shapes. In order to calculate these various formulas, a series of necessary repetitions must occur within certain steps. The most common of these steps are known as the simple patterns and in these patterns, the only repeating element is the series of numbers representing the data that will be used to represent that expression or symbol. A prime example of a repeating pattern in financial mathematics would be the application of the binomial tree.
Another way in which patterns are used in mathematics and finance is through the application of statistical methods. One example of a statistical method would be the binomial tree. Binomial trees are based on the mathematical theory called binomial probability. This theory states that given a specific set of real numbers, such as numbers that are involved in determining the odds of winning the lottery or making the grade in school that is somehow related to actual events that have occurred within the real world, there exists a certain probability that those numbers actually will produce the results that the creator of the pattern had in mind.
One can easily see the similarities between the patterns of symmetry and those of fractals. Symmetry in nature occurs in things that have both equal sides or equal heights and different degrees of slope on those same sides or same levels; while the patterns of Fractals are random or have no repeat patterns whatsoever. When we look at symmetry and the patterns of symmetries in nature we find them repeated in the creations that we observe every day. For example, the Fibonacci spiral, which is one of the most well-known and well-studied symmetrical patterns in the world of mathematics and finance, began life as the result of a mathematical equation that took the form of a spiral passing through the infinitely divisible naughtiness of space.