Why do patterns occur

Many mathematicians focus their attention on solving problems that originate in the world of experience. They too search for patterns and relationships, and in the process they use techniques that are similar to those used in doing purely theoretical mathematics.

The difference is largely one of intent. In contrast to theoretical mathematicians, applied mathematicians, in the examples given above, might study the interval pattern of prime numbers to develop a new system for coding numerical information, rather than as an abstract problem.

The results of theoretical and applied mathematics often influence each other. Studies on the mathematical properties of random events, for example, led to knowledge that later made it possible to improve the design of experiments in the social and natural sciences. Conversely, in trying to solve the problem of billing long-distance telephone users fairly, mathematicians made fundamental discoveries about the mathematics of complex networks. Theoretical mathematics, unlike the other sciences, is not constrained by the real world, but in the long run it contributes to a better understanding of that world.

Because of its abstractness, mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences.

The relationship between mathematics and the other fields of basic and applied science is especially strong. This is so for several reasons, including the following:. Using mathematics to express ideas or to solve problems involves at least three phases: 1 representing some aspects of things abstractly, 2 manipulating the abstractions by rules of logic to find new relationships between them, and 3 seeing whether the new relationships say something useful about the original things.

Aspects that they have in common, whether concrete or hypothetical, can be represented by symbols such as numbers, letters, other marks, diagrams, geometrical constructions, or even words. Whole numbers are abstractions that represent the size of sets of things and events or the order of things within a set. And abstractions are made not only from concrete objects or processes; they can also be made from other abstractions, such as kinds of numbers the even numbers, for instance.

Such abstraction enables mathematicians to concentrate on some features of things and relieves them of the need to keep other features continually in mind. As far as mathematics is concerned, it does not matter whether a triangle represents the surface area of a sail or the convergence of two lines of sight on a star; mathematicians can work with either concept in the same way.

From a biological perspective, arranging leaves as far apart as possible in any given space is favoured by natural selection as it maximises access to resources , especially sunlight for photosynthesis. Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature.

There is a relationship between chaos and fractals—the strange attractors in chaotic systems have a fractal dimension. Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects.

Smooth laminar flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the viscosity of the fluid. Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend.

The outside of the loop is left clean and unprotected, so erosion accelerates, further increasing the meandering in a powerful positive feedback loop. Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium — air or water, making it oscillate as they pass by. Wind waves are sea surface waves that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour can be predicted with wind wave models.

As waves in water or wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create dunes , sometimes in extensive dune fields as in the Taklamakan desert. Barchans or crescent dunes are produced by wind acting on desert sand; the two horns of the crescent and the slip face point downwind. Sand blows over the upwind face, which stands at about 15 degrees from the horizontal, and falls on to the slip face, where it accumulates up to the angle of repose of the sand, which is about 35 degrees.

When the slip face exceeds the angle of repose, the sand avalanches, which is a nonlinear behaviour: the addition of many small amounts of sand causes nothing much to happen, but then the addition of a further small amount suddenly causes a large amount to avalanche.

A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one. A foam is a mass of bubbles; foams of different materials occur in nature. For example, a film may remain nearly flat on average by being curved up in one direction say, left to right while being curved downwards in another direction say, front to back.

Structures with minimal surfaces can be used as tents. No better solution was found until when Denis Weaire and Robert Phelan proposed the Weaire—Phelan structure ; the Beijing National Aquatics Center adapted the structure for their outer wall in the Summer Olympics. At the scale of living cells, foam patterns are common; radiolarians , sponge spicules , silicoflagellate exoskeletons and the calcite skeleton of a sea urchin , Cidaris rugosa, all resemble mineral casts of Plateau foam boundaries.

The skeleton of the Radiolarian, Aulonia hexagona , a beautiful marine form drawn by Haeckel , looks as if it is a sphere composed wholly of hexagons, but this is mathematically impossible. The Euler characteristic states that for any convex polyhedron, the number of faces plus the number of vertices corners equals the number of edges plus two. A result of this formula is that any closed polyhedron of hexagons has to include exactly 12 pentagons, like a soccer ball, Buckminster Fuller geodesic dome , or fullerene molecule.

This can be visualised by noting that a mesh of hexagons is flat like a sheet of chicken wire, but each pentagon that is added forces the mesh to bend there are fewer corners, so the mesh is pulled in. Tessellations are patterns formed by repeating tiles all over a flat surface. There are 17 wallpaper groups of tilings. While common in art and design, exactly repeating tilings are less easy to find in living things. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples.

Among animals, bony fish, reptiles or the pangolin, or fruits like the Salak are protected by overlapping scales or osteoderms, these form more-or-less exactly repeating units, though often the scales in fact vary continuously in size. The structures of minerals provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a crystal, defined by crystal structure, crystal system, and point group; for example, there are exactly 14 Bravais lattices for the 7 lattice systems in three-dimensional space.

That appealed to me. I always liked subjects that don't respect those traditional boundaries. But I think also it was the visuals.

The patterns are just so striking, beautiful and remarkable. Then, underpinning that aspect is the question: How does nature without any kind of blueprint or design put together patterns like this?

When we make patterns, it is because we planned it that way, putting the elements into place. In nature, there is no planner, but somehow natural forces conspire to bring about something that looks quite beautiful. Perhaps one of the most familiar but really one of the most remarkable is the pattern of the snowflake. They all have the same theme—this six-fold, hexagonal symmetry and yet there just seems to be infinite variety within these snowflakes.

It is such a simple process that goes into their formation. It is water vapor freezing out of humid air. There's nothing more to it than that but somehow it creates this incredibly intricate, detailed, beautiful pattern.

Another system we find cropping up again and again in different places, both in the living and the nonliving world, is a pattern that we call Turing structures. They are named after Alan Turing, the mathematician who laid the foundation for the theory of computation. He was very interested in how patterns form. This is also an effective form to use in business infographics when communicating the parts of an organization or project progress.

People gather around a resource because of its importance or they are forced into that situation by constraints which are typically physical. The location of a food source, mode of transportation, or job opportunities are all reasons why people come to a certain spot. When there is no more room to expand outwards, then people will grow upwards when it is financially feasible as we do with our skyscrapers and cities. Clusters can be linked. At a very large scale, as in multiple cities on the East Coast of the United States, they can be seen as part of asymmetrical system.

The advent of data-mining and statistical learning techniques has enabled scientists to better understand groups of patterns.

Learning what is causing the clustering to occur can enable people to learn how to take action and what levers to pull to create change in this kind of system. It can also be understood as a detailed pattern that repeats itself. Fractals are useful for understanding natural patterns of growth to include the fault lines, mountain ranges, craters, lightning bolts, snowflakes, heart rates and sounds, blood vessel and capillary networks, crystals, and DNA.

They also have applications in technology and man-made constructs: urban growth, music, architecture, computer graphics, digital imaging, video game design, and price analysis. By becoming aware of fractal patterns, we can see that all things are connected. What happens at a small scale impacts what happens at a large scale. Small changes in a pattern at a micro scale and have an impact at the macro scale. As a template, having an understanding of what might happen in a situation may give you the opportunity to create a different outcome than what you have seen before.

This approach is the foundation of architecture, design, and science that can also be applied to other disciplines and fields. This originally appeared on my blog at scottjancy. You can also find me exploring various contexts and capturing moments, patterns, and perspectives on Instagram.