The role of mathematics in understanding nature
Mathematics is a discipline of thought. It helps to develop our way of thinking and is an exercise in improving our intelligence. Mathematics can be considered as another kind of language, a language very different from that of a spoken language. When it is hard to convey our thoughts in terms of words, or our words become insufficient to express our thoughts, mathematics may be used as an alternative. On some occasions, expressing ideas via mathematics might be more concise, much clearer and more understandable. Although mathematics is considered to be a separate branch of science, in fact it is related to all branches of science. Nowadays, even in biological and social sciences, extensive studies are being conducted using mathematics.
Engineering was one of the earliest application fields of mathematics. It has strong links with mathematics as well as physics. Many engineering problems can be considered as an application of mathematics and hence applied mathematicians and engineers share common research areas. Engineers try to improve the quality of life by designing new products and in the design process, geometry and mathematics play a vital role.
Since the first day of existence on the world, mankind has tried to understand and formulate the surroundings and events that take place around them. They have investigated the world and the cosmos and accumulated knowledge. Each question that was answered yielded more questions to be answered and the more the knowledge that was acquired the better the extent of our ignorance about the universe was understood.
The universe has been established in a very complex orderly manner. The magnificent order observed cannot be expressed well in words, but may also be expressed using mathematics. A person who develops their knowledge of mathematics can understand more about this supreme order. For example, the universal gravitational law, which describes the movement of planets, can best be understood through mathematical equations, while the solutions of the equations yield the well-known elliptic paths. The concept of infinity that is attributed to the Creator can be realized through the concept of infinity that is frequently used in mathematics. So mathematics is an essential tool in developing our understanding of the nature and universe. It is essential also in applying the principles of physical laws in nature to improve our quality of life. The design of an airplane requires extensive mathematical calculations and applications of physical laws.
Finally, it should be noted that mathematics also has its limits, as it is something that has been developed by human beings and may not be sufficient to express the total order and all physical laws. Chaotic motion, a very complex order, was developed recently to understand some phenomena that do not obey the rules of deterministic motion. A daily example of such motion would be atmospheric motion. With even super computers and satellite technology, the path of the hurricane Katrina could not be predicted precisely due to its largely chaotic behavior and these errors cost thousands of lives.
Basic engineering principles and their applications in nature
First, let’s briefly describe some of the fundamental engineering courses and their aims. Dynamics is the science of motion. It models motion, describing the relation among displacement, velocity and acceleration. The specific type of motion and its causes, such as forces, movements, impulses etc. are examined. Dynamics deal with solid bodies while fluid mechanics basically deals with liquids and gases.. In the context of fluid mechanics the rest states of fluids as well as their motions are investigated. The strength of materials deals basically with the design of structures and mechanical parts to loading conditions. Under a given loading condition, what would be the best design for withstanding the loads while using the minimum amount of material? Materials science deals basically with the mechanical properties of various materials and the causes (microstructure etc.) of those properties. Proper selection of the materials to perform the required task is another important issue.
Living organisms can also be considered as some sort of design, but of course they are different from man-made designs. Living organisms, whether they are plants, animals or human beings, are designed to perform a specific predetermined task. The organism has to move, find food, safely operate and resist the forces that act on it throughout its life, and it must reproduce. Therefore, organisms have to be designed (or more precisely created) according to the principles of engineering. The development of technology drew attention to creatures and the underlying engineering principles in their structures. Extensive research on living creatures revealed a clear conclusion: Designs applied in nature are much more sophisticated then the ones humans come up with.
Bernoulli’s principle is a fundamental principle in fluid mechanics. Basically, the principle states that when the velocity of fluid increases the pressure drops and visa versa. The lift force generated in the wing of a plane is explained with this principle. Air separates in front of the wing and reattaches at the back. When the upper surface of the wing is slightly curved and the bottom flatter, the air particles in the upper part travel a further distance at a higher velocity and meet the particles traveling under the wing at the back. The relatively higher velocity on top causes a pressure difference in the lift direction and this lift force balances the weight of the plane. Many applications of Bernoulli’s principle can be found in living organisms. A fish moving in water is a good example. In particular, fish that swim at great speeds, like the tuna, have distinctive body shapes: The mouth of the fish is at the front where the fluid comes to rest and the pressure is very high, making the fluid intake of oxygen easier. The heart is located at the minimum pressure point to make it easier for it to beat. The eyes are located on a precise saddle point, a place which is not affected by velocity changes. Since the pressure is constant for all ranges of velocities, vision is not distorted by movement. Another example is the human body. When one breathes in the fluid velocity in the nose increases and pressure drops. The outer pressure is higher than the inner pressure and the walls tend to collapse. If bones were found at the tip of the nose, they might easily break when excessive force was present. We need some other material to sustain the shape yet be elastic enough not to break down. Cartilage is the best choice in this case, as it has both strength and elasticity. Our ears are also made from the same material. If bones were used instead of cartilage in our ears, resting our head on one side would be painful or even cause damage to the ears.
Insect flight is another important issue and has attracted considerable research recently. Fluid scientists now realize that insect flight is much more developed than our flight techniques. Turbulence is the main issue. In turbulent flow, the fluids move in erratic paths colliding with each other, forming eddies and irregularities. This is a dangerous state, especially for planes, and increases the friction forces between fluid and structure. Therefore the maintenance of a regular flow (laminar flow) over the wings is advantageous. However, all insects benefit from turbulence and some portion of their lift is gained from eddies that are formed over their wings. Mechanical insect robots are built to understand insect flight. Insects have movable elastic wings, but aircraft only have immovable rigid wings. Movable elastic wings would certainly improve the flight of planes and their maneuverability, but extensive research has to be done before these designs can be safely implemented.
The bumps on the fins and heads of some whales are not accidents of nature. They were given to them by the Creator for some very special purposes. They decrease the friction (drag) force by 10% and increase the lift by 5%.1 When some have the effect of decreasing drag, they can also decrease lift and visa versa. This effect of both decreasing drag and increasing lift, which can be observed in whales, is very uncommon in fluid mechanics.
Streamlining is a very important issue for an object that moves in a fluid. Fluid particles move around an object that follows a path. Roughly speaking these paths are streamlines (in a steady motion) and it is a general rule that abrupt distortion of these streamlines should be avoided. Smooth changes in the streamline help to reduce the friction force between the object and fluid. All organisms, particularly those that move at greater speeds, have been created in accordance to streamlining principles. In these you can find many species of birds and fish, such as dolphins, sharks, whales etc. The friction reduction caused by the shape of a dolphin is still a controversial issue in science and the underlying mechanism has not yet been well understood.
An example of the strength of natural materials can now be given. Our bones are optimum structures, combining strength with lightness. In modern buildings, 60-70% of the buildings consist of the skeletons, which carry the loads and moments. In our body, our skeleton is only 1/7th of our body weight. Bones have inspired a new generation of lightweight structures. For instance, a bridge inspired by the backbone was recently designed.2 When a longitudinal cross-section is taken from a femur, some curved lines are observed. Recent numerical simulations revealed that these lines are to be found in one exact place and their configuration increases the strength of the bone. Our backbone and the muscles around it withstand very high loads, equivalent to 7,000 Newtons or approximately 700 kilograms of weight.3 The bones of mammals are hollow inside to increase strength. The inner to outer ratio of the radii is at the optimum range, between 0.4 and 0.7.4
Hardness is another important issue in some applications. Seashells are the leaders in this issue. Their microstructures are being investigated under electron microscopes to invent new materials with extreme hardness properties. Micro-cracks inside a material grow over time, finally leading to failure. This is a major problem in turbine blades and this phenomenon is responsible for some plane crashes. In seashells, micro-crack inhibiting mechanisms are inserted to prevent crack growth. Inspired by spider silk and the microstructure of bird feathers, a new generation of bullet-proof waistcoats has been developed.
Owls are very silent flyers; they need to be so in order to approach rodents as rodent ears are highly sensitive to sound. Recent investigations have shown that the special geometry of their wings results in this silent flight. Their feathers are placed to form fringes on their wings. The technology might be mimicked to reduce the noise generated in planes.5
A recent engineering discipline is robotics. There are industrial robots, which are designed to perform some very special tasks. But there are also robots inspired by living organisms. A new robot is designed to mimic caterpillar motion so that it can be stable enough in a hazardous region, pass through small gaps and detect humans who are alive under debris.6 By mimicking the motion and body of a scorpion, a military robot was designed with a camera and sensors to safely operate in a battle region.7 Of course there are human-like robots that are designed to mimic our motion and activities. The developments in robotics teach us a very important lesson: All animals are much more sophisticated in their locomotion, actions, and behavior and it is extremely hard to mimic those. A robot that can move freely like a cat and climb a tree yet maintain its balance has not yet been produced. Our robots are very slow in motion, and their stability in movement is an important technological issue that requires extensive sensors and control designs.
Newly developing engineering branches
As mentioned above, one of the newly developing branches of engineering is robotics. Day by day, better robots are being designed and those designs try to better mimic animals and humans. Some 50 years ago, a human walking might be considered a simple issue, but now we know that comfort in walking and excellent balance in such movement are very complex issues.3 Each new design in robotics adds to our knowledge of understanding animal locomotion and behavior and how miraculous their designs are. Some people think that robots may take control of the world in the future. Yet this is simply not possible: If humans are to design them, there is no way that such machines can be superior to the designers.
Other promising new fields are the MEMS (Micro-electrical machinery systems) and nano-technology. These are design attempts on extremely small scales which actually mimic some micro biological systems and micro-physics. A vertebrate consists of an enormous number of cells, while the chemical and physical events that take place inside the cells and their establishment as a system are crucial parts of staying alive. It is extremely hard to design at the micro and nano scale and it is likely that research in this field will reveal more about understanding the art of God.
1 M. Le Page, “Speed Bumps Give Humpbacks a Surprise Boost,” New Scientist, 13 January 2001, p. 22.
2 I. Sample, “A Bridge with Backbone,” New Scientist, 16 September 2000, p. 7.
3 R. Mc Neill Alexander, The Human Machine, Colombia University Press, 1992.
4 R. Mc Neill Alexander, Optima for Animals, Princeton University Press, 1996.
5 C. Seife, “Deadly Hush,” New Scientist, 6 march 1999, p. 10.
6 C. Zandonella, “Wriggle into Rubble,” New Scientist, 10 November 2001, p. 22.
7 D. Graham-Rowe, “Walk Like a Scorpion,” New Scientist, 21 April 2001, p. 18.