Cauchy and the Beauty of Abstract Math

3/20/2011 02:27:00 AM / Posted by Iron Lung /

It seems at times human arrogance can stifle our own evolution and mental growth. There are instances when we must step back, remove our assumptions and accept the flagrant facts.  Our own beliefs can hinder our success if we are not willing to let go of them to seek truth. Some of history’s greatest minds had to over come such obstacles, struggling to balance out new knowledge with old habits. Einstein struggled with the contradiction of his personal beliefs and his theory of relativity, which also served to promote the idea of a dynamic universe. Cauchy had to endure some of the same as his political beliefs were more often then not, highly unpopular. Through Cauchy’s unique upbringing, he was able to have a mindset for logic and engineering, allowing him to propose the stress tensor and theory of elasticity, which have helped to shape our modern world.
Born in 1789 to affluent parents, Agustin-Louis Cauchy and his family relocated to Arcueli just one month after the start of the French Revolution. His schooling was domineering at best and undoubtedly shaped his formative thinking and logic. After placing second out of 293 students for the Ecole Polytechnique entrance exams, Cauchy had to put his own outlooks aside in order to excel, as the school function under military discipline. It was at this point when Cauchy turned his attention to civil engineering, and the math world would never be the same.
After a brief engineering stint and two published manuscripts, Cauchy began to shift his attention to more abstract forms of mathematics. Among the most noticeable milestones of his early work, is his theory of series, which was instrumental in developing the idea of convergence and other principles in limits. He soon moved onto complex functions and rigorous notations.
As humans, we sometimes forget to think about our own frailty. In fact we tend to think of ourselves as rigid bodies, a thought we tend to project towards other mediums. Cauchy helped usher in a new way of thinking in mechanics. He proposed that when a body meets an external force, it deforms, and starts a process to attempt to regain its previous form. This would mean that bodies are not rigid, but instead are malleable to an extent. This was just the tip of the iceberg for Cauchy, but it was the beginning of the theory of elasticity. A theory that spawned in the midst of continuum mechanics, it began by ignoring the fact that matter is composed of atoms, and indeed not continuous, a continuum concept assumes that the substance of the body is distributed uniformly throughout, and completely fills the space it occupies. Continuum mechanics studies objects with plotted properties in all points of space. For example, the movement of a cricket ball would be described as a single velocity vector at a single point in space. However, its velocity in every point in space describes the motion of a fluid. The force experienced at every point in space calculates stress.
Before diving deeper into the theory, we begin to build an immediate relation between stress and force. The irony is that if it were not for math and this formula, both of these elements would remain completely intangible. Essentially, stress is defined within this theory as the measure of internal forces as a result of body forces reacting to surface forces. If we had a cube and laid a bowling ball on top of it, the bowling ball would be applying surface force. The initial attempt to balance out this surface force would impact the cube directly hence body force. If we were to hypothetically divide the cube along a plane through the cube, then a new plane of internal forces becomes visible. Here we can see the actions brought forth to maintain equilibrium, better known as stress. 
As a direct result of surface and body forces colliding, the body may warp internally. The amount of, and severity of warping is defined as strain. Hooke’s Law is an imposing example of a linear correlation between stress and strain. It was first published as a Latin anagram in 1678 that read “Ut tensio, sic vis”. Translated, this means “As the extension, so the force”.  This is where we get the direct link between the two; the extension or distortion produced, is directly proportional to the load or applied force. Objects that are of a high elasticity often obey Hooke’s Law, but there are of course instances where it may no longer apply. For instance, Hooke's law is only valid for a portion of the elastic range in some materials such as aluminum. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible. This helps us everyday whether we realize it or not.
A proportional limit, or yield strength, is a common factor in modern engineering. More likely then not, steel beams are supporting the very building you are on. The placement of those beams along with the knowledge of where and how to place them in accordance to weight and support is what is keeping you from crushing your downstairs neighbor. A yield, in engineering terms, is the amount of stress that can be applied before the object surpasses elasticity and moves into the realm of plasticity. The difference being that with plasticity, the deformation caused due to stress is not reversible.
It is not farfetched to say the without Cauchy, some of the world’s most beautifully engineered buildings would not exist. The practice of designing buildings around these concepts has become known as structural engineering, which started with the Egyptians and the pyramids, and continues to this day. The Gateway Arch in St. Louis, the Sydney Opera House in Australia, the Millennium Dome in the UK, the Burji Khalifa in Dubai; all of these beautiful creations would not be possible without an inquisitive mind and structural engineering, a knowledge of stress loads on materials.
Without fully realizing it, Cauchy paved way for modern architecture, and also inspired me to apply his principles to another structure, the human being. Bringing us back in to the human realm, we can now apply these principles to life. There is an outside force, or a surface force, which is something that we have little or no control over, that is affecting our bodies. Our bodies which in some cases may also be considered as the mind, then reacts by putting out equal force to attain equilibrium. The strain and result of the two forces is what we would consider stress. Therefore, the difference between plasticity and elasticity is exactly how much stress we can take before we break. Bringing in the human element only makes the equations that much more complicated.
In physics, other factors aside from force are variables in defining stress. Thus, when speaking about stress in the biological sense, we must also take into account the timing of the occurrence in relation to the subject’s current life events and surroundings. Once all the other variables are in place, we can then define the amount of the force applied with regard to the individual. The elasticity of the subject will be due to the resiliency and toughness of character, as this will determine how quickly the person is able to return to a balanced state. Once we can establish this, we can now compare the two on a case-by-case basis and further study the effects of stress and forms on manipulating it.
In the grand scheme of things, Cauchy could not have seen how important his work would be to our modern world. His meticulous formulas paid off in theories that helped shape our world. Despite his upbringing and the challenges he faced in a politically changing France, he managed to produce vast amounts of publications. There are instances when we must step back, remove our assumptions and accept the flagrant facts.  Our own beliefs can hinder our success if we are not willing to let go of them to seek truth. Some of history’s greatest minds had to over come such obstacles, struggling to balance out new knowledge with old habits. Luckily for us, Cauchy was able to do so. His stress tensor helped the theory of elasticity, which allowed us to improve structural engineering. The applications seem endless. By stripping away my own ideals and beliefs as Cauchy did, through rigorous testing until it is infallible, I intend to apply the theory of elasticity to the human version of stress, in hopes to better our own structural composition. 


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