Spring, 1998Exploration of
INTRODUCTIONIn everyday life, the most common case of symmetry is associated with mirror reflection; a figure is said to be symmetric when you can draw a line through it and thereby divide it into two parts, which are exact mirror images of each other. Higher degrees of symmetry are exhibited by patterns, which allow several lines of symmetry to be drawn, like the following pattern used in Buddhist symbolism (diagram 1). A figure is said to be symmetric if it looks the same after it has been rotated though a certain angle.
D’Arcy Thompson, the author of ON GROWTH AND FORM, mentioned symmetry only in a sense of pure geometry that he found in symmetrical forms such as the hexagonal shape of bee’s cell, which is packed close and tight, minimizing the empty space. Vogel (LIFE’S DEVICES) seemed never to mention symmetry. However, I found symmetry a lot in the book LIFE’S OTHER SECRET by Ian Stewart. First he discusses spiral patterns in the BZ reaction (page 37) and says that mathematicians can explain the rings and spirals using simple but subtle reactiondiffusion equations. He continues that such chemical reactions and a million other patterns speak volumes for the ability of the inorganic world to generate elaborate and beautiful forms. And he mentions a notable idea which attracts me a lot: these abilities, in large part, come from a simple but fundamental principle which governs the deep structure of the physical universe: symmetry. "It was Albert Einstein, above all, who emphasized the importance of symmetry in our universe. Einstein argued that truly fundamental laws of nature must be the same at all times and in all places: that is, the laws must be perfectly symmetrical"
My conceptualization of symmetry began as it was a geometrical form in the program Springtime in Science. But my other perception always interfered with my thought, which I learned last quarter that the other concept of symmetry was phenomenal. As Stewart says, "modern physics has confirmed that at its deepest levels, the universe runs on symmetric lines. Principles of symmetry govern the four forces of nature (gravity, electromagnetism, and the strong and weak nuclear forces that act between fundamental particles); the quantum mechanics of elementary particles; the nature of space, time, matter, and radiation; and the form, origin, and ultimate destiny of the universe. I do not understand all of the above particle physics, the principle of symmetry and so forth, but I enthusiastically can agree with Stewart’s notion. I am interested not in Stewart’s emphasis on mathematics but rather the symmetry of the laws that lie behind the patterns, forms and natural phenomena.
I definitely want to study thoroughly more about the relationship of mathematics and symmetry, and I will be able to do so when I study in the program Matter and Motion next. Until then, I focus on observing patterns and shapes that I can find not only in nature but also human world which, I believe, will help me to comprehend in some aspect and enjoy the true nature of science. I also believe that science is an essential desire of inquiry of human beings.
WHAT IS SYMMETRY?
In addition to the brief account of symmetry as I mentioned in the introduction earlier this paper, let me write some description of symmetry that I found in a dictionary. It says, exact correspondence of form and configuration on opposite sides of a dividing line or plane or about a center or axis, an arrangement with balanced or harmonious proportions. Another dictionary says, an arrangement marked by regularity and balanced proportions.
SYMMETRY IN GEOMETRY AND IN A TOUCH OF PHYSICS
Symmetries in a touch of physics
In particle physics, symmetries are associated with many other operations besides reflections and rotations, and these can take place not only in ordinary space and time but also in abstract mathematical spaces. This abstract meaning is used in particle physics. Physicists developed the abstract meaning of symmetry into a powerful tool, which proved extremely useful in the classification of particles. I just can grasp a very thin first layer of the concept of applied symmetry in particle physics.
The author of THE TAO OF PHISICS, Fritjof Capra, says in his book, "The reason that these symmetry operations are so useful lies in the fact that they are closely related to conservation laws.Whenever a process in the particle world displays certain symmetry, there is a measurable quantity which is conserved: a quantity which remains constant during the process. This quantity provides elements of constancy in the complex dance of subatomic matter and are thus ideal to describe the particle interactions. Some quantities are conserved in all interactions, others only in some of them, so that each process is associated with a set of conserved quantities. Thus the symmetries in the particles’ properties appear as conservation laws in their interactions."
I do not have an intention to write further about particles physics or conservation laws, and to be honest, I do not understand much of the notion above. I just want to introduce a touch of physics to myself and remember that symmetries have a relationship with conservation laws. But also I am very happy to let myself to explore the world of physics. So readers must know that although I write about some physics in this paper, it does not necessary means that I understood them all.
Timothy Ferris, the author of THE WHOLE SHEBANG used Steven Weinberg’s quotation in his book, "The task of the physicist is to see through the appearances down to the underlying, very simple, symmetric reality." Ferris discuss a whole lot of symmetry in the chapter Symmetry and Imperfection. He says, "Wise hunters stalk the ultimate theory by searching for signs of symmetry. The laws of nature are all expressions of symmetry, and all physics is in some sense a search for symmetry."
I think we’d better broaden our sense by exploring symmetry in various ways. Here is the other account where Ferris defines symmetry as representing a quantity that remains unchanged through a transformation. This is quite straightforward, so symmetry represents the law of conservation of energy which describes a energy (quantity) that is conserved (remains unchanged). Thus the fundamental conception of symmetry is the conservation law.
Stewart, the mathematician, made a further step with implementing the principle of symmetry to approach and study growth and forms. He says if symmetry is one of the regimes underlying the universe, then everything could be explained by math. But he also admits that there are many circumstances in which qualitative information is what really counts, and quantitative measurements are a rather poor route toward finding that information. And he adds more by saying, qualitative means "features that are conceptually deeper than mere numbers."
I think it would have been better if our class took more time to discuss this fundamental concept and principle of symmetry before we began to explore shapes and patterns that Stewart and Thompson talk about.
The principle of symmetry is used to explain such a tiny world of particle physics, and it expanded to explain not only natural phenomena but also organisms’ growth and forms in books. So how I can apply these ideas to observe and explore the world? We can find a lot of shapes and pattern that represent symmetry in nature, but they are not perfect symmetry rather they are broken. (Reference, Plate 1)
 Coffee break note: Let me introduce a noteworthy quote of Einstein from Ferris’s book: "Einstein remarked, the scientist looks to an outside observer like an unscrupulous opportunist." I like it!
The meaning of symmetry I found in a dictionary was "an arrangement with balanced or harmonious proportions." If I take this meaning of symmetry and apply to the natural environment, it will be very sound. What breaks an arrangement with balanced or harmonious proportions? Steven Vogel discusses a gradient such as temperature, pressure, concentration, velocity. He says, "the relationship between heat loss and temperature gradient is an enormously important constraint in the design of animals that maintain body temperatures substantially different from those of their surroundings." Because, according to the conservation law, wherever there is a temperature gradient the heat must flow from the higher region to the lower region. The both entities must behave or act toward equilibrium. So I would say that the animal which must keep their body temperature in certain degree all the time have to fight against thermodynamics. I wonder why those animals have to keep their body temperature, which include us human beings. The body temperature (heat) follows from our metabolism, so it seems not to be a cause for our organism and it is a result or effect. If we always have to keep our body temperature in not less or more but a certain degree, then what is the reason for that? Why do we have to maintain our body temperature in a certain degree? Why do we have to fight against thermodynamics? This is a broken symmetry. We had to develop a mechanism to maintain our body temperature against thermodynamics. The reptiles may be more efficient creatures in terms of energy conservation. Other mammals and we human beings need more energy to maintain not only our life but also the body temperature. This must be a broken symmetry in a sense.
I find many symmetrical shapes and patterns in nature, but somehow they are not so perfect (Plate 1). I do not know yet how to explain, though I feel that there must be some gradients like pressure (wind), temperature (sunshine), moisture (rain), and chemically or physically, something cause a distortion or malformation.
Symmetry in geometry
Spatial (geometric) symmetries:

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