**Ugly Symmetry, Beautiful Diversity**

**Shu-Kun Lin**

MDPI, Molecular Diversity Preservation International, Matthaeusstrasse,
Basel CH-4057 Switzerland
(address updated on 27 November 2005)

E-mail: lin@mdpi.org, http://www.mdpi.org/lin/

June 12, 1999

My theory of structural stability and process spontaneity is a new and a quantitative relation of the five concepts: higher symmetry, higher similarity, higher entropy, less information and less diversity; and they are all related to higher stability.

In many books on symmetry, symmetry is regarded as a measure of beauty. Why symmetry is beautiful? My answer is that symmetry contributed to beauty only because symmetry is related to higher stability. However, symmetry is in principle ugly because it is related to information loss and diversity loss.

In many modern arts museum, you can always find a couple of paintings which have nothing on them but same pure white color everywhere. It has obviously the highest symmetry. Symmetry is beautiful? OK, that's it!

Why not? The emperor is topless and bottomless, even though many great intellectuals can cheerfully enjoy his most beautiful clothes. The most symmetric paintings are the emperor's new clothes.

Symmetry is in principle and generally ugly because it is associated with the information loss or entropy increase.

Misunderstanding of the entropy concept (information loss) associated with the second law of thermodynamics is well known, even among some scientists. However, the concept of symmetry or similarity is very easy to understand. Mathematics regarding symmetry is group theory which has been well established. This new relation provides new and broader mathematical approaches to study entropy and related topic of structural stability and process spontaneity. The second law of thermodynamic can be much more clearly expressed by symmetry and similarity concepts. Meanwhile, symmetry studies have a new approach.

Higher symmetry is directly associated with higher similarity. Semantically, symmetry is "sameness measure" in Greek. The "same" is the "most similar". We can use similarity to define the sameness measure. If a subject is the same on many directions, scientists say there is rotational symmetry. If you always find the same molecules and same orientations when you go certain distances in a crystal, you have translational symmetry.

According to Gibbs, the so-called Gibbs paradox (see the website http://www.mdpi.org/lin/entropy/gibbs-paradox.htm), all the textbooks give the relation of higher similarity-lower entropy. The new and the correct relation is the opposite of this conventional relation. "Birds of a feather flock together." As a result, different things separate. Scientists say this phenomenon is phase separation when the same (or most similar) substances go together. Many phenomena of spontaneous process like this can be explained only if the new relation of "higher similarity (higher symmetry)-higher entropy" is recognized. Traditionally these are regarded as discrete or "yes-or -no" properties. According to Gibbs, the properties are either the same (indistinguishability) or not the same. According to classical textbooks on symmetry, a system under consideration is either symmetric or not symmetric. In my theory, symmetry, similarity and entropy are all continuous measures.

A gas of high entropy is more symmetric because it is isotropic (and homogeneous), compared with a solid (such as a crystal). This might be difficult to understand. It would be easier to understand if you compare a dynamic system with another dynamic system (e.g., homogeneous and nonhomogeneous fluids) or a static system with another static system (e.g., a crystal and a noncrystal solid). Therefore, I categorize symmetry, similarity, entropy and information loss concepts into two kinds: static and dynamic, to explain all kinds of natural phenomena.

Rapid dynamic motion leads to information loss. You cannot find the beautiful face of a girl if she rotate rapidly around a straight line to become a cylinder, or rotate around a point to become a sphere. Suppose a tank of gas is divided into two parts. One is full while the other part is in vacuum. Now the door connecting these two parts is opened, the situation (or state, in physical chemistry definition) is not symmetric (the two parts are not the same) and has low entropy. Dynamic motion leads to high symmetry (the two parts are the same) and higher entropy. This is an example of dynamic symmetry, similarity and entropy. At higher temperature, any system would like to have the highest possible dynamic symmetry, similarity and entropy, so that the concentration and the other properties are the same (most similar) on all directions and all accessible parts. If everyone in a country claims he or she is the president, has the equal power and moves around in a same way, this would be very high symmetry. The dynamic situation would be chaotic. Crystal might be more beautiful than fluids because solid structure obviously has less dynamic symmetry. In a crystal, isotropicity does not exist: the properties along different directions are diferent. That is why we like chocolate or ice cream of certain figure and may not like the melt chocolate or running ice cream soup.

Symmetric static structure is also stable but not necessarily beautiful. A beautiful model, whether she stands or sites, never poses herself in a symmetric style before the crowd. A fit and beautiful body should differ from a most symmetric, spherical shape of over weighted body. A Chinese empress would sit in a symmetry style because stability was more interesting to her. The pyramid in Egypt is stable and old. But it is by no means the beautiful construction. Static symmetry is ugly.

Static symmetry is also related to information loss. Information will be lost if the ink is extremely faint or the same as the background. Crystal is more stable than noncrystal solid because the former has high static symmetry, the perfect symmetry of a paper without any information. Children understand it: On the walls of children's classrooms and bedrooms a visitor may find a lot of paintings, drawings and even scrawls done by the kids. If you do not put some colorful drawings there, children will create some there. The innocent children would like to destroy the ugly and boring symmetry surrounding them. If your kids destroy symmetry on the walls, they are normal and doing well.

Why do we appreciate diversities, such as biodiversity, cultural diversity, etc., and why is diversity beautiful? My theory gives a clear answer. If a collection of 10000 stamps have extremely high similarity, which means you have them all of the same figure and size, even though this kind of stamp is a very interesting one. Scientists call it the permutational symmetry: If you exchange the positions of any two stamps, your album of stamps is still the same. You cannot recognize the replacement. This collection is not beautiful and not precious because it lacks diversity. A library of 10000 copies of the same book is not nice either. Based on this idea, as a chemist, I initiated MDPI, an international project of collecting diverse chemical samples to build up molecular diversity (See the website http://www.mdpi.org). I will not collect 10000 samples of the same compound, even though it might be very interesting, e.g., the famous C60 molecule. C60 is beautiful not because of its symmetry, but because of its distinct structure and property compared to many other molecules and because this contributed to the diversity of molecules we studied. Many derivatives of C60 have been synthesized by organic chemists. These derivatives have symmetry reduced, they are more difficult to prepare and should be more interesting. Very few drugs or bioactive compounds are very symmetric. For similar reason, it is true that we may feel a symmetric object beautiful if we have many other objects of less symmetry in our collection or in our experience. This symmetric object contributes the diversity.

Diversity is beautiful. Symmetry is not. Coffee with sugar and cream is an interesting drink because of its diversity in taste (sweet and bitter) and color (white and dark). Diversity in a mixture makes the so-called high throughput screening of bioactivity testing possible. USA is considered as a nice country because of its tolerance to all kinds of diversity (racial diversity, cultural diversity, religious diversity, etc.). Without such appreciation of diversity, this country would become much less colorful and less beautiful. If everyone behaves the same as you, looks the same as you, and there is a lot of symmetry, the world would be truly ugly. Democracy is a sort of social diversity, I believe.

I would like to comment that Dr. Ilya Prigogine's dissipative structure theory of symmetry and entropy is erroneous, misleading, and possibly useless. His theory is based on the tacit and wrong conception that symmetry is order (and reduction of entropy). It has been presented by most of my respected teachers as unbelievably important, beautiful and useful. Therefore, 20 years ago as a young student of chemistry, I wanted to understand Prigogine's theory and I studied all kinds of related mathematics and physics, including several graduate courses in physics, to prepare myself. Now, after more than 20 years, first 10 years of theoretical investigation, then, several years of diverse experimental practice in chemistry laboratories, I have a clear opinion regarding his symmetry and entropy theory. Its main problem is that it does not conform with the second law of thermodynamics. The famous second law says "chaos out of order"; Prigogine says "order out of chaos". Therefore, it is not a surprise that an honest chemist (among any other educated chemists, physicists, biologists, etc.) will tell you that he has never found an application of his theory in chemistry (or in biology, physics, engineering, ...). For more information and on-going discussions, please visit http://www.mdpi.org/entropy/htm/e1010001.htm website. Prigogine's theory and the public opinion (including that of many otherwise respected scientists) regarding his theory for so many years might be a perfect story of the emperor's new cloths. Think about it.

The second law of thermodynamics is correct. Symmetry is related to
information loss which is entropy. Any spontaneous process will lead to
an equilibrium state which certainly has the highest values of static
and
dynamic symmetries. Following the Clausius statement (in German) "*Die
Energie der Welt ist constant. Die Entropie der Welt strebt einem
Maximum
zu*" (The energy of the universe is constant. The entropy of the
universe
tends toward a maximum), http://webserver.lemoyne.edu/faculty/giunta/clausius.html),
we may claim that the universe evolve towards a maximum symmetry.

Biographic summary for Shu-Kun Lin: http://www.mdpi.org/lin/lin-bi.htm

Recent publications are listed at: http://www.mdpi.org/lin/lin-rpu.htm

Retuen to: http://www.mdpi.org/lin/uglysym1.htm

Updated on 11 September 1999