To page 1                                                                                                Page 2

---

This is a description of the block:

This block is a square with four connection points shown as red and blue dots, according to the picture. The block has its top and bottom, as the fill shows.

  
 Here is another block:

It differs from the first in the location of the dots. Here is yet another  block:

       1

The blocks and the rules of connection define a space for all possible combinations and  form a kind of a creative system for producing combinations. We will call the system of the above three blocks and the rules of connection SYSTEM 1. It has a certain complexity that we don't know how to measure. We can compare, however, two systems that do not differ much, i.e., one is produced from the other in a small step.

This is how it could be done.

Let us form a new system by adding four new blocks, without changing the rules. 

    2

Although we add more blocks than there were initially, this is a small step. If it seems big, we can add them one by one. If we changed the blocks and rules at the same time, it would be a bigger step. But we can always divide a change into minimal steps (an assumption). The newblocks (not the system yet) are more complex than the previous three because they have more kinds of dots: three colors instead of two. Therefore, the new system, let us call it SYSTEM 2, is more complex. SYSTEM 2 with seven blocks is more complex than the system with either three or four blocks because it uses more different types of blocks, some of them more complex.

         This is just one example of what  we can make of the seven blocks.
 

        Naturally, the number of  combinations in SYSTEM 2 is larger than in SYSTEM 1.

        Now let us try something quite new: a mutation of the shape. We add two identical triangular blocks:

 ......................                                       ...        3
 

        There is only one way we can connect such triangle with only one of  the seven:

This SYSTEM 3 of nine blocks is more complex than SYSTEM 2 because it has more types of blocks: square, as well as triangular.

 SYSTEM 4 comes next, in which the rule of matching colors is relaxed and the following combination, for example, is possible (it does not use all nine blocks):

    4

SYSTEM 4 is less complex than SYSTEM 3  because it has less rules, even though it generates more combinations.  If dots of all colors are equally  connectable, then they can be reduced to just one color, which makes the blocks much simpler.

We can have more combinations by allowing rotation of the blocks in plane (SYSTEM 5). We will require, however, the sides of the blocks be approximately parallel.

In fact, with the relaxed rules of connection, we can eliminate not only the colors, but also the fill that distinguishes between the top and bottom of the squares, and even the dots, so that the above combinations look very trivial:

 A completely chaotic system, without any rules and with simplest blocks, is very simple. The billiard balls form such a system on the pool table.

 One can try designing various Lego-like systems and studying their complexity. The Microsoft Draw, which is a Microsoft Word function, is very convenient for this. It allows for a rather high complexity of combinations. I drew a picture.... (120K) made of two kinds of blocks: closed lines and fills. At this point I let the reader guess what the rules for the combination were.

To summarize, instead of trying to evaluate complexity of a single object, we do it for the system that generates it. All the objects generated by the system are of the same complexity, however they look.  It seems to contradict our intuitive concept of complexity because large combinations look more complex than a single block or a couple of blocks. We should not mix up size and  complexity, however. Still, we can take the number of different blocks in combinations within the same system as a measure of a partial complexity within the same system.

I don't believe it is possible to find a universal numerical measure of complexity for everything in the world. Moreover, it is not necessary. Instead of measuring complexity of an individual object by a number, we compare any pair of systems and by transforming one into another, we can trace the number of steps that reduce or increase complexity. We may not have an exact measure in complicated cases, but we can still have a good idea about the difference between two systems.   We might have even a scale of complexity by selecting a zero point.

In this concept of complexity I use the same principle that Confucius used to quantify human virtues: by comparing two selected individuals and thus establishing a partial order in moral values (see Essay 13. On Numbers ). It is the same approach as with the beauty contest. There is no absolute numerical measure of beauty but we still can run a beauty pageant by ordering the contestants.

 This concept of complexity, which I would call pattern (or chemical) complexity, is based on fundamental ideas of Pattern Theory developed by Ulf Grenander. It is not limited to static structure and can be applied to dynamic systems where transitions from one static structure to another take place according to a separate set of dynamic rules.
I came to Pattern Theory from chemistry, which can be considered as an application of Pattern Theory.

The main idea of Pattern Theory is that most (if not all) of our knowledge about the world can be presented as blocks connected with bonds. This atomistic principle comes from deep antiquity, from Democritus, but Ulf Grenander developed the simple ancient idea into a rigorous mathematical (and by no means simple) edifice. The ideas, implications, and applications of Pattern Theory are inexhaustible. I tried to outline some in my manuscript The New and the Different.

Ulf Grenander and I attempted to apply Pattern Theory to history in our History as Points and Lines, with no prospects of publishing until history changes its course.

Pattern theory covers not just ethical systems, beauty contests, and Lego, but also social relations, biological forms, digital anatomy, molecules, genealogies, language, philosophical ideas, personal relationships, and Everything in the World.
Pattern Theory presents a universal abstract language for describing any system of any complexity. Therefore, it is also a language for describing evolving complex systems.   It does not discriminate between sciences and humanities.

One way to solve a problem is to appoint a czar. A town meeting is another one. The third one is to ask a sage.

 NOTES
 

1. Researchers involved in measuring complexity can compare the pattern complexity  with Kolmogorov’s complexity and its controversial implications concerning random sequences. In pattern complexity, anything without rules is inherently simple.
 

2. In authoritarian society some blocks have many copies and are interchangeable, and others have a few or one. In liberal democracy, ideally, all blocks are unique, but the function of bureaucracy is to keep the uniqueness down.
 

3. The following is the inventory of the individual blocks used in this Essay. They don't have either names or numbers. They are their own symbols, like pictograms in hieroglyphic script. By writing particular terms on the blocks and specifying rules, we can construct particular systems in various fields of knowledge, from poetry to molecules.

4. Pattern Theory:  Ulf Grenander, Elements of Pattern Theory, 1996. Baltimore and London: Johns Hopkins University Press.
Ulf Grenander, General Pattern Theory, 1993. Oxford: Oxford University Press.

P.S. (2016) It occurred to me while re-reading that complexity, like energy, does not have any absolute measure and can be only compared for two objects. It is, probably, an evidence of the fundamentality of the concept of complexity. But I can only raise the question whether a unit of complexity exists. I doubt it does. In Pattern Theory, it depends on a subjective knowledge and choice of an observer of images, templates, and configurations. Complexity is a bridge between sciences and humanities. It is mathematics with an anthropic principle: it makes sense only because humans exist.

---

To page 1

   Page created: 2001                                                     Revised: 2016