Concept borrowed from mathematics, symmetry is the systematic presence, between two groups of peoples, of certain traits of any nature (historical events, geographic landforms, linguistic or cultural phenomena) which constitute a pattern that is repeated several times and is due neither to natural determinism, nor kinship nor direct imitation, but must, perhaps, have its explanation connected with the phenomenon of synchronicity, such as proposed by Carl Gustav Jung.
Consider two families Alpha and Beta formed by populations A and B, and C and D, respectively, as in the following diagram.
| Family Alpha | Population A (A1CQ3RJC4P) | Population B (F1BR3MNT4J) |
| Family Beta | Population C (I2HG6PUY8D) | Population D (E2WS6KAQ8B) |
If we analyse the set of common and differential traits between A and B, we will reach a pattern of similarities and differences (if, supposedly, between A and B there were only similarities, A and B would be the same population; if there were only differences, A and B could not belong to the same family). It is quite possible that we find more differences than similarities between A and B, but, if these similarities form a pattern that we will also find analysing populations C and D, of family Beta, so we can say that there is symmetry between A and C, as well as between B and D, what leads to the conclusion that there is also symmetry between families Alpha and Beta.
In this example, each population has a set of traits that we are representing by letters and numbers. If we confront the traits of A and B, of family Alpha, we will have:
A = A1CQ3RJC4P
B = F1BR3MNT4J
What is there common between these traits? In both cases, the second, fourth and eighth characters are digits (in boldface), while the others are letters. And the digits are 1, 3 and 4, whereas among the letters (which form most of the traits) there seems to be no relation at all.
Well, if we now confront the traits of populations C and D, of family Beta, we will have:
C = I2HG6PUY8D
D = E2WS6KAQ8B
Again, in both cases the second, fourth and eighth characters are digits, and the others are letters. Once again, the letters do not seem to make any sense, but the digits are 2, 6 and 8 in both populations. So there is a pattern in family Alpha (second character = 1, fourth character = 3, eighth character = 4) and another pattern in family Beta (second character = 2, fourth character = 6, eight character = 8).
But what else can we discover? Confronting sequences 1-3-4 of Alpha and 2-6-8 of Beta, we soon notice that the digits of Beta are exactly twice the correspondents of Alpha. Therefore, also and especially between Alfa and Beta there is a pattern of symmetry.
When comparing the hypothetical populations A, B, C and D under many aspects, we will reach many sequences of characters and, if there really is symmetry between families Alpha and Beta, we must find among all these sequences a very larger amount of repetitive patterns than what would be expected if such sequences were casual (that is, the number of similarities between families Alpha and Beta is too big to be attributed to fortuitous coincidences). Possibly, in most of the comparisons no patterns will be found; sometimes, we will find patterns between A and C, B and D; other times, we will find them between A and D, B and C. What matters is that the found patterns, even though are minority in our sample, are too persistent to be explained as chance.
If most symmetries take place between A and C on the one hand, and between B and D on the other, we say that A-C and B-D form primary (because they are majority), direct or vertical symmetries (given the disposition in our diagram). In contrast, symmetries A-D and B-C, being frequent but minority, are called secondary, cross or diagonal symmetries. One can also speak of horizontal symmetry when we have A-B and C-D, that is, the repetition of a pattern among all the populations of family Alpha as well as among all of family Beta. But this kind of symmetry is usually trivial, since, in most cases, it can be explained by the very kinship between the populations of each of the families.
When a pattern shows in a reversed way between two families, we say that there is reverse symmetry. For example, if all the Latin countries were republics, except for one, and all the Germanic countries were monarchies, except for one, we would have a kind of reverse symmetry, that is, the specular image would appear as a photo negative. The reverse symmetry is a combination of non-simultaneous vertical and diagonal symmetries. (For further information, consult the article The laws of supersynchronicity.)
Finally, symmetries can be strong, medium or weak. We have strong symmetry when there are synchronicities both in history and in language. On the other hand, we have medium symmetry when there are many synchronicities in history, but few in language, or few synchronicities in history, but many in language. Last of all, there is weak symmetry when we have few synchronicities both in history and in language, or many synchronicities in history, but almost none in language.