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Greek mythology, pt. 2: archetypes and Punnet squares

October 18, 2011

[The impetus for this series of posts came from some thoughts I had after writing about the Milosz poem “Bypassing rue Descartes” in the post titled “Empire, City, and Earth”. The first post took the form of a series of diagrams, with commentary, laying out the relationships between various archetypal figures. This post will carry the inquiry slightly further as well as comment on the form of the inquiry and how it can be infinitely extended. A third post will discuss whether that inquiry is valuable and what it tells us about language.]

Here is the diagram with which I ended the previous post:

man

body

spirit

god

sky

HERACLES

APOLLO

sky

earth

DIONYSUS

ORPHEUS

earth

god

body

spirit

man

There, I was talking about the actual concepts and archetypes which this diagram deals with. But here, I want to talk primarily about the form of this diagram: what is it to make such a diagram, and to point out, as I did, the importance of the diagonal lines? I’ll still make some observations about the diagram’s content, but they will be tangential to the main thrust of the post. I will also from here on ignore the fact that we previously placed Christ at the center of the above diagram, in the place of chiasmus, even though I think that is a good way to think about it, because it would detract from the following discussion, which is about extending this mode of inquiry beyond the first dimension. In a related omission, I will ignore the concept of “analogy,” which I suspect has something to do with both Christ and the square, in favor of talking about dichotomies.

*

Before I do any of that, though, let us go on a biological excursion–let’s talk about genetics. A Punnet square, as you man recall, is used in biology to describe the probability of a child have a particular genotype, based on the genotypes of his parents. A cross between two parents who are heterozygous in a particular gene looks like this:

(y)

(Y)

(Y)

Yy

YY

(y)

yy

yY

Where Y is the dominant trait, y the recessive, and YY, Yy=yY, and yy are the four possible offspring of a pair of heterozygous (Yy or yY, vs. homozygous, YY or yy) parents.

Now, the first diagram in this post is in form quite similar to the Punnet square. Both cross two dichotomies with each other and show all possible results; that one crossed spirit-body with sky-earth, while this one crosses the same dichotomy, dominant/recessive, with itself. Moreover, both result in a further dichotomy, a dichotomy between the two diagonals; that one offers us god/man, this one homozygous/heterozygous. Which is to say, our archetype square could be rewritten as follows:

(b)

(B)

(A)

x2=Ab

x1=AB

(a)

x4=ab

x3=aB

Where the emergent dichotomy is C=(AB,ab) and c=(Ab,aB).

But our archetype square differs from the Punnet in important ways. First, it begins with two different dichotomies, not the same one “squared.” Second, its entries are not primarily the combinations AB, Ab, aB, ab, but the entities x1, x2, x3, x4 that are associated with them. Third, its emergent dichotomy is different in nature from the homozygous/heterozygous one. The question is: different how?

Perhaps it is this: while the Punnet square offers us a dichotomy of a different sort than the original one–heterozygous/homozygous describes allele pairings, while dominant/recessive describes individual alleles–the archetype square offers us one of the same sort as its original two, namely, a dichotomy between two nouns whose associated adjectives can describe both archetypal figures and the ways of life they embody. That is, since x3=a and x3=B, then x3=c, in the same sense of “=”.

Or perhaps it is this: while it would be nonsensical to ask if there could be a product of Y and y which was homozygous, it is not nonsensical to ask if there could be a product of spirit and earth which was divine; Orpheus happens to be human, but we could have found a divine figure to place there instead. x3=aBc, but it is conceivable that some X=aBC.

*

Three dichotomies means eight possible combinations, and the original four plus the following four cover all of them:

god

body

spirit

man

sky

APHRODITE

SIBYL

sky

earth

MAENAD

PERSEPHONE

earth

man

body

spirit

god

This is the result of a half an hour of thought on the issue as I attempted to go to sleep a few nights ago. It’s not perfect, but it is more interesting than I expected it to be, for a number of reasons. First, because of how natural it seems. Aphrodite and Persephone, especially, are obvious opposites. Second, because of how closely it corresponds to the previous square; Maenads are the servants of Dionysus, Sibyls the servants of Apollo, Persephone aided Orpheus in the underworld, and while Aphrodite is as far as I know not associated with Heracles, both are associated with erotic prowess. And finally. because it suggests a further dichotomy: this square is entirely female, while the previous one was entirely male. We could proceed to ask: can we come up with two squares were each figure is replaced with one of the oppose sex, and everything else stays the same? Chances are we could, but I’m not going to try.

*

Two 2×2 squares together form a cube (though a Punnet cube would only be useful to biologists if there were a species which required three parents and three copies of each gene). More importantly, our archetype cube generated new dichotomies in the same way as the archetype square, simply scaled up to three dimensions: instead of looking at diagonals, we looked at complementary tetrahedrons. This generalizes. Given n dichotomies Aa, Bb, Cc, …, give each figure a point for each capital letter associated with him. Each figure with an odd point totals is placed on one side of the dichotomy, those with even totals on the other. (Note: this always divides the vertices of the n-cube into two equal sets; it’s easy to show why using Pascal’s triangle.) So, to get from 2 to 3, x1=AB gets 2 points; x2,x3=Ab,aB get 1 each; and x4=ab gets 0; x1 and x4 are one side, and x2 and x3 are the other. And so on.

This is, of course, simply to say that from a pair of dichotomies a third can always be generated, though it will not always be coherent; from the three, a fourth; from the four, a fifth; and so on. This is not the case with a single dichotomy; a single opposition cannot generate a second. What we might call a “square investigation” must begin with two dichotomies, which we “cross” with each other, that is, place them perpendicular to one another, consider the dichotomy between the diagonals, and repeat.

But if we can always generate another dichotomy, should we?

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