Fourfolds and Double Duals, Part 2

In addition to the symbolic fourfold presented in the previous entry Fourfolds and Double Duals, which includes the alchemical symbols for the four elements, here is another generic representation. This fourfold reminds me of the fourfolds Bright-Light-Dim-Dark shown in the early entry with the Four Elements, as well as the newer fourfold of The One and the Many.

If one considers the outer ring and the inner circle to be “outer-as-inner”, then this fourfold is related to the One and the Many by letting One be white and Many be black.

Additionally, the symbols can be seen to map to the fourfold Bright-Light-Dim-Dark by noting that White-as-White is Bright, White-as-Black is Light, Black-as-White is Dim, and Black-as-Black is Dark.

For fun, this figure can be tweaked to reveal a deconstructed yin yang. It reminds me of John Dee’s Hieroglyphic Monad. Or someone holding a Vulcan Lirpa from Star Trek’s Amok Time.



Amusingly, I’m starting to think of the lower figure as a crossed fork and spoon: a fork to separate and a spoon to combine. Note the handles of the utensils (the circles) appear to be switched, but that is just the way it is.

Also: a c & t’s mouth and c t’s eye; a d g’s nose and d g’s eye. These are the binary operators of Linear Logic!


Fourfolds and Double Duals

For every aspect of the world that someone has thought to analyze into its components, it probably has been suggested to divide it into four parts. I suggest that many of the things that have a fourfold form, are a fourfold in the same way. Not in the trivial cardinal sense, but in a deeper structural sense. They are a combination of two dualities, a double dual if you will, such that one dual operates as interior and exterior, or true and false, a duality of opposites, and another (dual) dual operates as one and many, or unity and multiplicity, or discreteness and continuity, a duality of numeracy.

I have been gathering fourfolds for a time, and have written about some more than others. Some have been around a long time, and others I’ve been inspired to fashion. I have tried to orient them all in the same way to accentuate their common deep structure. For example, everything in the left position in the diagrams have a commonality across fourfolds, as does everything in the lower, upper, and right positions. The four ancient symbols shown in the diagram above represent the four elements of alchemy: fire, earth, air, and water.

Because of these relations between the fourths and the halves of these fourfolds, I have choosen the name “Equivalent Exchange”. In addition, the fourfolds themselves might be “equivalently exchangable” with each other because of their common deep structure. For many reasons, I believe that the best exemplar for these fourfolds is the recent logical system known as Linear Logic, which has two combining binary operators and two dividing binary operators.

Others before me have reached similar hypotheses about fourfolds in general, and I am grateful for their scholarship. I hope others will follow, and I’m sure they will present their findings more eloquently and convincingly than I have.