A Game of Four-folds


One thing those that love four-folds (quadraphiles?) like to do is compare and contrast them. A solitaire card game based on four-folds might be fun for some individuals obsessed by tetrads and such. But how would it work?

If two four-fold cards are chosen at random, then insight into their relationship might be obtained. For example: Structure-Function and Matter-Energy-Space-Time (MEST). The simple observation is that each quadrant of Structure-Function requires a corresponding quadrant of MEST. Actions require Energy. Parts require Matter. Structures require Space. Functions require Time. Energy is necessary for Actions, etc.

One problem is that not everyone agrees how the quadrants of two different four-folds can correspond, or be “lined up”. In general, the only things that are available to explain the reasoning for the arrangement are analogical thinking and argumentation, so it is at least half subjective. This isn’t science, after all.

Note that the different poles of each four-fold can be ordered in six different ways. If we have a square four-fold of A B C D, we can always place A at the Left or West position. Then the remaining 3 letters can be arranged in six different ways.


If one just has a fixed card for each four-fold, then the other five permutations are not available. You could have six versions of each card, but I think that would be a poor solution, since you are only picking one of them at any time.

To acheive flexibility in arrangement, each card could be divided into its quadrants, for example by right triangles. But then you’d have triangular cards. Plus the fact that you couldn’t combine them well on a web page.

Or you could have the four-folds turned to be X’s, and then the quadrants could be squares. That’s somewhat appealing, since this blog is titled “Equivalent eXchange”, after all. For example:



However, if you are picking two cards at random, then the four quadrants won’t be together as a group.


Is Mahjong called “Game of Four Winds” or is it just a name of a computer version of it? From what I can tell, the players are named after the four winds (i.e. cardinal directions). But there are also four flowers and four seasons.



The Quadralectics of Marten Kuilman

sq_quadralectics Marten Kuilman has written extensively on four-folds and what he calls quadralectics, division-thinking, or four-fold thinking.

Publishing in the Netherlands, his books aren’t available on Amazon. Graciously, he has made several of his works available on the internet via his blogs Quadralectics and Quadriformisratio. Quadriformisratio presents Four – A Rediscovery of the ‘Tetragonus Mundus’, a treatise of four-folds through history, and Quadralectics is his two volume work on Quadralectic Architecture.

Not only does Kuilman expound at length on various four-folds throughout the ages and how they affected the intellectual and artistic developments of the time, his work unifies many of them into his four aspects of visibility: invisible invisibility, invisible visibility, visible visibility, and visible invisibility. Above, I’ve arranged these four aspects by my positions for the four elements. Unfortunately, they aren’t in the same sequence as Kuilman’s quadrants.

Because Kuilman emphasizes a recurring association of  his four-fold of visibility with communication, it is also reminiscent of Hjelmslev’s Net. Then, invisibility could be understood as content, and visibility as expression.

Interestingly, my four-fold of Bright-to-Dark (here or here) is most relatable to this four-fold of visibility, but in the reverse sense that the invisible invisibility is bright, and the visible visibility is dark. One could quickly reconcile this opposition by considering the empty circle as most invisible, and the full circle as most visible.

Another interesting result of Kuilman’s investigations is to derive his four-fold of Unity, Muun (Multi-unity), Part, and Whole, which I believe has important associations with my four-fold Structure-Function.


Marten Kuilman / Four – A Rediscovery of the ‘Tetragonus Mundus’

Marten Kuilman / QUADRALECTIC ARCHITECTURE – A Panoramic Review




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Notions of Equivalence

sq_equivalencesRecently I have been wondering what different notions of equivalence are possible. Thinking about one of my favorite fourfolds — Structure-Function — with help from the fourfold of The One and the Many, I have (naturally) come up with four notions: Identities, Isomorphisms, Confluences, and Indiscernibles.

Identities: One as One. In mathematics, an identity is an equivalence between two (or more) things that are really just the same thing. One can say there is an equivalence relation between the things and they are part of the same equivalence class. In equations, each thing can be substituted for the other thing because they are really just the same thing! In my consideration, Actions have these kind of identities, and so it is a reasonable term to use for Actions in the Structure-Function fourfold.

Isomorphisms: One as Many. In mathematics, an isomorphism is an equivalence between two (or more) things that have the same (mathematical) structure. Another way to consider this is to say that there is a paradigm or model that is representative of all the things that have this same structure. Thus it is a good term to use for Structures in the Structure-Function fourfold.

Confluences: Many as One. In logic, computer science (rewriting theory), and mathematics, a confluence is an equivalence between two (or more) things that can each be transformed into the same, maybe different, thing. Confluences can be used for this notion of equivalence because if one says that two rivers are confluent, then that means that they both flow into another larger river. Thus I think it is a valuable term to use for Functions in the Structure-Function fourfold.

Indiscernibles: Many as Many. In philosophy and perhaps physics, an indiscernible is an equivalence between two (or more) things where one cannot tell the difference between them. Thus it is a useful term to use for Parts in the Structure-Function fourfold. Another term to consider using is Substitutivities. Thus I do not believe I agree with the principle of the Identity of Indiscernibles since I think that would collapse my two notions of Identities and Indiscernibles into one. For instance, atoms of gold may be indiscernible from one another but that doesn’t mean they are the same atom.

To Do:

A recent foundational project for mathematics starts with the following:

Univalence Axiom: (A = B) ~ (A ~ B): Identity is equivalent to equivalence.

At this time I do not understand the implications of this axiom and how it might impact my four notions of equivalence.


Perhaps Extensionalities would  be a better choice than Confluences.






The HoTT Book: Homotopy Type Theory: Univalent Foundations of Mathematics





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