Archive for the ‘sixteenfolds’ Category

The Arcane Arts of Ramon Llull : the Dignities

July 7, 2019

Oh, Ramon Llull, where have you been all my life? I’m sure he’s been there all along, death now over seven hundred years in the past, just like always. His legacy seems at first glance to be quite the essence of medieval religion and scholastic philosophy, but still significantly and obscurely different to be enticing to this one. And on further examination, much more.

My schema above has little to do with his grand elaborate figures, except for listing the sixteen attributes he called “dignities”. Llull’s diagrams are full of clock-like wheels within wheels, complicated tableau, and combinatorial patterns. He wished to create a universal model to understand reality, and who wouldn’t want to discover the same? It is said that his methods are akin to an early computer science, and I’m just now starting to understand why.

The magister based the substance of his methods on his Christian faith, although he converted in midlife from Islam. Living in Barcelona, it was probably a good place to make such a change, but felt his calling was to convert others as well, so traveling he went. The methods he developed to convince others of their errors in belief were quite remarkable, as were the volume of his writing.

Like Gottfried Wilhelm Leibniz, who lived four hundred years later and was influenced by him, Llull wished to automate reasoning. But instead of building mechanical devices, Llull built computers from paper and ink, rulers and drawing compasses, scissors and glue. And instead of numbers as the smallest tokens of his computer, he used abstractions (i.e. words) that he felt would be understood by everyone in exactly the same way.

For example, he enumerated these sixteen dignities or aspects of his Christian diety, although sometimes he used the first nine. His constructions allowed one to pose questions and then obtain answers mechanistically that would be convincing to all observers of the correctness of the result. Too bad he was ultimately stoned to death while on his missionary work, although he lived to be eighty two.

Llull’s devices remind me of some of my pitiful charts and diagrams, and make me wonder if I may either adapt some of his techniques to my own use, or be inspired to develop others. I suspect I have locked myself into limitations by my approach, or are these constraints to my advantage? It might be hard to have spinning elements, but I can envision sliding elements like Napier’s Bones, origami-style folding and pleating, and even physical constructions like linkages and abacuses.

Now a martyr within the Franciscan Order, Llull’s feast day is June 30, which I’ve now missed. I hope to remember him to repost or improve on this by next year.

Further Reading:

“Let us Calculate!”: Leibniz, Llull, and the Computational Imagination

The memory wheel



Richard McKeon’s Aspects of Knowing, Part 3

July 1, 2019

Previously on this blog:

Each method can be associated with a discursive process: operational with debate, dialectical with dialogue, logistic with proof, and problematic with inquiry. Each method is also associated with a mode of thought which in turn has two moments and one dependency or assumption: the operational method is debate by discrimination and postulation dependent on chosen theses, the dialectical method is dialogue by assimilation and exemplification dependent on changeless models, the logistic method is proof by construction and decomposition dependent on indivisible constituents, and the problematic method is inquiry by resolution and question dependent on discoverable causes.

For this diagram, the four dependencies or assumptions are in the center, and the associated methods are adjacent to them. Filling out the outer edge are the four pairs of moments. Listed, these facets are:

  • Methods (associated discursive process): Operational (Debate), Dialectical (Dialogue), Logistic (Proof), Problematic (Inquiry)
  • Assumptions: Chosen Theses, Changeless Models, Indivisible Constituents, Discoverable Causes
  • Modes of Thought: discrimination and postulation, assimilation and exemplification, construction and decomposition, resolution and question

The second diagram comes from a chart in McKeon’s “Philosophic Semantics and Philosophic Inquiry”. Here, the four methods are in the upper left corner (Universal) and lower right corner (Particular), and four principles are in the lower left corner (Meroscopic) and upper right corner (Holoscopic). Four interpretations are in the center (the vertical pair is Ontic, and the horizontal pair is Phenomenal), and four selections are adjacent to them. Listed, these facets are:

  • Methods: Operational, Dialectical, Logistic, Problematic
  • Principles: Simple, Actional, Comprehensive, Reflexive
  • Interpretations: Existentialist, Entitative, Ontological, Essentialist
  • Selections: Knower (Types), Knowable (Matters), Knowledge (Hierarchies), Known (Kinds)

Note that the Archic Matrix of Watson and Dilworth is essentially derived from this, and even has many of the same terms. However and obviously, the sixteen-fold arrangements of the two diagrams are different.

Further Reading:

[*5.184, *5.185, *6.20, *11.102, *11.106]




The Mouse’s Tale

June 7, 2019

No apologies to Lewis Carroll.

Further Reading:

[* 11.100]




The Tangram

June 1, 2019

Further Reading:



The Chromatic Scale of Music

May 28, 2019

All art constantly aspires towards the condition of music.

— Walter Pater

I really don’t know much about music theory, but I know I like this chromatic “circle” with solmization center.

Also, if you count out seven triangles clockwise around the rim starting from C you get G, D, A, E, B, F#, etc., which is the Circle of Fifths for chords. (But you get that with any chromatic circle.)

And I imagine if you go counter-clockwise you get the Circle of Fourths, which is important for Jazz chords!

Further Reading:

Music of the Squares: David Ramsay Hay and the Reinvention of Pythagorean Aesthetics

[*11.68, *11.69]


Combogenesis: a constructive, emergent cosmos

May 22, 2019

I just finished reading Tyler Volk’s “Quarks to Culture: how we came to be”. In this book Volk outlines an interesting model for what he calls combo-genesis, a “great chain of being” leading from basic physical law up through the highest organizational structures that we know of, human societies. He traces a path through human knowledge: physics, chemistry, biology, zoology, sociology, etc., and I am reminded of E. O. Wilson’s “Consilience: the unity of knowledge” that argued for some of the same things.

But Volk’s work has some good new ideas. He details twelve hierarchical levels, where each level is constructed on a “lower” previous level, and the new “higher” level has new things and different abilities than its predecessor. These levels range from the level of fundamental quanta (the quarks of the title), to geo-political states (the culture of the title).

  • QUA: fundamental quanta
  • PRO: nucleons, which are protons and neutrons
  • NCL: atomic nuclei
  • ATM: atoms
  • MOL: molecules
  • PCL: prokaryotic cells
  • ECL: eukaryotic cells
  • ANI: multicellular organisms, including animals
  • ASG: animal social groups
  • HUM: human tribal meta-groups
  • AGR: agro-villages
  • STA: geo-political states

These levels are within three dynamical realms, the first realm being of physical laws and then those realms of biological and cultural evolution. Each of these realms has a base level that has a capability for great constructive and emergent potential via an “Alpha-kit”. An alpha-kit has two facets, an element set and a cornucopia set, that operate like an alphabet and the myriad combinations that that alphabet can produce.

Dynamical realms:

  • Realm of physical laws: QUA -> MOL
  • Realm of biological evolution PCL -> ASG
  • Realm of cultural evolution HUM -> STA

Base levels and their Alpha-kits:

  • QUA, means for chemistry and molecules (atomic)
  • PCL, mechanisms for biology and its evolution (genetic)
  • HUM, faculty for culture and its evolution (linguistic)

As Volk’s model has each higher level based on or constructed from the previous lower one, I make the following suggestion utilizing my four-fold Structure-Function. The structures of each lower level serve as the parts of the next higher level, and the functions of each lower level serve as the actions of the next higher level. In this way a chain of actions and parts, structures and functions are built giving different entities and capabilities to different operational domains.

In the diagram shown, the sets of structures (S) and functions (F) of level i are used for the sets of parts (P) and actions (A) of level i+1, so S(i)=>P(i+1) and F(i)=>A(i+1). Not all structural information or functional abilities are necessarily accessible in the higher level of parts and actions, similar to the information and method hiding in object-oriented programming, and so reducing overall complexity. And as I have argued before, parts are combined to create the structures and actions are combined to create the functions of each level, so P(i+1)=>S(i+1) and A(i+1)=>F(i+1). In this way we have a bottom-up combo-genesis leading from quarks to culture.

Are we now entering another dynamical realm, perhaps based on some technological or computational alpha-kit? But, unfortunately we have to ask, will it take us forwards or backwards?

Further Reading:

Tyler Volk / Quarks to Culture: how we came to be

Tyler Volk and Robert Wright discuss:


Edward O. Wilson / Consilience: the unity of knowledge

[*11.98, *11.99]



Fire and Ice

May 16, 2019

Apologies to Robert Frost.

Further Reading:

[*8.128, *11.96]



The Hero’s Journey

May 11, 2019

Professor of Literature Joseph Campbell [1] popularized the monomyth of the “hero’s journey” [2], a recurring template for the plot arc or cycle of many heroic characters found in mythologies and even popular modern stories. From his most famous book, “The Hero with a Thousand Faces”:

A hero ventures forth from the world of common day into a region of supernatural wonder: fabulous forces are there encountered and a decisive victory is won: the hero comes back from this mysterious adventure with the power to bestow boons on his fellow man. [3]

Quite a few diagrams are available showing the generic journey of the hero or heroine; some are more complete than others, and the one I present here is no exception. Campbell’s schema is said to have seventeen basic parts to it, but not all elements need to be present in any specific narrative. I list four basic “acts” (separation, descent, ascent, unification) and twelve basic events, including four thresholds: 1st) crossing into the special world, 2nd) surmounting the supreme ordeal, 3rd) returning to the normal world, and 4th) journey’s end and / or beginning.

  • Call and Refusal
  • ——–< Separation >
  • Meet the Mentor
  • Crossing Over (1st Threshold)
  • Tests, Allies, Enemies
  • ——–< Into the Abyss / Descent >
  • Inmost Cave / Whale Belly
  • Supreme Ordeal (2nd Threshold)
  • Ultimate Boon / Reward
  • ——–< Magic Flight / Ascent >
  • Road Back / Refusal
  • Crossing Back (3rd Threshold)
  • Return with Elixir / Resurrection
  • ——–< Unification >
  • Master of Two Worlds
  • Freedom to Live (or Die) (4th Threshold)

Any hero’s journey must be transformational, and so has associations with alchemical change. It must be of such difficulty that it utterly changes the nature and the mindset of the traveler. Even though many supporting characters may influence, help, and even hinder our hero, it is ultimately about their journey, their sacrifice, and their reward. But what the hero brings back with them is also key to the story, because it is for the benefit of others as well as for themselves.

An interesting dissertation by Richard Warm that I ran across recently proposes that leaders, like heroes, need ordeals and trials in order to bring back the inspiration and wisdom to motivate others. From the abstract:

This dissertation will explore leadership as a mytho-poetic transformational journey toward self-knowledge, authenticity, and ultimately wisdom; the power to make meaning and give something back to the world in which we live; and the necessity of transformation. I view leadership as a transformative process and a transformational responsibility. As leaders we must undergo our own transformation in order to lead change on a larger scale. The dissertation will be both philosophical and theoretical, exploring how the threads of the hero’s journey, transformation, wisdom, and leadership intertwine. …[4]

These ordeals may also be of an intellectual or moral kind, and not just feats of strength or stamina, although those are important too. Warm also has a company [5] for coaching you on your leadership, life, and legacy.

One of my favorite books that I think of when the hero’s journey comes to mind is “Figures of Earth: a comedy of appearances” by James Branch Cabell. It is a bit of a spoof on fantasy tropes, well before its time. However, I’m not quite sure what the “magic elixir” is that the hero returns with except for his story: that life is what you make of it, or what it appears to be to others, or what you can convince yourself that it is, or that it is what it is and must be sufficient to you in the end.




[3] Joseph Campbell / The Hero with a Thousand Faces



Further Reading:

“There are one or two things that I do not fancy the looks of in this torture-chamber.”

[*6.38, *6.133, *6.149, *9.200, *11.70]


Four Forms Make a Universe, Part 2

March 17, 2019

This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:






And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed:







Further Reading:

[* 11.50, *11.58]





Four Forms Make a Universe

March 12, 2019

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)


Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

[*9.145, *11.50]




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