Archive for the ‘sixteenfolds’ Category

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]




The Rosetta Stone of Arthur M. Young

January 27, 2019

In his book “The Geometry of Meaning”, Arthur M. Young developed a system which he called his “Rosetta Stone” because he claimed that it associated mental attributes to some physical measurements, giving a kind of map to mind-body dualism. In the above diagram the expressions and names of these twelve objective values or physical units are shown. Examine the list below to see the twelve subjective values or mental attributes (shown in parentheses) that Young ascribed to them.

I’ve rearranged the sequence of these somewhat because I don’t quite understand the reasons for the organization Young gave them. I’ve also relabeled the units using the more standard derivative notation L’ = dL/dT, L” = d^2 L/ dT^2, etc. His notation is given at the end of each line below in parentheses for comparison (essentially he removes the d’s from the derivatives, writing dL/dT as L/T, which seems confusing to me when then multiplied by L).

Given the measurement Position or Length (L), Moment is Mass (M) times Position giving ML, and Moment of Inertia is Moment (ML) times Position giving ML^2. Velocity (L’), Acceleration (L”), and Control (L”’) have similarly related pairs of values by multiplying each by M and by ML. This gives four sets of similar expressions differing by their common derivative. I should have probably named the 1st set to be 0th, 2nd set to be 1st, 3rd set to be 2nd, and 4th set to be 3rd after the order of the common derivative.

  • Position (Observation): L
  • Moment (Significance): ML
  • Moment of Inertia (Faith): ML^2
  • Velocity (Change): L′ (L/T)
  • Momentum (Transformation): ML′ (ML/T)
  • Action (Impulse): MLL′ (ML^2/T)
  • Acceleration (Spontaneous act): L” (L/T^2)
  • Force (Being): ML” (ML/T^2)
  • Work (Fact): MLL” (ML^2/T^2)
  • Control (Control): L′′′ (L/T^3)
  • Mass Control (Establishment): ML′′′ (ML/T^3)
  • Power (Knowledge): MLL′′′ (ML^2/T^3)

How does Young justify associating, for example, the mental attribute “Faith” with the physical unit “Moment of Inertia”? It seems to have something to do with his original cyclic order of these twelve measurements and the associations he gives them to the Astrological Signs of the Zodiac. This mapping is not shown but can easily be found in the links below.

Further Reading:

Arthur M. Young / The Geometry of Meaning

The only early entry in my blog without a leading figure (and poor tables as well):

[*7.78, *7.79, *8.2, *11.38]


The Archic Matrix

September 13, 2018

Now that I am presenting sixteen-folds, let me briefly return to a subject I’ve mentioned previously. The Archic Matrix of Walter Watson and David Dilworth is a four-by-four matrix representing different aspects of the “personalities” of philosophers, determined by their writings. It is adapted from the Philosophical Semantics of philosopher Richard McKeon.

The four aspects of the Archic Matrix (also called Archic Variables) in this diagram are Perspective (upper left), Reality (lower left), Method (upper right), and Principle (lower right). The archic variable Perspective can have values Personal, Agonistic, Existential, and Creative, and similarly for the other three variables.

Each of the values of each of the variables is conditioned by one of the variables. For example, Personal is only conditioned by the Archic Variable Perspective, even though it is already a value of that variable. The value Agonistic is conditioned by the Archic Variable Method, Existential by Reality, and Creative by Principle.

Whereas the Myers-Briggs Type Indicator is a two-by-four matrix giving sixteen different combinations for its four personality “variables”, each having two values, the four-by-four Archic Matrix has four values for each of four variables and so gives 256 different combinations. It would be interesting if someone mapped the larger scheme into the smaller.

Further Reading:

Archic Matrix: Perspectives

Archic Matrix: Realities

Archic Matrix: Methods

Archic Matrix: Principles


The only work that seems to mention both the Archic Matrix and Myers-Briggs is as follows:

Mondo Secter / The Architectonics of Culture: A Critique, Modification, and Extension of Hofstede’s Study of Societal Culture with a Chinese-Based Typology, Ph.D. Dissertation, Simon Fraser University, August 2003

Secter is elsewhere mentioned to be completing an adaption of this dissertation (albeit long ago), called “The Architectonics of Culture and Personality: Six Core Dimensions of Who We Are”. It would useful to know anything else about this work. Updates, anyone?

[* 10.151]





The Myers-Briggs Type Indicator

September 7, 2018

I am not a number, I am a free man!

— Number Six from the 1960’s television series, The Prisoner

The Myers-Briggs Type Indicator (MBTI) is usually shown as a four-by-four grid of “personality types”. Rebel that I am, here is a triangular version for your personal edification or confusion.

Interestingly, some say that there is an association between the MBTI and Astrology. If they are being positive, they are seeing parallels between the personality pigeonholes of the two schemas. If they are being negative, they are saying that both classification systems are pseudo-science rubbish.

One might think that this is part of the larger discussion of a person being a “blank slate”, and whether we are determined by nature or nurture or a combination of the two, or are free to be who we desire to be.

Further Reading:

More Images of the MBTI:

Comparisons of MBTI and Astrology to chew on:






The Standard Model of Particle Physics

September 5, 2018

Until a few years ago, the Standard Model of particle physics had sixteen particles verified. Recently, the Higgs Boson was added, so just imagine another triangle hanging off somewhere. Or, alternately, one may combine the Z and W Bosons when adding the Higgs, so a sixteen-fold scheme is maintained.

Also, the Standard Model is much more than a list of these seventeen particles and their attributes. Not shown are a plethora of equations that describe the properties of these entities and their interactions.

I admit that this diagram is not very useful for scientific knowledge, but the reader may contemplate it as they please.

Further Reading:




The Diurnal Cycle and Chinese Hours

August 23, 2018

Long ago in a distant land, I, Aku… wait… let me start over.

Long ago in China, the diurnal cycle of day, dusk, night, and dawn was divided into twelve two-hour hours and each hour was named after an animal in the Chinese Zodiac. The Hour of the Rat was from 11 P.M. to 1 A.M., and so on. These animals were also assigned directions: Rat being North, Ox being NNE, etc. Many are familiar with the animals having “years”, so if you were born in a certain animal’s year, you might have certain attributes. These twelve animals are certainly multi-purpose!

Further Reading:




The Seasons and the Zodiac

August 22, 2018

This isn’t a bad little diagram of the four seasons along with the twelve zodiac names and symbols. However, it might be oriented wrong by convention or going clockwise instead of counter-clockwise.  Interestingly, old horoscope charts that show what was in the sky (the positions of the zodiac stars and the eight or nine planets in regards to the twelve “astrological houses”) at the time of a person’s birth were shown using the outside ring of twelve triangles instead of the more familiar circle that is used today. The inner square might be for notes or some nice drawing.

Further Reading:






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