Archive for the ‘fourfolds’ Category

Seven Sermons to the Dead, V2

March 19, 2019

We must, therefore, distinguish the qualities of the pleroma. The qualities are pairs of opposites, such as—

The Effective and the Ineffective.
Fullness and Emptiness.
Living and Dead.
Difference and Sameness.
Light and Darkness.
The Hot and the Cold.
Force and Matter.
Time and Space.
Good and Evil.
Beauty and Ugliness.
The One and the Many. etc.

— Carl Jung, from Seven Sermons to the Dead

I previously mentioned two fourfolds which are perhaps better combined as an eightfold. These concepts or entities are from Carl Jung’s “Seven Sermons to the Dead”.

To the left are the “principal gods”, who are said to be in one-to-one correspondence with the “world’s measurements”. What then are these measurements? Are they the coordinates of relativistic Space-Time or something else?

  • The God-Sun: The beginning
  • Eros: Binding of two together, outspreading and brightening
  • The Tree of Life: Filling space with bodily forms
  • The Devil: The Void, opening all that is closed, dissolving all formed bodies, and destroyer of everything

And to the right we have:

  • The Pleroma: The spiritual universe as the abode of gods and of the totality of the divine powers and emanations.
  • The Creatura: The living world, subject to perceptual difference, distinction, and information
  • Abraxas: The supreme power of being transcending all divinities and demons and uniting all opposites into one
  • Philemon: Jung’s spiritual guide and the narrator, also said to be Simon Magus or Basilides

Further Reading:

[*10.76, *11.49]




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 the Boolean True (A or ~A), A, B, and 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)




The Visions of W. B. Yeats

March 8, 2019

William Butler Yeats published two editions of his book, “A Vision”, the first in 1925 and the second in 1937. They are interesting amalgamations of metaphysics, poetry and esoteric speculations. After years of neglect I return to mention these works briefly.

Previously I showed the fourfold of the Faculties (Creative Mind, Body of Fate, Mask, and Will), but ignored the complementary fourfold of the Principles (Spirit, Celestial Body, Passionate Body, and the Husk). Here I present an eightfold of both Faculties and Principles, and it makes a bit more sense with all its accompanying structure.

Each of these eight elements has certain dual properties which I have indicated by the second diagram: Primary (+) vs. Antithetical (/), Solar (Sun) vs. Lunar (Moon), Static (=) vs. Active (~). Primary is also called Objective, Antithetical is also called Subjective, Solar is also called Core, and Lunar is also called Changing.

In my diagram I have the Faculties as down-pointing triangles (+ and /) and the Principles as up-pointing triangles (Sun and Moon). Elements that are Active (~) are in the upper-left and lower-right squares and those that are Static (=) are in upper-right and lower-left. Since explanations of these elements usually involve interpenetrating cones and gyres and such, whether this diagram has any value is left for a later time.

I also just found out that Neil Mann, the author of the excellent and enduring web site about “A Vision”, has a new book out. It’s called “A Reader’s Guide to Yeats’s ‘A Vision'” and is published by Clemson University Press. It’s a bit pricey but I’m sure it’s full of valuable insights into these works. Perhaps I’ll just have to raid my piggy bank.

Further Reading:;

[*7.199, *7.200, *8.1, *11.56]


Education is Not a Product

March 1, 2019

A recent meme popular in education:

  • Education is not a product.
  • Students are not customers.
  • Professors are not tools.
  • The University is not a factory.

I’m not sure who originally came up with this interesting saying.

Examining the anti-analogy, you have “customers buy products made in a factory with tools”. Perhaps a better fourfold would be to change this to “workers use tools to make products in a factory”, so you could say “Students are not workers.” But then you lose the capitalistic “consumer culture” geist.

Or you could say the Professors are (not) factory workers and the Students are (not) the raw material. As a homework assignment, write an essay or develop a mathematical theory comparing this anti-analogy fourfold (or a close derivative thereof) to Aristotle’s Four Causes. Please show your work.

Further Reading:

Note the similarity with some of the elements of the following (which also reminds me for the Four Causes):



Four New Synthetic Base Pairs Added to DNA

February 27, 2019

All life on earth uses DNA with four types of base molecules arranged in a “double helix” to encode information as sequences of Adenine-Thymine (A-T) and Cytosine-Guanine (C-G) pairs. Because A pairs only with T and C only pairs with G, DNA can be split into two halves and then each half may be completed so that two double helices with the same information can be obtained. Nature also uses the “genetic code” to translate the information in DNA into specific amino acids via its molecular machinery that assembles proteins in the correct sequence.

Now scientists have doubled the number of base pairs of DNA by adding four synthetic ones to the original four natural ones. Four new base molecules S (a Pseudo-T), B (a Pseudo-B), Z (a Pseudo-C), and P (a Pseudo-G) have been created so that S only pairs with B and Z pairs only with P. Now you have twice as many options as you had before and so this “hachimoji” (eight letter) DNA can have much greater information density. Plus if you re-engineer the molecular machinery to use these new bases you can have a brand new extended genetic code to build all sorts of synthetic proteins.

Further Reading:



The Four Horse-persons of the Apocalypse

February 26, 2019

In preparation for the “Good Omens” TV (specifically, Amazon Prime and BBC Two) series, I started reading the 1990 novel by Terry Pratchett and Neil Gaiman. Concurrently, and after watching too much TV news, I started to think (again) about the Four Horse-persons of the Apocalypse. In the book, personifications of these entities appear as characters working conscientiously to bring about the end of the world (but so far without their trusty steeds). They are (by one interpretation):

  • War (or Conflict) (said to ride a Red Horse)
  • Famine (said to ride a Black Horse)
  • Disease (said to ride a White Horse)
  • Death (said to ride a Pale Horse)

These four riders famously make their appearance in the biblical book of Revelation, after the Four Living Creatures (essentially, the Tetramorph) say “Come” or “Allons-Y!”, depending on your translation. And indeed the Four Horse-persons are a powerful “meme” and are referenced in many popular entertainments and real-life derivative nomenclatures. I’m looking forward to the video series as I enjoy the main actors playing the “odd couple” of angel and demon, and after viewing the various trailers and photos.

Who says Armageddon (or the struggle against it) can’t be fun? Ok, I admit it’s scary but sometimes you just have to laugh!

Further Reading:

Here Death is the end, but it’s only the beginning of the end (or is it the end of the beginning?) in the four-fold “The Four Last Things”:





February 25, 2019

Octonions are the fourth, the last, and the greatest of the so-called normed division algebras based on the Real numbers. When I first learned about the Octonions way back when, I didn’t like them because they weren’t associative like the Quaternions, the Complex numbers, and the Reals. But now I’m fine with that, and they may be important for new theories of physics!

Octonions have the general form:

a0e0 + a1e1 + a2e2 + a3e3 + a4e4 + a5e5 + a6e6 + a7e7

Where the coefficients ai are Real and the bases ei have (something like) the following relations:

e0 = 1 (and -1 = e12 = e22 = …)

e1 = I = e2e3 = e7e6 = e4e5

e2 = J = e5e7 = e3e1 = e4e6

e3 = IJ = e1e2 = e6e5 = e4e7

e4 = K = e5e1 = e6e2 = e7e3

e5 = IK = e7e2 = e1e4 = e3e6

e6 = JK =e5e3 = e1e7 = e2e4

e7 = IJK = e6e1 = e3e4 = e2e5

In addition, if any of the products like e2e3 = e1 are reversed you get the negative, so e3e2 = -e1.

Non-associativity is demonstrated by going through the list of triples:

(e1e2)e3 = e32 = -1

e1(e2e3) = e12 = -1

(e1e2)e4 = e3e4 = e7

e1(e2e4) = e1e6 = -e7

(e1e2)e5 = e3e5 = -e6

e1(e2e5) = e1e7 = e6

so it is hit or miss I guess. Also note that e7 = (IJ)K = -I(JK). And for all (eiej)ek and ei(ejek), if they are not equal, is one equal to the negative of the other? And do I have to multiply them all out to find out?

Further Reading:




February 22, 2019

The Quaternions are a number system that enlarges the Complex numbers, just as the Complex numbers enlarge the Real numbers. In fact, Quaternions can be thought as special pairs of Complex numbers, just as Complex numbers can be thought as special pairs of Real numbers.

Quaternions can be used for all sorts of wonderful things, such as rotations in 3D space, instead of using matrices. Above is a pitiful diagram (although better than my last one) of the Quaternion units 1, i, j, and k used in the typical representation a + b i + c j + d k. Please read about them in the links below and be amazed!

Further Reading:





Spiral Dynamics

February 21, 2019

Spiral Dynamics is an intriguing model for the different levels of development and interaction that individuals (and even cultures) may exhibit towards each other and the world. It has interesting parallels to Erikson’s Eight Stages of Psychosocial Development and even Leary’s Eight Brain Circuit Model of Consciousness.

Below is a list of these levels by (Catchy) Name, Social Structure, Motives, and Means of Coping:

  • SurvivalSense, Loose Bands, Survival, Instinctive
  • KinSpirits, Tribes, Magic/Safety, Animistic
  • PowerGods, Empires, Power/Dominance, Egocentric
  • TruthForce, Pyramidal, Order/Morality, Absolutistic
  • StriveDrive, Delegative, Autonomy/Achievement, Multiplistic
  • HumanBond, Egalitarian, Approval/Equality/Community, Relativistic
  • FlexFlow, Interactive, Adaptability/Integration, Systemic
  • GlobalView, Global, Compassion/Harmony, Holistic

And maybe another, further step if we don’t destroy ourselves and the Earth by just being stupid.

Further Reading:

Don Edward Beck, Christopher C. Cowan / Spiral Dynamics: mastering values, leadership and change

Don Edward Beck, et. al. / Spiral Dynamics in Action: humanity’s master code

[*11.41, *11.45]



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