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:

https://equivalentexchange.blog/2019/03/12/four-forms-make-a-universe/

[* 11.50, *11.58]

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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” andÂ “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)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

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)

https://en.wikipedia.org/wiki/Spacetime_algebra

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

[*11.50]

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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:

http://www.yeatsvision.com/Yeats.html

https://en.wikipedia.org/wiki/A_Vision

https://blogs.clemson.edu/press/2018/02/13/forthcoming-a-readers-guide-to-yeatss-a-vision/

https://equivalentexchange.blog/2013/10/23/a-vision-by-w-b-yeats-faculties/

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

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]]>- 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):

https://equivalentexchange.blog/2015/03/11/j-y-girards-transcendental-syntax-v2/

[*11.18]

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]]>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:

https://www.scientificamerican.com/article/four-new-dna-letters-double-lifes-alphabet/

http://science.sciencemag.org/content/363/6429/884

https://gizmodo.com/freaky-eight-letter-dna-could-be-the-stuff-aliens-are-m-1832823430

[*11.46]

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]]>- 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:

https://en.wikipedia.org/wiki/Good_Omens

https://en.wikipedia.org/wiki/Good_Omens_(TV_series)

https://en.wikipedia.org/wiki/Four_Horsemen_of_the_Apocalypse

https://en.wikipedia.org/wiki/Living_creatures_(Bible)

https://publicdomainreview.org/collections/the-four-horsemen-of-the-apocalypse/

https://en.wikipedia.org/wiki/Four_Horsemen_of_the_Apocalypse_in_popular_culture

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”:

https://equivalentexchange.blog/2017/09/06/the-four-last-things/

[*11.20]

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Octonions have the general form:

a_{0}e_{0} + a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + a_{4}e_{4} + a_{5}e_{5} + a_{6}e_{6} + a_{7}e_{7}

Where the coefficients a_{i} are Real and the bases e_{i} have (something like) the following relations:

e_{0} = 1 (and -1 = e_{1}^{2} = e_{2}^{2} = …)

e_{1} = I = e_{2}e_{3} = e_{7}e_{6} = e_{4}e_{5}

e_{2} = J = e_{5}e_{7} = e_{3}e_{1} = e_{4}e_{6}

e_{3} = IJ = e_{1}e_{2} = e_{6}e_{5} = e_{4}e_{7}

e_{4} = K = e_{5}e_{1} = e_{6}e_{2} = e_{7}e_{3}

e_{5} = IK = e_{7}e_{2} = e_{1}e_{4} = e_{3}e_{6}

e_{6} = JK =e_{5}e_{3} = e_{1}e_{7} = e_{2}e_{4}

e_{7} = IJK = e_{6}e_{1} = e_{3}e_{4} = e_{2}e_{5}

In addition, if any of the products like e_{2}e_{3} = e_{1} are reversed you get the negative, so e_{3}e_{2} = -e_{1}.

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

(e_{1}e_{2})e_{3} = e_{3}^{2} = -1

e_{1}(e_{2}e_{3}) = e_{1}^{2} = -1

(e_{1}e_{2})e_{4} = e_{3}e_{4} = e_{7}

e_{1}(e_{2}e_{4}) = e_{1}e_{6} = -e_{7}

(e_{1}e_{2})e_{5} = e_{3}e_{5} = -e_{6}

e_{1}(e_{2}e_{5}) = e_{1}e_{7} = e_{6}

so it is hit or miss I guess. Also note that e_{7} = (IJ)K = -I(JK). And for all (e_{i}e_{j})e_{k} and e_{i}(e_{j}e_{k}), 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:

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

http://mathworld.wolfram.com/Octonion.html

http://homepages.wmich.edu/~drichter/octonions.htm

https://ncatlab.org/nlab/show/normed+division+algebra

[*11.33]

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]]>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:

https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://theanalyticpoem.net/quaternions/

http://mathworld.wolfram.com/Quaternion.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

[*10.160]

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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

https://en.wikipedia.org/wiki/Don_Edward_Beck

http://www.cruxcatalyst.com/2013/09/26/spiral-dynamics-a-way-of-understanding-human-nature/

https://www.biznews.com/good-hope-project/2017/09/04/understanding-spiral-dynamics

https://scottjeffrey.com/spiral-dynamics/

https://en.wikipedia.org/wiki/Integral_theory_(Ken_Wilber)

http://www.awaken.com/2014/04/ken-wilber-summary-of-spiral-dynamics-model/

[*11.41, *11.45]

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]]>- Intra-personal
- Interpersonal
- Bodily-kinesthetic
- Nature-existential
- Visual-spatial
- Verbal-linguistic
- Musical-rhythmic
- Logical-mathematical

Further Reading:

https://en.wikipedia.org/wiki/Theory_of_multiple_intelligences

https://www.tecweb.org/styles/gardner.html

http://www.institute4learning.com/resources/articles/multiple-intelligences/

Howard Gardner / Frames of Mind: The Theory of Multiple Intelligences (1983)

Howard Gardner / Multiple Intelligences: The Theory in Practice (1993)

Howard Gardner / Multiple Intelligences: New Horizons in Theory and Practice (2006)

[*11.41]

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