Category Archives: eightfolds

The Endless of Sandman

The Endless are merely patterns. The Endless are ideas. The Endless are wave functions. The Endless are repeating motifs. The Endless are echoes of darkness, and nothing more… And even our existences are brief and bounded. None of us will last longer than this version of the Universe.

Destruction of the Endless

The Endless are:

    • Destiny
    • Death
    • Dream
    • Destruction
    • Desire
    • Despair
    • Delirium

Further Reading:

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

https://en.wikipedia.org/wiki/Endless_(comics)

https://sandman.fandom.com/wiki/The_Endless

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Charles Howard Hinton

Another early author of note about the fourth dimension, Charles Howard Hinton, invented the terms “Ana” and “Kata” to denote the two movements along a fourth-dimensional direction. He was also the inventor of the term “tesseract” for a four-dimensional cube.

Further Reading:

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

https://publicdomainreview.org/essay/notes-on-the-fourth-dimension

https://en.wikipedia.org/wiki/Four-dimensional_space

https://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Hinton%2C%20Charles%20Howard%2C%201853%2D1907

https://www.sciencefocus.com/science/the-four-dimensional-life-of-mathematician-charles-howard-hinton/

https://paw.princeton.edu/article/charles-howard-hinton-he-wrote-science-fiction-genre-existed

https://en.wikisource.org/wiki/What_is_the_Fourth_Dimension%3F

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

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The Circumplex Model of Affect

James Russell developed a circumplex model of emotion, not to be confused with the interpersonal circumplex. (There are a variety of circumplex models for various subjects, since they are merely circular and continuously scaled in nature.)

This model is also called Emotional Valence and Arousal, where Valence ranges from Negative to Positive (or Unpleasant to Pleasant) and Arousal ranges from Low to High (or Mild to Intense). We get the fourfold partitions of

  • Mildly Unpleasant
  • Mildly Pleasant
  • Intensely Unpleasant
  • Intensely Pleasant

or

  • Low & Negative
  • Low & Positive
  • High & Negative
  • High & Positive

There are several way to discretize this circumplex into eight parts, but here is one from Russell (1980).

  • Aroused
  • Excited
  • Pleased
  • Contented
  • Sleepy
  • Depressed
  • Miserable
  • Distressed

Compare with the concept of Flow, where the variables are challenge and skill, instead of arousal and valence.

Further Reading:

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

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

https://www.google.com/search?channel=cus5&source=univ&tbm=isch&q=circumplex

James Russell / A Circumplex Model of Affect, J. of Personality and Social Psychology 1980, Vol. 39, No. 6, 1161-1178

James A. Russell, Maria Lewicka, Toomas Niit / A Cross-Cultural Study of a Circumplex Model of Affect, J. of Personality and Social Psych. 1989, Vol. 57 No. 5, 848-856

Jonathan Posner, James A. Russell, Bradley S. Peterson / The circumplex model of affect: An integrative approach to affective neuroscience, cognitive development, and psychopathology
Dev. Psychopathol. 2005, 17(3), 715-734

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2367156/

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Eight Days a Week

Eight days a week
Is not enough to show I care

— The Beatles

No, I’m not trying to convince you that we should all change to an eight-day week, and join all the other myriad and failed attempts at calendar reform. But just look at what we have in the Gregorian Calendar: seven day weeks, months of various sizes, days of the month falling willy-nilly on random week-days, and even extra days thrown into the calendar year because of day and solar year mismatch. And the moon’s phases also have conditioned the rough size of months, without helping to show us where they occur.

Instead I’m thinking about a change in calendar representation. It all started with the diagram above showing the days of the week and the sun, moon, and planets associated with them. Ordinary calendars show weeks as seven rectangles in a row for the days of the week labeled by the days of the month. But each day looks the same, except for their place in line along the row. It’s hard to remember the dates of each day, week after week.

Having some success using the non-circular diagram to help remember the Zodiac and their houses (which is pretty old anyway), I thought I might represent each week of Sunday through Saturday as an eight-fold with a blank, with the days of the week in specific places. Since months consist of 28 days or more, four eight-folds (E) could show the bulk of the days for each month (E1 through E4). Some days might come before the first Sunday (in E0), and some days might come after the last Saturday (in E5). Sometimes E1 and E4 may be short a few days as well.

For example, here is January 2022. By my nomenclature, E0 has just the 1st. E1 has the 2nd through the 8th, then E2 has 9-15, E3 has 16-22, E4 has 23-29, and E5 has 30-31.

Months may be joined end to end in a descending series, with four extra weeks (as Es) occurring here and there as need be. After all, for 7 day weeks, having 13 months of 28 days each would be ideal since 28*13 + 1 = 365. Why go to all this trouble when it is easier to look at a regular calendar? I think it is a bit easier to remember dates with this method, but I’m really just beginning with this idea. I’ll try to post a picture of February 2022 soon.

Further Reading:

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

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

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

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

https://en.wikipedia.org/wiki/Symmetry454

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

https://encyclopedia.pub/1433

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The Eight Legged Essay

Further Reading:

https://en.wikipedia.org/wiki/Eight-legged_essay

http://www.chinaknowledge.de/History/Terms/baguwen.html

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

https://ctext.org/confucianism

http://www.chinaknowledge.de/Literature/literature.html

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The Four Elements of Thinking

While searching for something else I chanced upon “The Four Elements of Thinking: Reasoning, Creativity, Synthesis, Evaluation” by Benjamin Cheung, Ph.D., published in 2019. Any book about the four classical elements being used as metaphors for other things is of personal interest, plus that the things being four aspects of thinking was immediately intriguing. Dr. Cheung divides thinking into the four aspects of the subtitle, plus he divides each aspect into four sub-aspects or skills as follows:

    • Reasoning (Earth Thinking)
      • Evidence
      • Inductive Reasoning
      • Deductive Reasoning
      • Abductive Reasoning
    • Creativity (Air Thinking)
      • Investigation
      • Incubation
      • Insight
      • Innovation
    • Synthesis (Water Thinking)
      • Linking
      • Perspective
      • Synthesis
      • Pivots
    • Evaluation (Fire Thinking)
      • Decisions
      • Judgments
      • Contingency Plans
      • Validation

Note his choice in assignment of the Classical Four Elements to each of these aspects of thinking. I would have assigned them differently: Reasoning to Fire, Creativity to Earth, Synthesis to Air, and Evaluation to Water. The price of the e-book is reasonable, so I may investigate the “thinking” behind his alignments by purchasing and reading further.

He also has an interesting book on collections of ideas which he arranges into a “periodic table,” which might be analogous to a more modern scientific elemental assignment. Additionally, he has had Kickstarters on playing or flash cards for both books, which is a notion somewhat dear to my heart (See A Game of Fourfolds).

I have mentioned thinking or thought often in this blog, and believe that poor thinking or irrational thinking is greatly to blame for many of our current ills. Blame can also be attributed to poor communication skills. The best thinking can be obscured by poor communication. What is the best theory of the linkage between thought and language?

Further Reading:

Benjamin Cheung, Ph. D. / The Four Elements of Thinking: Reasoning, Creativity, Synthesis, Evaluation

Link to Amazon

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

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Maxwell Relations

Maxwell Relations are commonly known as a set of four partial differential equations between four thermodynamic quantities or potentials: pressure (P), volume (V), temperature (T), and entropy (S). So for example the expression (∂T/∂V) |S  means the partial derivative of T with respect to V while keeping S constant.

    • (∂T/∂V) |S = -(∂P/∂S) |V
    • (∂P/∂T) |V = (∂S/∂V) |T
    • -(∂S/∂P) |T = (∂V/∂T) |P
    • (∂V/∂S) |P = (∂T/∂P) |S

In my diagram above, the expressions that are equal are on either side of the common leg of adjacent isosceles right triangles.

Further Reading:

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

https://johncarlosbaez.wordpress.com/2021/09/17/maxwells-relations-part-1/

https://johncarlosbaez.wordpress.com/2021/09/18/maxwells-relations-part-two/

https://johncarlosbaez.wordpress.com/2021/09/22/maxwells-relations-part-3/

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

Thermodynamics and the Four Thermodynamic Potentials

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John Buridan’s Octagon of Opposition

Medieval logician John (or Jean) Buridan was a scholar of Aristotle, and wrote many works of commentary and elaboration on Aristotelian philosophy. Several items in logic and philosophy are tied to Buridan (such as Buridan’s Bridge and Buridan’s Ass) but he may now be more widely known for his Octagon which combines Aristotle’s Square of Opposition with a Square of Modality.

Below and to the right is a fourfold diagram of Aristotle’s Square of Opposition. The modern universal and existential (or particular) qualifiers are ∀ (meaning All) and ∃ (meaning Some), respectively. Also in these diagrams, ¬ means logical Not.

  • ∀ S are P
  • ∃ S are P
  • ∀ S are ¬ P
  • ∃ S are ¬ P

Next I show a fourfold of modal operators and their equivalents. The modern modal symbols are (meaning Necessarily) and or ◊ (meaning Possibly).

  • P ≡ ¬ ◊ ¬ P
  • ◊ P ≡ ¬ ¬ P
  • ¬ P ≡ ◊ ¬ P
  • ¬ ◊ P ≡ ¬ P

 

Further Reading:

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

https://plato.stanford.edu/entries/buridan/

https://plato.stanford.edu/entries/medieval-syllogism/#JohnBuri

The Art of the Syllogism

https://1000wordphilosophy.com/2018/12/08/possibility-and-necessity-an-introduction-to-modality/

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

https://plato.stanford.edu/entries/logic-modal/

https://plato.stanford.edu/entries/modality-varieties/

https://lirias.kuleuven.be/retrieve/517366

Click to access Demey_DWMC2015_Buridan_Avicenna_slides.pdf

Click to access Buridan_Octagon.pdf

Click to access hughes-buridan.pdf

https://en.wikipedia.org/wiki/Buridan%27s_ass

https://en.wikipedia.org/wiki/Buridan%27s_bridge

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

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Humankind, Unbound

Many books detail how we are our own worst enemy, how we are too smart for our own good, or not smart enough in the right way. This one traces the origin and evolution of eight technologies that make us who we are today, for better or worse.

  • Fire
  • Tools (incl. Digging tools, Weapons)
  • Language (incl. Writing, Music, Art, Symbolic Communication, Ethnicity, Culture)
  • Clothing
  • Shelter
  • Farming (Agriculture, Husbandry)
  • Machines (incl. Ships, the Wheel, and on to Precision Machinery)
  • Computers

Instead of having two technologies, “Clothing” and “Shelter,” Currier combines them, and lists “technologies of interaction” which includes writing, water-craft, the wheel, etc.

Can we overcome the myriad conflicts that threaten our survival? Please check back in a hundred years to see how we’re doing! It’s not very long, compared to how far we’ve come.

Further Reading:

Richard L. Currier / Unbound: how eight technologies made us human, transformed society, and brought the world to the brink

Some reviews:

https://www.tandfonline.com/doi/full/10.1080/25729861.2019.1596211

Book Summary: “Unbound: How Eight Technologies Made Us Human” by Richard L. Currier

Click to access Volume8-Number1-Article7.pdf

Also:

https://equivalentexchange.blog/2016/12/05/the-anatomy-of-technology/

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Hyper-analogies

I’ve written a few times about relating two different fourfolds and juxtaposing or comparing them in one eightfold diagram. A normal analogy is usually written as A:B::C:D, which means A is to B as C is to D. So if you have two such fourfolds, one way to write it as an eightfold is A:B::C:D:::E:F::G:H. I think a good name for this is a hyper-analogy. I couldn’t really find anything of merit about this concept, but perhaps it goes by another name.

I’m not sure what diagram arrangement shows this the best. Some ideas are 1) A-D are arranged vertically (as well as E-H) and both are arranged horizontally (side-by-side), or 2) A-D are the triangles pointing down, and E-H are the triangles pointing up. Or, 3) A-B can be the upper left and C-D the lower right squares, and E-H can be the upper right and lower left squares. Or, one can use the radial eightfold schema, and have 4) A-D are the triangles on the inside, E-H are the triangles on the outside (or vice-versa).

For example, here I show the Four Causes combined with my Structure-Function. Also, please examine my Eightfold Metaphysics, which combines Space-Time-Matter-Energy with Structure-Function. Additionally, another I like is the one for Tanabata, which relates the astronomical with the symbolized. I could go on and on, and probably will.

Further Reading:

https://equivalentexchange.blog/2019/02/01/an-eightfold-metaphysics/

https://equivalentexchange.blog/2019/07/03/tanabata-v2/

https://equivalentexchange.blog/2019/04/10/noether-pauli-jung-v2/

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