Category Archives: eightfolds

The Main Sequence of Star Types

I will scan your star charts.

— Nomad, from Star Trek “The Changeling”

While reading “Entering Space” by Robert Zubrin, I chanced upon the subject of stellar classification. The Morgan-Keenan system goes from O-type, the hottest, to M-type, the coolest (Red dwarfs). Our own sun, Sol, is an example of a G-type star, which are on the cool side yet still hot and bright. Later, other types have been added, for example White Dwarfs are known as D-types. If humankind will someday journey to remote stars, it’s best to memorize this handy list!

The Morgan-Keenan system is as follows:

  • O-type
  • B-type
  • A-type
  • F-type
  • G-type
  • K-type
  • M-type

This is sometimes remembered by the mnemonic “O be a fine gal/guy, kiss me.”

Further Reading:



The Millennial Project: colonizing the galaxy in eight easy steps

The eight “easy” steps of the Millennial Project (MP) are:

  • Foundation (Institutions to promote MP)
  • Aquarius (Colonize and utilize Earth’s oceans)
  • Bifrost (To low Earth orbit and outer space)
  • Asgard (Build a large space station)
  • Avalon (Colonize and utilize the Moon)
  • Elysium (Colonize Mars)
  • Solaria (Colonize the Solar System)
  • Galactia (Then on to the Milky Way)

Further Reading:

Marshall T. Savage / The Millennial Project

Robert Zubrin / Entering Space: creating a space-faring civilization

Robert Zubrin / The Case for Mars

Carl Sagan / Pale Blue Dot: a vision of the human future in space

MP reminds me a bit of Olaf Stapledon:


Linear Logic and the Four Elements

Here is an alignment between two of my favorite topics, the four operators of linear logic and the four elements.

I’ve been wanting to create this eight-fold for a while, and so here it is. I think it looks rather nice.

At this point I should present my reasons for this symbolic amalgam, but I leave it up to you, dear reader.

However, I will write the names of the symbols starting with the upper left and going widdershins…

  • Fire / With
  • Earth / Plus
  • Water / Times
  • Air / Par

[*5.188, *11.26, *13.82]


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:




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:






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


  • 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:

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


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:





The Eight Legged Essay

Further Reading:




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



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:

Thermodynamics and the Four Thermodynamic Potentials

[*12.194, *12.195]