Archive for October, 2014

The Four Seasons

October 31, 2014

sq_four_seasonsbA four-fold of Time.

My diagram is so dull compared to others.

Links:

http://en.wikipedia.org/wiki/Season

Images of the Four Seasons.

[*7.194, *8.99]

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The Four Cardinal Directions

October 30, 2014

sq_cardinal_directionsFour-fold of Space.

There are so many images, why not add another?

http://en.wikipedia.org/wiki/Cardinal_direction

Other images of Cardinal directions.

Notes:

Also, the four corners of the world.

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

[*7.122]

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A Game of Four-folds, Part 3

October 21, 2014

geviert_cube

Here’s another idea for the Game of Fourfolds, instead of using hexaflexagons. Since each fourfold can be permuted six ways, how about arranging those in six squares on the faces of a cube?

Here are the permutations of Heidegger’s “das Geviert” arranged on the faces of a cube, if one cuts it out and folds it up properly. Not only is the square the regular polygon of materiality, the cube is the regular polyhedron of Earth, the most material of the ancient four elements.

It might be awesome to have a cube of every fourfold I’ve talked about (at least the good ones), to combine and compare them. Even more awesome are the cubes for the REAL elements I’ve seen when I’ve searched images for “element cube”.

What I want to know is whether the six permutations can be arranged in a symmetric way on a cube?

[*8.81]

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The Collatz Conjecture

October 17, 2014

sq_collatz

I thought I might be continuing my apparent effort to convert all triads into fourfolds, but I’ll take a little break to bring you this mathematical fourfold for the Collatz Conjecture. The Collatz algorithm starts with any specific nonzero integer, then iterates in the following way to turn the number into the next number to operate on, and so on. If X is even, then replace X with X/2. If X is odd, then replace X with 3*X + 1. If you get to 4, then you get 2, then 1, but you next go back to 4. In fact, if you have any power of 2, you quickly drop to 4 and so cycle. The Collatz Conjecture is that if you start with any number, then you will eventually reach 4. (Actually the conjecture is that you will reach 1, but since I’m partial to 4, I’ll state it in this way since it’s pretty much the same thing.)

The conjecture is not proven, but has been shown to be true for every number up to some very large numbers. Some numbers can jump around from larger to smaller to larger again, or up and down and back up, for quite a few steps before being reduced to the 4-2-1-4 sequence. That’s why these sequences are also called hailstone numbers.

One might think of the Collatz algorithm as a model of a toy universe in the following way. The procedure starts with a given number, which for the toy universe would be its initial state. As time moves from instant to instant, the procedure operates to halve the number, or multiply by three and add one. So instant after instant the procedure operates, making the number smaller, or larger as required. The conjecture, if true, would mean that no matter how it starts, our toy universe would always wind down and cycle endlessly through the sequence of 4-2-1-4-2-1-4-2-1-4…

http://en.wikipedia.org/wiki/Collatz_conjecture

http://xkcd.com/710/

[*8.88]

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Plato’s Chariot

October 13, 2014

sq_plato_chariotAt the risk of being labeled a neo-platonist, another triad of Plato’s that is often discussed is that of the Allegory of the Chariot. This analogy is supposed to bring insight into the workings of the human soul and consists of two horses, one good and one bad, and the charioteer whose duty is to control them. You never hear about the chariot itself (but it is always pictured), but it is required to have a chariot, after all. The charioteer isn’t just standing on the backs of the horses, like Jean-Claude Van Damme doing his epic split, although that would be cool. (They do this at the circus, and I know I’ve seen it in old gladiator movies when the chariot loses a wheel and the charioteer has to cut away the chariot.)

Thus, unless you want to change the nature of the analogy, the chariot is required for everything to be connected together. This fourth material component completes the triad into a fourfold, and I place it at the lowest, fundamental position where I added The Real to The Beautiful, The True and The Good.

And of course everyone knows that Bad Horse is the Thoroughbred of Sin!

http://en.wikipedia.org/wiki/Chariot_Allegory

[*8.86]

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Plato Redux

October 9, 2014

sq_plato_redux

What is, is The True
What is, is The Good
What is, is The Beautiful
What is, is The Real

— Anonymous

I recently finished reading Rebecca Goldstein’s Plato at the Googleplex: Why Philosophy Won’t Go Away. In it there was much talk of The Beautiful, The True and The Good.

Besides Plato’s Divided Line, which was mentioned in The Republic and consists of four parts, the threesome of The Beautiful, The True and The Good is mentioned in various dialogues.

Being the quadraphile I am, I thought adding The Real to the threesome makes the now foursome nicely balanced. Usually one hears of just the three, without the fourth, but why is that?

Some argue loud and long that The Real has no part in this threesome of Universals, that the three are sufficient among themselves. Others disagree. Which side would you say I’d be on?

http://en.wikipedia.org/wiki/Transcendentals

http://en.wikipedia.org/wiki/Analogy_of_the_Divided_Line

http://www.warren-wilson.edu/~dmycoff/plato.html

Henry Rutgers Marshall / The True, The Good and the Beautiful (The Philosophical Review, Vol. 31, No. 5, Sep. 1922 pp. 449-470)

Michael Boylan / The Good, the True and the Beautiful

[*4.82, *8.72, *8.82]

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Hayden White’s Metahistory

October 3, 2014

sq_metahistory

In 1973, historian Hayden White published Metahistory: The Historical Imagination in Nineteenth-century Europe. His key fourfold was that of literary emplotments due to Northrop Frye: Romance, Comedy, Satire, and Tragedy. White also derived a synoptic table that associated other fourfolds by Stephen Pepper (Organicist, Mechanicist, Formist, and Contextualist), tropes (Metaphor, Metonymy, Synecdoche, and Irony), plus various modes, ideologies, representational historians, and key philosophers.

I’ve had this fourfold sitting around for a while, and I haven’t written anything about it, because I don’t really agree with the synopticisms as given. For instance, I would pair Romance with Organicism, Comedy with Formism, Satire with Contextualism, and Tragedy with Mechanism. Why? Romance is the “drama of self-identification”, as the organism is self-identified, being that the “individual part of the whole is more than the sum of its parts”. Comedy is “harmony between the natural and the social”, as Formism is created by social “classifying, labelling, and categorizing” of natural objects. Tragedy is about the “limitations of the world”, as Mechanism is “finding laws the govern the operations of human activities”. Finally, Satire is the “opposite of romance — people are captives in the world until they die”, whereas Contextualism is “events explained by their relationships to similar events”. That last one isn’t very convincing, but I’ve only switched Romance and Comedy, and left Satire alone.

Again, as with my problems with the synoptic table of Arthur M. Young, it would be nice to play the Game of Fourfolds to see if I can find better arguments for my synoptic claims, or to convince myself that the claims of others are better. Of course, it would probably help for me to read the original works, instead of reading summaries. One might care to look elsewhere for better exposition.

http://en.wikipedia.org/wiki/Metahistory

http://www.lehigh.edu/~ineng/syll/syll-metahistory.html

Perhaps I could elaborate further at a later time:

Romance: The individual succeeds.
Comedy: Most succeed. Society wins.
Satire: Most fail. Society comes up short.
Tragedy: The individual fails.

[*3.172, *3.173]

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