All posts by Martin K. Jones

Complex Numbers and Quaternions

November 29, 2013 Martin K. Jones 2 Comments

Two conceptualizations of four directions in mathematics are Complex numbers and Quaternions. For complex numbers, two of the directions are the opposite or negative of the other two. Complex numbers are like Cartesian coordinates in that they combine an (x,y) coordinate pair of real numbers into one complex number x + yi. Plus, complex numbers can by manipulated by extensions of arithmetic operations, like addition, subtraction, multiplication, and division.
Quaternions are an extension of the complex numbers where the directions are all different, and each direction is perpendicular to the other three. Like complex numbers are a notion of numbers that cover a plane, quaternions are a notion of number that fill four dimensional space. Thus they combine a 4-tuple (w,x,y,z) into one quaternion number w + xi + yj + zk. Like complex numbers, quaternions can be manipulated by further extensions of arithmetic operations.

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

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

[*7.189]

fourfolds, Science, technology

Ohm’s Law

November 8, 2013 Martin K. Jones 1 Comment

Charts for the equations of Ohm’s Law usually consist of the fourfold relation between Voltage (Volts), Current (Amps), Resistance (Ohms), and Power (Watts). For each electrical quantity, there are three equations that represent it in terms of two others, making twelve equations total.

Ohm’s Law is really only between voltage, current, and resistance. Apparently, power is more correctly introduced by the formula for “Joule heating”.

Both voltage and current have been in two fourfolds previously (Four Basic Electrical Components and System Dynamics) but not power and resistance. Well, resistance did make an appearance as a relation between current and voltage.

The twelve equations can be generated by the following identities (where E = voltage, I = current, R = resistance, and P = power):

I*R/E = P/(I*E) = P*R/E^2 = R*I^2/P = 1

http://en.wikipedia.org/wiki/Ohm%27s_law

http://en.wikipedia.org/wiki/Joule%27s_first_law

Google search for images for Ohm’s Law.

[*8.4, *8.5, *8.8, *8.9, *8.10]

alchemy, fourfolds, Psychology

The Four Temperaments

November 8, 2013 Martin K. Jones 8 Comments

What would a list of fourfolds be without mentioning the Four Temperaments: Sanguine, Choleric, Melancholic, and Phlegmatic?

Indeed, these personality types from ancient medicine and philosophy are still a source of classification and inspiration for modern psychology, even though the reasons for the differences (the four humors) has been discredited. The four humors were blood, yellow bile, black bile (they were big on bile), and phlegm, and were also related to the Four Seasons and the Four Elements.

Below is psychologist David Keirsey’s modernization of the four temperaments, consisting of Idealist, Artisan, Guardian, and Rationalist.

References:

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

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

http://en.wikipedia.org/wiki/Myers-Briggs_Type_Indicator

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

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

https://en.wikipedia.org/wiki/The_Four_Temperaments_(ballet)

Images of the Four Temperaments.

[*6.122, *6.123, *7.194, *8.14, *8.94, *8.139]

alchemy, Esoteric, fourfolds

A Vision by W. B. Yeats: The Four Faculties

October 23, 2013 Martin K. Jones 2 Comments

A Vision by W. B. Yeats is the focus of a large amount of scholarship for a book on the occult. It looks interesting as a source of metaphor and symbolism, and I hope to return to it at another time. There is an excellent website devoted to Yeat’s Vision, the link of which is below.

Above I show the fourfold of the Faculties: Will, Body of Fate, Mask, and Creative Mind. It is not arranged as recommended by the four elements, but by my whim.

I saw a preview of the recent movie “Kill Your Darlings” which mentioned A Vision and showed the woodcut of the Cosmic Wheel.

Modal Things

October 11, 2013 Martin K. Jones 1 Comment

As a complementary fourfold to Things Happen, here is a combination of Structure-Function and Modal Verbs.

Actions may. Parts can. Structures must. Functions should.

Actions may occur. Parts can vary. Structures must maintain. Functions should perform.

[*7.192, *8.40]

fourfolds

Things Happen

October 9, 2013 Martin K. Jones 3 Comments

When someone says “Things Happen”, we think they are being superficial. That’s because we want to hear an explanation of why the thing has happened. In fact, the “Principle of Sufficient Reason” mandates that everything must have a reason or cause. Still, you can’t argue with the fact that things do happen.

Aristotle described four causes that explain why things happen. These four causes are called the efficient, material, formal, and final causes. This classification was all the rage back before the scientific revolution, but now it has been mostly abandoned. Still, with some modification, I believe it can be useful for the modern world.

Why do things happen? They happen by actions, from parts, into structures, and for functions.

This fourfold is a mixture of two others: Structure-Function and the prepositions I’ve associated with Aristotle’s Four Causes.

http://www.brainyquote.com/quotes/keywords/things_happen.html

http://plato.stanford.edu/entries/sufficient-reason/

Discourse, fourfolds

Modal Verbs

September 23, 2013 Martin K. Jones 4 Comments

This fourfold of modal verbs was mentioned in an earlier post A Story for Everyone, but was not shown explicitly. It consists of can, may, must, and should.

References:

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

[*6.6, *6.36,*6.60, *7.188]

fourfolds, Philosophy

Schopenhauer’s Fourfold Root of the Principle of Sufficient Reason

September 9, 2013 Martin K. Jones 1 Comment

Schopenhauer’s Four Classes that comprise his Fourfold Root of the Principle of Sufficient Reason are diagrammed above. Becoming, Knowing, Being, and Willing are these classes.

From Wikipedia:

In his Translator’s Introduction to Schopenhauer’s The World as Will and Representation, E. F. J. Payne concisely summarized the Fourfold Root.

Our knowing consciousness…is divisible solely into subject and object. To be object for the subject and to be our representation or mental picture are one and the same. All our representations are objects for the subject, and all objects of the subject are our representations. These stand to one another in a regulated connection which in form is determinable a priori, and by virtue of this connection nothing existing by itself and independent, nothing single and detached, can become an object for us. …The first aspect of this principle is that of becoming, where it appears as the law of causality and is applicable only to changes. Thus if the cause is given, the effect must of necessity follow. The second aspect [knowing] deals with concepts or abstract representations, which are themselves drawn from representations of intuitive perception, and here the principle of sufficient reason states that, if certain premises are given, the conclusion must follow. The third aspect of the principle is concerned with being in space and time, and shows that the existence of one relation inevitably implies the other, thus that the equality of the angles of a triangle necessarily implies the equality of its sides and vice versa. Finally, the fourth aspect [willing] deals with actions, and the principle appears as the law of motivation, which states that a definite course of action inevitably ensues on a given character and motive.

States of Matter

August 22, 2013 Martin K. Jones 1 Comment

The four most familiar states of matter are solid, liquid, gas, and plasma. Other states exist (such as Bose-Einstein condensates), but if I talk about them I couldn’t put them into a nice fourfold.

Note that the sequence of plasma, gas, liquid, and solid is arranged in the order of highest to lowest temperature (usually). Compare this to the classical Four Elements of Empedocles (Fire, Air, Water, and Earth) and to the made-up fourfold of Bright-to-Dark, based on light.

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

http://www.chem4kids.com/files/matter_states.html

http://gizmodo.com/there-are-way-more-than-three-states-of-matter-1182677930