The study of graphs is the study of nodes connected by edges. Another name for a graph is a network. Mathematically, graph theory is the field of understanding the structure of these graphs or networks.

* * * * *

The physical systems are very interesting because they help us think about how networks are embedded into space. The brain network is another embedded system so we’re very interested in understanding where the physical constraints inside a three-dimensional space inform the process of reconfiguring neural connections. Just as particles in a granular material exert force on nearby particles, parts of the brain tend to connect to other parts that are physically close by.

Image: Example of a neural network. Published 2006-12-07 | Author: Kjell Magne Fauske [Source]

**Loose ends: Routes, results and commutation**

(continued from here)

**As an example of a context in which commutation fails** consider the transforms of cube inversions. To clarify, we have already demonstrated that commutation in the narrowly focused mathematical sense *does* hold true in this context. But here we are considering the larger more inclusive case of objective reality that physics and the sciences in general are concerned with.

**Referring to the diagram above** and focusing here on the forward facing plane we can see that **WIND** times **MOUNTAIN** equals **FIRE** and also that **MOUNTAIN** time **WIND** equals **FIRE** as well. So the purely mathematical definition of commutativity is satisfied. But hold on a moment. Consider how we got from *here* to *there* and whether that makes a difference in the real world, the world we actually live in.

**When WIND as operator** acts upon **MOUNTAIN** as operand in the operation of dimensional multiplication it causes **MOUNTAIN** to move horizontally one step to the right to **FIRE**. When **MOUNTAIN** as operator acts upon **WIND** as operand, however, it causes it either to move first horizontally one step to the right to **HEAVEN** then down one step to **FIRE** or down first one step to **MOUNTAIN** then horizontally one step to **FIRE**. Another alternative, given sufficient force, would be to cause **WIND** to move along the diagonal down and to the right simultaneously to **FIRE.**

**Pure mathematics would have us believe** that none of this makes a difference. In other words, all these real world differences make no difference. Really? How is that? Solely because pure mathematics has decreed it so in its rule book which it is ever so careful to maintain as internally self-consistent.

**Mandalic geometry, however, is concerned with that real world** that pure mathematics would have us ignore because it is a hybrid discipline the allegiance of which is to the way things actually work rather than to the preconceived notions of a self-consistent book of tautological rules. For that reason it insists that the route taken in arriving at a destination can be as important or more so than the destination itself. Whether or not this is true in a particular case can only be determined by experiment and experience. There is no rule book that will reveal real world truth for every occurrence or all parameters of exploration.

**Science often marvels** at how mathematics so successfully applies to the real world. It would do well to take note of the many situations where this platitudinal truism falls far short of the truth that science seeks.

(continued here)

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Physicists have now shown how to encode three quantum bits, the kind of data that might be used in the computers of tomorrow, using just two photons

Image: Simple qubits. By Clemens Adolphs (Drawn by myself with Inkscape) [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

Uri Treisman, 1993

(via desparatus)

**Loose ends: ****Commutation’s dirty little secret**

(continued from here)

**If we accept as definition of the commutative property** the misleading simplistic statement that a binary operation is **commutative** if changing the order of the operands does not change the result then we too must number ourselves among the many sheep of the world who do not object to having the wool pulled over their eyes. Though the statement may well be true in a narrow mathematical sense it fails the more inclusive test of what in fact occurs in physical reality. *That* is the condition we must insist on if we wish all our prized philosophies not to degenerate into tautological redundancy.

**There is another definition sometimes given** describing commutation which at first seems to be saying the same as the definition above. It states that being commutative means having the property that one term operating on a second is equal to the second operating on the first, as a × b = b × a. The difference between the two definitions resides in the nuance of different significance between the terms “is equal to” and “the result”. This second definition is not even true in the narrow mathematical sense if by “is equal to” it intends “is the same as”. But in erring it does us the favor of revealing the conflationary sleight of hand performed by mathematics in its manner of usage of the concept of *commutation*.**(1)**

**We have already agreed** that all scalar numbers do indeed commute in the operation of multiplication. The two points mandalic geometry takes issue with are

- that vectors or dimensional numbers necessarily commute in the same manner either in multiplication or addition
- that the
*result*of an operation is the*only*factor of importance to take into consideration in the real world

**To clarify the second objection,** in physical processes the route(s) taken in arriving at a result are often as important as and at times more important than the actual result itself. Pure mathematics, unconcerned as it is with the world of objective reality, would have us deny this experiential truth. Physics and all the sciences in general must remain constantly vigilant and be wary about what particular aspects of mathematics creep into their intellectual disciplines.

**In the next post** we will demonstrate how *route* and *result* are related in inversion transforms of the cube and show how the mathematical definition of commutation fails in that context.

(continued here)

**Image:** *Illustration from “SLEIGHTS: A Number of Incidental Effects, Tricks, Sleights, Moves, and Passes (1914)” by Burling Hull* (found here)

**(1) Somewhat ironically** one of the definitions of *commutation* offered outside the field of mathematics is

*(n.)*A substitution, as of a less thing for a greater, esp. a substitution of one form of payment for another, or one payment for many, or a specific sum of money for conditional payments or allowances; as, commutation of tithes; commutation of fares; commutation of copyright; commutation of rations. [Source]

In other words *commutation* sometimes signifies the return of that which is inferior in place of the original. An interesting definition, one not specifically directed toward usage of the same term within mathematics but tantalizingly apropos nonetheless.

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

**Loose ends: Notation, notation, notation**

(continued from here)

**Before we begin to consider the transforms** of cube inversions there are a few loose ends we still need to address.

**First, we need to make perfectly clear** that with appropriate maneuvers everything that can be done with Taoist-notation mandalic coordinates can also be accomplished with Indo-Arabic-notation Cartesian coordinates. Just the difficulty involved is such that it seems hardly worth the while.**(1)** Use of the Taoist notation not only makes the necessary mental manipulations much easier, it also brings to light a considerable number of very important relationships of parts within the whole which otherwise pass unnoticed.**(2)**

**Both Cartesian and Taoist coordinate systems** are positional notational systems. They both assign different dimensions to different positions in the respective notational structure. Cartesian ordered triads, for example, order dimensions from left to right; Taoist trigrams, from below to above. If we view the horizontal x dimension as dimension one, the vertical y dimension as dimension two and the forward/backward z dimension as dimension three then Cartesian notation represents this arrangement of dimensions as **x,y,z** (e.g., **1,-1,1**) by convention while Taoist notation by convention represents the same by , the trigram **FIRE**. All eight correspondences are shown in the diagram above.**(3)**

**If one learns to associate the notational form** of a trigram with both its English name assignation and its family association in the group of trigrams mental manipulation becomes quite easy. The same cannot be said about the Cartesian ordered triads no matter what one does with them to attempt easily performed mental jugglery. Once the mechanism which deals with composite dimension is introduced and we begin to consider the hexagrams, the six dimensional figures of Taoist notation, the I Ching and mandalic geometry, this critical difference in difficulty of usage between Cartesian coordinates and mandalic coordinates becomes all the more apparent and significant as it increases, one might say, almost exponentially.

(continued here)

**(1) The caveat here is that some minds** work very differently from my own. Of these there may well be a few that could manipulate the Cartesian notation as easily as the Taoist. The point is that I believe most minds will find Taoist notation far easier for manipulation of dimensional numbers than Cartesian notation. This will become readily apparent as we begin to explore the inversion groups of the 3-D cube and symmetries and asymmetries of the 6-D hypercube or mandalic cube. As a historical parallel, if you are a merchant or banker living in 16th century Europe do you elect to do your sums and products with Roman notation numerals or Indo-Arabic notation numerals? (Hmm … MCMLIV times MDCLXXVI or 1954 times 1676?)

**(2) There are, for instance, important relationships** of symmetry and asymmetry existing among the eight trigrams which are overlooked when considering the eight Cartesian ordered triad analogues because of the difficulty in identifying them which is a direct result of the notation system employed. This is no slight matter as it has serious repercussions in the manner in which modern physics, particle physics in particular, proceeds with its investigations (or in some cases fails to do so.)

**(3) It should be noted here that the Taoist notation** natively deals only with the sign or direction portion of a vector, not with the magnitude portion. All the rectilinear relationships involved in the Taoist notation can be thought of as having a magnitude equal to unity or absolute value of 1. But where necessary or desired a combination of Taoist and Indo-Arabic symbols can be used to represent both direction and magnitude greater than absolute value of 1. The advantage Taoist notation offers pertains uniquely to the direction(sign) portion of vector numbers. It focuses one’s attention on the interrelationships of different dimensions in a way Cartesian notation fails to do or does so weakly as to pass unnoticed.

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)