The word cloud features a
sample of 100 word pairs (called bigrams) taken from Jake’s speech as
determined by how often each word appears next to each other. If a word
is very rare, it will make the two words it appears next to more unique
statistically as a bigram; thus a lot of very rare bigrams appear that
are often unexpected. Pairs are also more correlated (I’ll call this bigram measure “correlatedness” in future posts to avoid confusion with vocabuary term “distinctiveness”) if they occur in a
set phrase like “auto responder”.

Some of these are frankly hilarious, like “clean trousers” and “dream towels”. Also observe “weekend at” and “at bernies” which are obviously part of a three-word sequence.

Planar two-dimensional numbers exhibit three out of the four categories of multiplication found in the three-dimensional trigrams. Importantly, that is one more than the one-dimensional number line posesses. And the two-dimensional plane is significantly the domain in which squares and square roots naturally reside.^{[1]}

The two multiplication categories identical for the trigrams of the cube and the bigrams of the square are those of ortho-multiplication and auto-multiplication. Meta-multiplication and para-multiplication are, in a sense, telescoped in two dimensions into a single operation. This third category shares with para-multiplication of cubic trigrams the fact that it involves complementary pairs, but only of two-dimensional bigrams now because the third dimension of cubic space is suppressed.^{[2]}

In the planar context, the identity element of multiplication is OLD YANG, the bigram having two yang lines. The inversion element is OLD YIN, the bigram having two yin lines. These are complementary bigrams occupying meta-positions to one another. Two hybrid bigrams, YOUNG YANG with its yang line below, in the horizontal dimension, and YOUNG YIN with its yang line above, in the vertical dimension, are also complementary bigrams occupying meta-positions with respect to each other.

OLD YIN as operator produces inversion of both Lines of any bigram it is multiplied with, including itself. As the identity element, OLD YANG has no effect at all. YOUNG YIN as operator produces inversion of only the first dimensional lower Line of the bigram it acts upon, while YOUNG YANG produces inversion of only the second dimensional upper Line. The multiplication operations of planar unit vector numbers described here, functioning in an extended real plane, are the operations that are intended by mandalic geometry to supplant imaginary and complex numbers.^{[3]} Demonstration of the full power of this method though will not be unleashed until we cover the topic of composite dimension more completely.^{[4]}

In the next post we will consider the important ways in which all of this impacts the matter of square root.

[1] Notwithstanding the insistence of 17th to 21st century mathematics that they somehow inhabit the one-dimensional number line.

[2] The term suppressed is more correctly used here I think than any term suggesting absence because the two dimensions of the plane still exist within context of the three dimensions of the cube and, indeed, within the context of even higher dimensions. Referring back to the diagram here it should be clear that the four bigrams exist still within the trigrams of all six planar faces of the cube. They are hidden, so to speak, in plain sight. They play an essential role, however, in the transitional dynamics of the geometric structures of any dimension number of which they are a part, also of the transitional energetic dynamics of material particles they are intended to represent.

[3] The eight trigrams perform an analogous though slightly more complex role in the three-dimensional geometry of the cube. In that higher dimensional context, transitions occurring among eight octants of 3-dimensional space are coordinated; here transitions among four quadrants of 2-dimensional space are.

[4] Composite dimension extrapolates the real plane to a four-dimensional spacetime configuration which can, however, be constructed in a manner fully commensurable with the ordinary two-dimensional Cartesian plane.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

As part of the group project I’m doing on bigrams (two word sequences) in Homestuck dialog, I was comparing shared bigrams between Dave, John, Karkat, Vriska (the four big speakers), Jake, and Aranea. Aranea had the most bigrams (43) in the top 100 most frequent ones with Jake with the second most (17), while Dave had the fewest bigrams not shared (9). This is really fascinating because it suggests the uniqueness of Jake’s speech style in a way that lexical density (number of content word entries to total words) cannot do.

We previously introduced the wildcard symbol *. This wildcard stands for either yin or yang. Use of such a symbol enables conceptual placement of any geometric figure of a lower dimension within a higher dimension context without an exact specification of the vector direction (yin or yang value) of the Line(s) or dimension(s) where it is found. In bigrams, trigrams, tetragrams and hexagrams the symbol can be used in any or all of the Lines or Places.^{[1]} Its use in a Line is not conditional on what appears in any other Line of the figure. It represents a kind of fuzzy logic (1, 2, 3, 4, 5, 6, 7), which at times can be quite useful.

Here we are introducing a different wildcard symbol, one that is conditional. It is dependent upon the Line type in the second Line of the bigram containing it.^{[2]} The two must be complementary to one another. If one is yin, the second must be yang, and vice versa. A bigram with two such Lines can resolve into either of two possible variant forms, which we have already identified as YOUNG YIN and YOUNG YANG. Both of these can be viewed as zero alternatives (or as zero equivalents, but only in a vector direction sense.) The symbol we’ll use for this wildcard is - - - .^{[3]}

To be clear here, this type wildcard has application only in contexts of composite dimension where two Lines exist linked as a bigram. These include the basic two-dimensional bigram, having two Lines which refer to a single Cartesian coordinate in a one-dimensional line; the tetragram, a four-dimensional construct, having a pair of bigrams, which together refer to a two-dimensional Cartesian ordered pair; and the hexagram, a six-dimensional construct, with three bigrams that together reference a three-dimensional Cartesian ordered triple.

The critical idea here is that the two Lines are necessarily inversion complements of one another. In a sense the two Lines are entangled, and the bigram containing them is multipotent, capable of differentiating into either of two possible inversion forms, both of which occupy the identical Cartesian location and always a location designated partly or entirely by a Cartesian zero or zeros.^{[4]} This in effect transforms the Western null zero into a multi-potential element of a multiple-valued logic, more promising for use in quantum logistics than classical two- or three-valued logics.^{[5]}

The bare bones composite dimension line of mandalic geometry can then, using our new wildcard symbol, be diagrammed as

This line represents the mandalic geometry alternative to the Cartesian x-dimension horizontal axis. Here OLD YIN replaces Cartesian x = -1; OLD YANG replaces Cartesian x = +1; and the wildcard bigram in the center, representing both YOUNG YIN and YOUNG YANG, takes the place of Cartesian x = 0. This is a pure abstraction as it is taken out of context. In the next post, we will begin investigation of this line as well as other composite dimension lines in their native contexts. Woo hoo!!

(to be continued)

Notes

[1] The * symbol is not found in the I Ching, nor is it part of Taoist thought, at least never in any explicit manner. I believe though that such a usage is implied in the Taoist system, and have elected to use it in mandalic geometry because of its considerable practicality. See, for example, how the * symbol is used here to reference a third dimension without stipulating vector direction.

[2] The term bigram is used here in its holistic sense as being any two related Lines in a trigram, tetragram, or hexagram. In context of the hexagram the term generally refers to the pairs of Lines 1 and 4, Lines 2 and 5, and Lines 3 and 6 which are the Line pairs that are used to generate composite dimensions.

[3] This symbol is not found in the I Ching either, however it is used in a later related work, the Tài Xuán Jīng (“Canon of Supreme Mystery”, Chinese: 太玄經) which was composed by the Confucian writer Yáng Xióng (53 BCE-18 CE). The first draft of this work was completed in 2 BCE. This text is also known in the West as The Alternative I Ching and The Elemental Changes. The symbol is used differently here from the way it is in the Tài Xuán Jīng. In that Confucian classic it is used to represent man as distinct from heaven and earth. We use the symbol here, always in a set of two paired Lines, to denote the zero alternative of composite dimension.

[4] This would seem to indicate that the symbol or one similar could be used to represent the analogous quantum entanglement of physics. It might also find use in quantum computing as it is a kind of qubit. For that matter, OLD YIN and OLD YANG bigrams are as well, although of a different nature. Of interest here also are the Wikipedia articles on Three-valued logic, Many-valued logic, and Ternary computer.

[5] In any geometric line segment of six-dimensional composite geometry four discretized locations exist, determined by the resident bigram(s). Three-dimensional Cartesian coordinates admit just three (-1, 0, +1) corresponding locations similarly discretized that must suffice if the two geometric systems are to be made commensurate. Although YOUNG YIN and YOUNG YANG both appear at Cartesian zero (0), they exist at two different locations in six-dimensional terms.

It might help here to think of an analogous situation we are all familiar with in the macroworld, the annual cycle of the earth around the sun. We speak of the onsets of the four seasons, related to the angle of the earth’s axis with respect to the sun. These are the winter and summer solstices, which can be compared to OLD YIN and OLD YANG respectively, and the vernal and autumnal equinoxes which can be compared to YOUNG YANG and YOUNG YIN respectively.

We know that the relation of the earth’s axis to the sun is different at the two equinoxes, as is the earth’s position in space relative to the sun. Nevertheless, the two equinoxes are coincident in terms of length of day and night, one of the important dimensions or parameters by which we characterize seasons. Were a schema developed based on this parameter alone and none other, the equinoxes would then coincide in position on such a mapping as the result of loss of input related to other possible parameters.

Something similar happens when we superimpose six-dimensional coordinates on three-dimensional coordinates making them commensurate in order to form a composite 6D/3D mandalic geometry. Something is lost in translation, but that something seems inexpressible in terms of three dimensions. This simply means that six-dimensional reality is the greater reality.

For another take on an endeavor to synchronize disparate events that are not completely based on identical parameters read about the ongoing attempt to keep our measurement of time commensurate with the actual passage of time (1, 2, 3). Or rather, with what we think of as actual time. We have this traditional link of time to astronomy but the astronomical events we’ve traditionally linked time to are all Earth-oriented. Those correspondences would have little meaning to any inhabitants happening to live on a planet in the Alpha Centauri system, which is the closest star system to our own. This exposes again the fundamental truth that conventional standards are generally to some degree limited in their applicability to a particular contextual locality in time and space. Specific context, therefore, must always be taken into account when dealing with things we believe in some manner connected, particularly when conventions are involved.

Related here also is this excerpt from the Wikipedia article on Synchronization:

In electrical engineering terms, for digital logic and data transfer, a synchronous circuit requires a clock signal. However, the use of the word “clock” in this sense is different from the typical sense of a clock as a device that keeps track of time-of-day; the clock signal simply signals the start and/or end of some time period, often very minute (measured in microseconds or nanoseconds), that has an arbitrary relationship to sidereal, solar, or lunar time, or to any other system of measurement of the passage of minutes, hours, and days.

In a different sense, electronic systems are sometimes synchronized to make events at points far apart appear simultaneous or near-simultaneous from a certain perspective. (Albert Einstein proved in 1905 in his first relativity paper that there actually are no such things as absolutely simultaneous events.) Timekeeping technologies such as the GPS satellites and Network Time Protocol (NTP) provide real-time access to a close approximation to the UTC timescale and are used for many terrestrial synchronization applications of this kind.

The thoughts I’ve just expressed provide only a sketchy analogy and require further refinement certainly, yet they have, I think, some merit and are a start at least toward something that could in time be more effectively developed.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)