Proportionality via Salt

        One other set of salts involves several examples of sulphate compounds, all of whose ratios approximate the Pell ratio of 0.4142135 and 2.4142135. These compounds include: sodium ammonium sulphate (0.4270988 and 2.3413786) and calcium sulphate (0.4172071 and 2.3968911). A close derivative of the Pell ratio is two times the lessor Pell value and one half of the larger, namely: 2 * 0.4142135 = 0.8284271 and 2.4142135 / 2 = 1.2071067. It is reflected in di-potassium sulphate's ratios of: 0.8140171 and 1.2284754, and also in ferrous magnesium sulphate's ratios of: 0.8343517 and 1.1985353, respectively. These sulphate compounds occur naturally on this planet, but like our sodium chloride, they may also be the predominantly soluble substance on other planets. Gypsum (calcium sulphate) is the primary, soluble substance that rests on the Martian surface. Di-potassium sulphate is the soluble substance predominanting on Venus. I postulate that sodium ammonium sulphate was the predominantly, soluble substance on the mythical, fifth planet of our solar system, Maldek, which resided between Mars and Jupiter where now resides the asteroid belt. I further postulate that mammalian bodies on Maldek used hydrogen sulphide in a manner similar to the mammalian bodies on our planet Earth as a regulator of metabolic rate. [Mammalian bodies on Earth use hydrogen iodide that is contained within a compound called thyroxin that is produced by the thyroid gland.]

        This hypothesis describing a distinguishing characteristic among different biologic systems on various planets would not interfere with any attempt by any intelligent specie to contribute its genetic material to neighboring planets. The basic language of human genetics, DNA, would still remain the same. Only the predominant blood and oceanic salt, and a few other characteristics described in subsequent lessons, would differ. Of course, both blood and oceans implies the presence of water, the main precursor to water-based life as we know it. Regardless of the presence or absence of water or biology that is familiar to us, the study of a planet's predominantly occuring, water soluble, binary compound is the starting point for studying that planet's potential, or actual, living systems. From this relationship between a planet's predominant salt and its aesthetic system, numerous other archetypes can be deduced.

        For computational reference, I used the following weights for the atomic elements and their ions:

        You may have noticed that I do these computations as if no water were involved. Therefore, I am presuming that ionization is not occuring to the salt in question. Water would normally lend its hydrogen ion to the base of a binary compound (sodium, for example) and would also lend its hydroxyl ion to the acid side of a binary compound (such as the chloride ion of common table salt) when ionizing a solute. But this would upset the present hypothesis by diverging the computations away from their projected values. Maybe nature doesn't need water (a familiar solvent) to correlate these ratios with its archetypes. Maybe nature only needs a solvent when putting these compounds into its biologic systems?

POSTSCRIPT:
        Di-potassium sulphate may not be a fluke. It may be one example of a subseries of proportions derived from aesthetic ratios by way of integer multiplication (or division --- depending on which ion your proportional skew is favoring). Magnesium chloride may be another. Its proportion is: 2.9173421. This is 9.85% less than two times the golden ratio (3.23606797 minus 0.31872586 -- by the way, this multiple of the golden ratio is one plus the square root of five). Compare magnesium chloride's offset of just under ten percent to sodium chloride's offset of 9.38% (when multiplied by two). There is not much difference between the discrepancies of these two mathematical proportions and these examples of analogous salt proportions. So, maybe integer multiples of proportions are acceptable variants.

        These variants skew proportionality either towards more atomic weight existing with the negatively charged ion, or more atomic weight associatied with the positively charged ion. On Venus, di-potassium sulphate is skewing its multiple of the Pell ratio towards the weight of positively charged (cat-)ions of di-potassium. But here on Earth, if magnesium chloride salt were to be concentrated and extracted from out of seawater (forming a traditional Japanese tofu ingredient called nigari and also found predominating in the solutes of the Dead Sea along the Israeli/Jordanian border), then its proportionality skews itself toward the negatively charged (an-)ions of di-chloride.

        I'll make another presumption at this point. Hyperborea existed as the source for our visions of an originally, enlightened Earth before the fall from grace overtook humanity. [Compare our english word "paradise" to Sanskrit's equivalent of: paradeshya. It presumably existed within the pre-Arctic region where it was originally within the Earth's equatorial belt of that time and also possibly later between the Tigris and Euphrates rivers as depicted by the Book of Genesis. Why else would the Dead Sea be rich in magnesium chloride if the salt weren't used as an agricultural, hygienic, and nutritional enhancer during the Middle Eastern paradisical experiment?] My presumption includes their use of magnesium chloride salt instead of our present-day sodium chloride salt. Although a normal approximation of aesthetic proportions without integer multiplication or division may be balanced, physical balance may tempt humanity to overindulge in excessive weighted, positively or negatively charged, electrical tendencies mirroring the addage: "If there aren't any problems, then let's make some. It'll be more interesting that way." Counter-opposed to this may be the tendency that if a planet's proportional ratio is already skewed, then humanity may be more inclined to put things right within their private and social affairs instead of forever looking for exciting mischief to set themselves off-balance. Maybe the serpent's temptation (or its aftermath) within our Bible's Garden of Eden story involved utilizing sodium chloride instead of magnesium? Since approximately one-sixth of our seasalt is already magnesium chloride, it is possible that our Earth's extremely, early history was originally magnesium before transforming itself into a sodium biosphere. If sodium ions were lurking within the magma, all it would have taken would be a gradual movement of sodium ions out onto the planet's surface during the Earth's active, volcanic stage. This would have increased the level of sodium in the ocean's content while retaining the original magnesium. But this hypothesis would require a revision of when the Earth actually went through its cooling stage, putting the spewing of sodium ions during volcanic eruptions well after the Earth's initial formation, which would (possibly) be further back in time than we presently estimate. Estimation usually is derived from corresponding information, making one misunderstanding progenitor to many.

        In the lessons that follow, it will be shown how golden ratios are the irrational roots of golden polynomials containing single unknowns. It will be further shown how to speculate in numerous ways (but not yet prove --- I'll leave that to others) that golden polynomials, and hence golden ratios, are an infinite class. The degrees (orders) of these polynomials are a simple progression of integers from one to infinity. All of these polynomials exclusively resolve to irrational roots within the field of real numbers. But others following up on the work of Fermat, namely: Gauss and Wantzel, have shown and proved an idea whose consequence is a limit imposed on only five regular polygons (containing golden ratios of their respective polynomials) that are constructible by merely using a straight-edge and compass (the same way the ancient Greeks did it). [A polynomial-in-one-unknown's solvability involves attempting to get that polynomial's singular unknown to one side of an equation, and all by itself, after having first set that polynomial equal to zero.] The golden ratio of our planet emanates from a quadratic polynomial. The quadratic is the second in the series of solvable polynomials. Also, the quadratic is a second degree polynomial. It is the only known golden polynomial in which all of its (two) roots are reciprocals of each other. Being reciprocals, they both share the same proportion. This implies only one proportion exists within our golden world. It also implies the simplicity of attempting to understand how a chemical compound would need to be constructed in order to be this world's precursor. A simple binary compound suffices. To conceive of worlds wherein lie other types of more complicated chemical precursors, requires a little imagination. The Fermat primes are: 3, 5, 17, 257, and 65537. Their correspondingly golden polynomials would be of degrees: 1, 2, 8, 128, and 32768, respectively. [Since you may not have read yet the following lesson(s) that I keep referring to, here is a summary: The Fermat primes are being equated to the number of sides of regular polygons in which contain the irrational roots of their correspondingly, golden polynomials. A regular polygon's number of sides minus one and then divided by two gives the degree of a golden polynomial fashioned from irrational values obtained from proportions existing among certain pairings of diagonals with one another and the polygon's side.] All of these five degrees of golden polynomials are powers of 2 (namely, 2 raised to the power of: 0, 1, 3, 7, and 15 gives the degrees: 1, 2, 8, 128, and 32768, for fermat-primed polygons), so their binary compound(s), and hence electrical potential, will most likely be somewhat similar to our own world (positively and negatively charged ions), but slightly more complicated in their body's mechanisms for maintaining itself. Let's just look at the simplist case greater than our own, degree 8, to speculate its chemical composition. Maybe on such a world, its eight salts would be simultaneously composed of four different bases and four different acids? Each ion might be matched two ways. Each base would then have an affinity toward two acids and each acid would have an affinity toward two bases. Maybe this could be accomplished by a two-way electrical bonding? Or maybe more likely, four acids and four bases have valences (electrical potential) of one, while the other four acids and bases have valences of two? [This could be depicted by the blue acids and bases (in the diagram below) representing ions of a single valence, while ions of a double valence could be represented by the red acids and bases. Our world's salt of sodium chloride is an example of a compound composed of ions having a single valence, each. Epsom (bath) salt, magnesium sulphate,

The Heptakaidecagon, or more popularly, the Heptadecagon
of 17 sides, in which contains the 8-th degree polynomial.

A Two-Way System of Acids and Bases

        After staring at the above image on two-way bondings of four acids and four bases, it dawned on me a better way to represent their relationships is to use the sacred cut diagram of the octagon, also know as the octagram. I didn't want to confuse you (any more than I might have already ;-) by introducing it first, for I was afraid some of you might think that I was implying that the octagon is associated with the 8-th degree polynomial. Directly, it isn't, but for the purposes of this analogy, it is useful. The octagon doesn't contain an 8-degree polynomial of eight, irrational roots (also known as: f proportions). That relationship is shared by the 17-gon and any polygons of more numerous numbers of sides provided that those numbers of sides are a multiple of 17. Well --- without further ado, here is the better diagram of two-way pairings of acids and bases on a world ruled by the 17-gon's aesthetic system:

The Heptadecagon

        Here is a taste of one of the future lessons on the Sacred Cut, now that its image is staring at you from above. One edge-length of the inner square is in proportion to one edge-length of the outer square in the ratio of (1 + Ö2) to 1, or approximately 2.4142135.... [Also known as the Pell ratio. Its reciprocal proportion of 1 to (1 + Ö2) is approximately 0.4142135....] This is called a sacred cut (so named by a Danish observer of Roman archetecture) and a time honored tradition among Renaissance artists and builders, such as Leonardo Da Vinci, and Roman architects planning such buildings as a set of garden apartment complexes, built during the first and second centuries A.D., in the coastal town of Ostia, Italy. [See: Watts, Donald J. and Carol M.; A Roman Apartment Complex; Sci. Am., Vol. 255, no.6, 132-140, Dec. 1986]

        Such a world involving the Pell ratio would have: sixteen astrological signs, sixteen chromatic notes to their musical system of recognition and appreciation spanning a frequency range of 5 to 2 (a major tenth, or a major third plus a perfect eigth) from the (root)-tonic of their scale to the next so-called-octave above it, nine major chakras, 70 letters to their Sanskrit --- but more on these things later......

PS --- For comparison, our world of the quadratically, golden ratio-ed, Earth, has: twelve astrological signs, twelve (or some multiple of twelve, such as: 24) chromatic notes to our Western, and some Eastern style music, spanning a frequency range of two to one (a perfect eigth), seven major chakras, and 50 letters of Sanskrit.
 
 
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Further References:
Yet another internet version of an outake from Leonardo's sketch of Vitruvian Man
The Octagram; a star-polygon from which arises the Pell proportion
Pell Numbers
Search Google for Pell Numbers
John Pell, a short biography

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