Chapter 3: The Golden Class Subjects: introduction,algebra,non-Euclidean geometry,Euclidean geometry Introduction Golden numbers carry within themselves the identity of the set to which they belong: any element within a set of values can manifest all the other elements of its set. So it is not necessary to use all the elements of a set simultaneously. The use of any one element implies the use and existence of all other elements within a set. This holographic sensibility is evident only in golden numbers. Beautiful golden numbers exhibit this quality the most: both algebraically and geometrically. Powerful golden numbers display this only geometrically. If the generic term of beauty is equated with this sense of "belonging", then it makes sense that powerful golden numbers have less of this sense. That is why I sometimes like to refer to powerful golden numbers as being semi-golden in the traditional sense: golden as defined as being equivalent to beauty. Beautiful golden numbers are frequently difficult to define since most exist within irreducible polynomials---polynomials which cannot be factored. But general power numbers can always be written as an algebraic expression: an expression of numbers and mathematical symbols (such as: +, -, *, /) with an emphasis on the use of radical enclosures which make them easy to define---unlike beauty. Golden power expressions exclusively make use of the square root radical. The expanded Euclidean algorithm may be used to seek ratio approximations of a whole set of golden beautiful numbers at the same time. That way, their approximations are always kept relevant to each other. My initial response is to use the word "integrated"; but lest I confuse this topic with calculus (which it may, or may not, have any relation to), I will keep to the use of the term: "relevant". Golden numbers are always irrational, with one exception: the first golden beautiful number is the number 1. The question arises: is there a general class for beauty? Essentially. The difference between general and golden beauty is: the sense for "belonging" is not linked to a unit (of 1) expansion rate, but to a varied, or complicated, rate of expansion---the partial quotients of the Euclidean algorithm may take on any value. The Euclidean algorithm is the algebraic test for beauty. Its output will signify whether a value is golden, or merely generally, beautiful. ************************************************************** The Algebraic Method: Sandwiched-Glass-Panes-and-Reflecting-Light Model H.E.Huntley in his book, The Divine Proportion, presents an imaginary lightbeam model to generate the Fibonacci series. I have taken the liberty to flex the model in order to produce an infinite class of golden (beautiful) numbers. It is done by making the number of sandwiched-glass-panes a variable, rather than a constant of two. His model has only two glass panes and produces the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, ...&. The ratio between consecutive members of the series converges on the golden ratios, 1.6180339... and 0.6180339...: 55/34 = 1.617647... and 34/55 = 0.6181818. The traditional golden ratio is either one of two quadratic irrational solutions of the equation: x^2 +- x - 1 = 0. The +- sign means that the sign for that term doesn't change the absolute values of the solution. Its solutions are: (+-1 +- <5>) / 2. There are other models for generating the Fibonacci series. Fibonacci's rabbit breeding model and the genealog There is one other model as good as this one for having the capacity to express golden numbers as an infinite trend of an algebraic model. That is the expanded Euclidean algorithm. Fibonacci's rabbit breeding model is a mathematical statement that doesn't allow for the production of any other golden numbers without changing the biology of rabbits. There is a genealogical model of the male honeybee which is an exact duplicate of this light model except that it also doesn't allow for expansion to infinity unless something drastic were done to honeybees. Light is so simple a concept, that it allows for a lot of flexibility. This model is one of two for generating golden numbers. A geometric model, which follows shortly after this one, is the second. In chapter ?, in section two, the parallel method for qualifying beauty is a recapitulation of this first method. Using a method for generating numbers does not automatically qualify a number's subjective value. But I am going to try and reason my way through this. ?only The ?first claim is the expansion of tradition. If 1.618 is a golden number, then any expansion of the territory over which this number is created must also be golden if the specific means used to create this number has not been altered. Again, if the expansion is at a global level of the method where something assumed to be fixed (namely, degree of a polynomial, or number of series produced simultaneously by said method) is actually variable, then we have been mistakenly assuming all along that the grand set of numbers known as golden has been restricted by an overlooked constant within a formula. Nobody ever said that one number was "it" for goldenness, but then no one ever had alternate ways to check a claim (two generation methods: algebraic and geome- tric). Also, no grand solution was ever sought because no problem was known to exist. Isn't one golden number enough? The expansion of a vision takes on water the minute people question its utility and connection to what is already known about numbers. But a true vision of formulation for mathematical ideas has always been sought to include variables, exclusively, whenever possible. Now to get on with the first method. It will generate golden numbers in sets of elements from one to infinity whose magnitudes within a set will range from one half to infinity. These will be only beautiful golden numbers. The second method will generate both beautiful and powerful golden numbers. A set of one golden element, [1], is produced by using one glass pane and having light entering from one side (always the same side for simplicity). A table is produced showing the relationship between the number of reflections from zero to infinity off the interior sides (never the exterior), the number of unique pathways the lightbeam may take within each number of reflections, and a genealogical lineage showing how each pathway came into being by linking the interior reflection points as a zig-zag pathway. The lightbeam may exit any glass pane it chooses and then enter and reflect within another, or it may choose to remain within a glass pane to bounce around for any length of time up to its limit, or any combination of the above. Once its reflections are used up it must exit in any direction, but without any further reflections. The interior sides are labeled in the direction of the traveling lightbeam (for the sake of convention): a, b. Exterior sides are not labeled, because the lightbeam is not allowed to bounce off of exterior sides. This particular example will be a simple linear lineage of reflections----linear, in the sense that the lineage does not branch into a family tree, but is more like an ever widening palm tree: Reflections(index) Pathways Lineage 0 1 ------- 1 1 b 2 1 ba 3 1 bab 4 1 baba 5 1 babab 6 1 bababa etc............................................. The series of numbers under the category "Pathways" is our primary series. The series of numbers under "Reflections" may be used as an index to locate any element within the "Pathways" series. The ratio among consecutive elements within the primary series is the primary ratio, or factor of a polynomial: the number one in this case. Since the lineage does not branch, no other auxiliary series may be produced making this a linear golden polynomial of just one factor. Composed in one unknown, the transformation from a set of factor of one element into a polynomial of integer coefficients is: y = the index for an element within a series P = the primary series x = a single unknown for constructing a polynomial The y-th element within series "P" divided by the previous element equals one: P(y) / P(y-1) = 1 An unknown value equals one: x = 1 A beautiful golden linear polynomial: x - 1 = 0 It is so easy to overlook 1 as the first golden number. We do that a lot in mathematics, such as with primes, but that is another story. To continue this trend into the two element set domain, we now enter the example offered by ?, by way of H.E.Huntley. The only addition to this model has been the lineage diagram. This diagram comes from other sources as a modification of the honeybee lineage model turned upside down. But more on bee lineages will follow a little later on. A set of two golden elements, [1.618..., 0.618...], is produced by using two sandwiched panes of glass. Light is still coming in from one side, the left in these examples, and moving towards the right. The interior sides are now labeled: a, b, for the left pane and b, c, for the right pane. Adjacent interior sides always share the same label. The lineage will now branch and each smaller network will be a replica of the whole. That is one reason why these numbers are held to be golden, they have something in common with holograms----their whole being is synonymous with every part. If two lineages are split from the first occurrence at reflections "1", then the total number of pathways may be split into two between the "b" and "c" lineages creating two parallel series of reflections: (index) Total "b" "c" Refl. Paths Line Line Lineage 0 1 0 0 ------- 1 2 1 1 b; c 2 3 1 2 ba; cb, ca 3 5 2 3 bab, bac; cbc, cab, cac 4 8 3 5 baba, bacb, baca; cbcb, cbca, caba, cacb, caca 5 13 5 8 babab, babac, bacbc, bacab, bacac; cbcbc, cbcab, cbcac, cabab, cabac, cacbc, cacab, cacac etc............................................................................. Notice that each of the two sublineages start off their lists with the old familiar palm tree lineage from the previous example: babab... and cbcbc..., which uses a completely repetitious pathway. Other pathways contain smaller limited repetitious patterns. Notice that every "a" ending to a pathway may go to either a "b" side or cross "b" to bounce off a "c" side. The same is true with "c" endings, but in reverse. They may go to either "b" or "a". "B" endings must go back to where they came from, either "a" or "c", forming a mini- repetition. These two sublineages form two parallel series whose parallel index values we will now use to approximate the two factors of a polynomial: y = the index for an element within either series "b" or "c" P = the primary series, series "c" a = the auxiliary series, series "b" x = a single unknown for constructing a polynomial The y-th element within series "P" divided by the y-th element of series "a" equals the primary(first) factor: P(y) / a(y) = 1.6180339... An unknown value equals it: x = 1.6180339... The y-th element within series "a" divided by the y-th element of series "P" equals the auxiliary(second) factor: P(y) / a(y) = 0.6180339... An unknown value equals it: x = 0.6180339... Now we must distinguish among the ordinal elements within the set of factors by alternating the signs: positive for the first and all odd ordered elements, negative for the second and all even ordered elements. It could just as easily be the other way around, negative for odd and positive for even. Not even convention will help here, because this structuring of signs is just for keeping the "polynomial books" straight so that the polynomial's coefficients are all integers. Signs would only get in the way for any usage that we might want to put ratios to, because these values get used in a geometric(multiplica- tive) way, not in an arithmetic(additive) way: x = +1.6180339... or x = -0.6180339... x - 1.6180339... = 0 or x + 0.6180339... = 0 (x - 1.6180339...) * (x + 0.6180339...) = 0 x^2 + (-1.6180339... + 0.6180339...)x + (-1.6180339... * +0.6180339...) = 0 results in a beautiful golden quadratic polynomial: x^2 - x - 1 = 0 Because of the partial arbitrariness of bookkeeping practices, the polynomial is just as valid in the form: x = -1.6180339... and x = +0.6180339... resulting in: x^2 + x - 1 = 0 so: x^2 +- x - 1 = 0 would be a more complete way to put it. Assuming a convention for dividing consecutive elements within a "pathways" series (the "b" or "c" line, for instance) determines which factor is primary. There can only be one, because both "b", "c", and the "total" series tend towards the same geometric(multiplicative) growth rate. It is mostly a question of: do we divide the latter by the former or vice versa? Convention is actually already established as: the latter by the former: P(y)/P(y-1) and a(y)/a(y-1). I don't want to overburden you with redundancies, but to quote a precedence: when we want to establish a length in terms of some unit of measure, we divide(measure) the unknown length in terms of the known unit----not the other way around. When a series is evolving, its latter(unknown) elements will be computed in terms of its former(known). By dividing elements within a series in a similar manner, we are in effect taking an accurate statistical survey of the results of computing that series. Although these series cannot be strictly considered as geometric, they can be thought of as a geometric approximation of an arithmetic, or more pointedly as a algebraic, series. But more on that point when we come to chapter ? on the substitution method, and its consequent serial method, for approximating factors of a polynomial. The structure of these present computations is more akin to a parallel, or simultaneous equations, method for approximating a polynomial's factors----a topic for chapter ?. Both topics will be found in section two. One more example with this model should help fix in our minds the major portion of this method for computing beautiful golden numbers. Within the three element set domain, [2.247..., 0.802..., 0.555...], are the three golden factors of a cubic polynomial. Three panes of glass are sandwiched together with light coming in from the left side. The interior sides are labeled: a, b, b, c, c, d, from left to right. Or more simply: a, b, c, d. It is as if the lightbeam is alternately recognizing and not recognizing if a gap exists between glass panes. If it assumes a gap exists, it will bounce off of it. If it assumes a gap does not exist, it will cross it into the next pane. So what we are really labeling is the air space to the left of the sandwich, the air gaps inbetween the panes, and the air space to the right of the sandwich. It is as if the air is sandwiching the glass, not the other way around: (index) Total "b" "c" "d" Refl. Paths Line Line Line Lineage 0 1 0 0 0 ------- 1 3 1 1 1 b; c; d 2 6 1 2 3 ba; cb, ca; dc, db, da 3 14 3 5 6 bab, bac; cbc, cab, cac 4 31 6 11 14 baba, bacb, baca; cbcb, cbca, caba, cacb, caca 5 70 14 25 31 babab, babac, bacbc, bacab, bacac; cbcbc, cbcab, cbcac, cabab, cabac, cacbc, cacab, cacac etc............................................................................. From the progression of the model and its calculation of golden numbers we can make some speculation at this point as to what the pattern is: y = the index for an element a series s? = a series s1 = the primary series; all others are auxiliary series Honeybee Genealogy Model.Algebraic Calculation of Golden Ratios. Review of Fibonacci's Bunny Model.Further Expansion of the Golden Set of Numbers by Calculation From Angular Classes of Non-Eucli- dean Polygons.Reformulation in Terms of Euclidean Geometry: Linear Aesthetic Relations Within the Plane.?Modularity of ?Golden ?(General Beauty) Numbers