Spiral 935: The Holy Grail Spiral of Life
A theory of the function of prime numbers in spiral formation and in life in general.
The Spiral of Life is a number spiral which forms a cross in the form of a vertical and horizontal axis of numbers. Emerging from a set of alignments 1, 2, and 3 enter like Magi bearing gifts.
Before I continue, I should explain that I am describing a 3-dimensional object in largely two dimensions - as a tetraskele of overlapping spirals - so there is some distortion in this simplified description. The graphic highlights the sequences involved in expansion and does not layout an exact physical pathway. Alignments along the south column extend to the finial circle and give rise to a new spiral whose West arm values retrace the steps of the parent spiral's south column. In other words, West describes where the new spiral emerges from the south column after the parent spiral has been rotated 90 degrees clockwise.
Viewed in three dimensions, 1, 2 and 3 enter from above as a stem joins an apple; 2 and 3 are so tightly bound they share virtually Identical shaped paths like those of twin stars. Because they share the same location and shape at the spiral center their values are combined to 5 only when considering the spiral's southbound formula for once applied at the start it is felt at every subsequent location. 3 and 4 are the respective centered numbers of the column and the crossbar. They exist one above the other with the interval path of 4 shaped like a near circular spiral staircase.
Like the journey of the Magi, the path on which they embark requires them to return along a different route to avoid a collision with the powers that be. Two essential paths emerge. The first path is the spiraling set of natural numbers which may either expand as a disc or move along as a helix.
The second path is the straight sequence on a Southern heading which gives birth to the spiraling natural numbers; she is a master chef and more formulaic than the flowing set she spawns. This straight path can be described using two recursive sequences - the Golden Egg sequence and the Fibonacci sequence. Upon reaching the finial circle, her offspring's arm returns in the direction she came but on a different route. The labyrinth offers a visual aid if we can imagine the straight path to her center being connected to a return path at a different elevation. Those are the two ways of describing the paths: one as rotation, and the other as ray. Taken altogether, what appears from the West are the stepping stones to a spiral which begins as a tight central ring and emerges with two more rings before finishing off with the Finial Circle.
Spiraling from the center loop at 3 and 4, a pattern of growth is reached in each compass direction by incrementally summing two terms along each axis and adding one. It should be noted that when the formula x + y + 1 = z is reached in one axis it is guaranteed to exist in the other three axes. As a result, four rays of whole number sequences extend outwards from center. Multiply this equation by three and twelve alternating radii of whole number sequences result. (For a look at the six-fold symmetry resulting from this model, explore the purple image "The Pillars of the Holy Cross" in this photo set.)
From this arrangement, a Fibonacci sequence (times six) emerges alongside the South column: 6, 12, 18, 30, 48, 78....
The Golden Egg Sequence defines the Spiral of Life's path of origin. She is born from the West near the spiral's center and returns Westward (on a different plane) to 88 at the Finial Circle.
The formula for the spiral's path of origin, the Golden Egg Sequence, begins with two zeros or "goose eggs" (0 + 0 + 1 = 1 ). (0, 0, 1, 2, 4, 7, 12, 20, 33, 54, 88...) The rays headed East and West form the composite crossbar of Spiral 935. The North and South constitute her prime column. In each compass direction the rays follow the formula x + y + 1 = z.
Prime numbers dominate the vertical column between finials 77 and 99. Composite numbers dominate the horizontal crossbar between finials 88 and 112. 7 is the only prime on the crossbar and 9 is the sole composite on the column. Their product (63) marks the start of the second half of the Spiral of Life; 1 to 62 marks her first half. Altogether the two halves underscore a rhythm of 124 intervals punctuated by sets of 76 and 48.
The South Column marks the location of key triple composites 77 and 125. The spiral gives rise to successive generations born from a unique relationship among the cross numbers leading to a specific location of regeneration as the South Column passes through the Finial Circle. Triple composite 125 marks this interval; it is an odd composite integer which when divided into two parts separated by one, yields an even part and a composite odd part. Therefore, all three properties - her obvious value and her two parts when separated by one, are all composite. Even-odds are odd numbers which can be divided into an even number and the same even number plus one. The first several triple composites among the even-odds are as follows:
49 = 24 + 25
65 = 32 + 33
69 = 34 + 35
77 = 38 + 39
125* = 62 + 63
Of these triple composites only 69, 77 and 125 are cross numbers. They are distinguishable by an additional composite feature; the sum of their respective digits are composite since 6 + 9 = 15; 7 + 7 = 14; and 1 + 2 + 5 = 8. Because they possess four composite qualities it seems appropriate to shorten their name from cross-number-triple composites to tetra-composites. (Use Ctrl F "framework" to read more about the triple composites.)
* 125 has the property of being in position 1 of 124 in a second generation spiral. A new spiral emerges 90 degrees offset from the previous one. Thus 496 intervals are required to complete one full gyration of the Spiral of Life's wave motion viewed in cross-section as a tetraskele of overlapping spirals.
From 77 to 125 the spire known as the Finial Circle has 48 intervals. Presuming a circle (in cross-section) is formed by this arrangement - for instance, by particles arriving at an orbit - there would be a very slight shift to the left of cross and crossbar with the South interval 77 remaining stable. 101 would emerge as the North column finial and 89 and 113 would mark the new East and West crossbar finials. 77 and 125 would emerge one above the other. The point of this is to demonstrate that if we sum the values of the intervals which form two distinct religious symbols - the Star of David, and the Cross of Lorraine - (particularly as shown on the tampion of the submarine Rubis) we find their values both equal 707. In fact, any two bars horizontally crossing the circle at any height will satisfy this result. I include both 77 and 125 in both calculations of 707. A cross with a single crossbar (at any height) will produce a value of 505. The trinity of the sole finial primes 89, 101, and 113 sum to 303.
I'd like to stress first and foremost that this spiral is meant to demonstrate mathematical relationships, not mere curiosities. However, if this model can lead to increased pattern recognition then I would be remiss not to point out areas of similitude with real-world objects and symbols. Furthermore, like viewing celebrity look alikes, it may not only provide entertainment but serve to improve memory and perception skills.
Mathematics also stands to benefit from this model particularly in the area of sums. The sum of 1 - 100 was a problem which the young Carl Friedrich Gauss found a solution to by adding 1 to 100, then dividing 100 by 2 and finally multiplying 101 x 50 to reach 5050. In the spiral we find this sum is equal to the sum of the finial circle plus 101. Finial circle 77 ~ 125 = 4949 + North finial 101 = 5050. It is also the value of the cross (505) x 10. When Descartes invented the Cartesian coordinate system it opened a new chapter in the history of mathematics. The Spiral of Life is poised to do the same.
In ascribing values to symmetric shapes like the cross or the Star of David the end values not only come together but exist one above the other. I refer to this type of sum as a k-sum, or knotted sum since the ends are not unlaced but overlap to form a node, or knot.
Are the k-sums of symmetrical points along a spire mathematically relevant? I would argue they are. When we count our fingers, for example, we do not count every joint making up our fingers. If our ancestors had considered the radius and ulna as the two parts of a node at our wrists then they may have counted to 7 using each hand. for a total of 14, the sum of which (1 - 14) would be 3 x 5 x 7 or 77 in base 14. We could easily have ended up with a base 14 system! And yet when you examine your hand, you soon realize all the five digits extend through the palm to form an aligned node of parts, most especially the thumb and pinky.
The good news is that considering this model in a different base system would have little but cosmetic impact. A key position marker of the Finial Circle, for instance, does not rely on the "prettiness" of having double digits but in reaching the same ratio of 8/13 in the un-shifted spiral model at each end of the crossbar in relation to the next spire (88/143 and 112/182). Furthermore all the key characteristics of a number such as primality or compositeness remain despite their conversion to any base system. In other words, the numbers representing the Spiral of Life's intervals might look different in a new base but their function and properties would remain exactly the same.
Goose eggs and knotted sums. Later in this essay I will be describing the goose eggs as the points of equivalence along the south column where it meets the Finial Circle. I have explained the concept of knotted sums where the ends of sums form a knot or a node along the South column. I have pointed out that if we add 101 to the sum of the Finial Circle (from 77 ~ 125) it equals the sum of 1 ~ 100. However I have not yet given an explanation as to how adding 101 to the sum of the Finial Circle may occur. From a symmetric standpoint it stands to reason if one considers 77/125 as a knot whose sum totals to 202 then an opposite node of a circle connecting them would have the same sum value, 202. We already find 101 at the North but where specifically might an additional 101 come from? One possibility is that it might come from a shared electron path, or another spiral model up-side down in orientation to the first thereby creating a situation where the sum of 1 ~ 100 would be equal to the double-knotted sum of the Finial Circle (sum of 77 ~ 125 plus 101).
Introducing the Pythagorean Localization - a Method for Justifying Migration of Cross bar Finial Values to 89 and 113 from 88 and 112.
Below we consider column values expressed as an interval of 3 (for example: 36~38 or 37 x 3). Summing two, stacked, column-centered, three-interval arcs and dividing by 2 identifies matches (shown in parenthesis) with the cross bar values on the right. Averaging the localized values with the spiral values leads to the precise adjustment (to 89 and 113) at the exact moments where the cross bar meets the Finial Circle.
(37 x 3 + 23 x 3) / 2 = 90 -- 88 (average 89)
(23 x 3 + 13 x 3) / 2 = (54) -- 54 match
(13 x 3 + 9 x 3) / 2 = (33) --- 33 match
(9 x 3 + 3 x 3) / 2 = 18 -----------
(3 x 3 + 5 x 3) / 2 = (12) ------ sum of 3~5; product of center knot 3 & 4.
(5 x 3 + 11 x 3) / 2 = 24 -----------
(11 x 3 + 17 x 3) / 2 = (42) ---- 42 match
(17 x 3 + 29 x 3) / 2 = (69) -- 69 match
(29 x 3 + 47 x 3) / 2 = 114 ----112 (average 113)
54, 33 and 42, 69 are the symmetric cross bar values leading to the Finial Circle. I have also marked the value 12 which in this chart lies midway between the four intersecting crossbar values. Perhaps not coincidentally 12 is also the value of the product of column 3 and crossbar 4, the knot at the center of the spiral. It is also the sum of the first three cross values: 3, 4 and 5 - a Pythagorean triple. The values separating 33, 42, 54, and 69 are 9, 12, and 15 also a Pythagorean triple.
Mapping out Pythagorean triplets on the Spiral of Life produces either right triangles or straight lines. Here are some examples of right triangles:
3, 4*, 5
5, 12*, 13
7, 24*, 25
8, 15*, 17
9, 40*, 41
11, 60*, 61
12, 35*, 37
13, 84, 85
15, 112*, 113
33, 56*, 75
36, 77, 85*
39, 80*, 89
48*, 55, 73
The Pythagorean triplet values are mapped out such that each value represents the location of an angle formed by a triangle connecting the values. An asterisk is used to mark the location of the right angle. We find that most Pythagorean triplets produce an additional right triangle but of different dimensions. In the case of mapping spiral values 8, 15*, and 17 on the Spiral of Life the resulting shape is a 3-4-5 right triangle. Is this an illusion based on a differing perspective? These values promise to be very useful for bringing the shape of the Spiral of Life into clearer focus.
For example the cross values seem to indicate nodes and near-circular orbits at the spiral center. 1|2|3|4, 5|11, 7|12, 9|13, and 17|29 are locations where overlap may occur if we consider the three circular knotted spires: one at 1, 2, 3, 4, 5; another at 9, 10, 11, 12, 13 and the third at 17, 20, 23, 26, 29. One Pythagorean triplet - 20, 21, and 29 stands out because it does not appear to form a square triangle on a flattened out Spiral of Life. Two explanations emerge: the shape of the spire at that location might be different as highly indicated by the cross numbers; or not every Pythagorean triplet forms a right triangle on the Spiral of Life.
The justification for the Pythagorean Localization lies in the circular spires and nodes which appear at spiral center. It was a surprise to find the values from 33 to 69 were unaffected by considering the 3-interval column-centered arcs where the overlapping nodes appear. The localization is so-named because it preserves the Pythagorean Triple between intervals 33 and 69.
Comparing the Spiral of Life to the string-petaled Passion Flower: the flower has an anchor petal where the south column meets the Finial Circle at 77 and 125 marking the highly composite zone of the spiral. Upon inspection of the flower (in a method and location described below) three grooves may be observed: north, one appears between two petals; east, marking an offset location of the East petal; and west where a notch exactly bifurcates one petal. The notches exist in only 3 of the 4 compass directions while a unique "anchor" petal exists South where there is no visible groove. I will demonstrate mathematically how the south column is a special location of The Holy Grail Spiral of Life, capable of blending, birthing, and other neat stuff. It is the most likely location for a central anchor petal to emerge. On the passion flower it is one of five green leafy petals and is the sole petal which has what can best be described as the flower's eyelash. It is easy to overlook because the eyelash structure is surrounded by a full ring of lashes which we may call her brow. This feature may only be seen after cutting the stem from bottom and clipping the stamen on top; applying pressure to the cut stem from the bottom will invert the bulb revealing the eyelash.
A memorable moment of David LaPoint's Primer Fields occurs when he presses a metallic Ping-Pong ball through the hole in his magnetic bowl causing it to shoot out the other end. It was a surprise to find that the flower's hard stamen popped in and out in a similar manner but not completely unexpected as the mathematics of the cross indicates strong congruence when values near center are flipped. Adjacent petals along the horizontal crossbar form a strong triangle. The flower's axis runs above the right petal's midpoint (bottom-view) which corresponds to the gradual counterclockwise shift in Spiral 935's axes with each generation of 124 intervals. Two more petals are located symmetrically on each side of the North Column making a total of five symmetric petals in a strong triangular base. Of course, to view the leafy petals, one must turn the blossom upside-down since her top-view is dominated by a profusion of white and purple strings numbering 88.
The Goose that laid the Golden Egg
The Spiral of Life's path of origin begins with two goose eggs. These goose eggs represent key points of equivalence. And what we find is that from these two goose eggs emerges a spiral. And from this spiral a column of primes. And on this column sets of equivalent points emerge - one highly overlapping pair at 17 and 23 and a second mature, divided set at 47 and 77.
How it works: The sets may be thought of as stages of an emerging wave where division occurs as expansion unfolds. The first sets are immature and overlap considerably and don't precisely align along the column but the final sets are where alignment on the south column occurs and it's here we find our goose eggs. The sets share but one number - 47. Let's take a closer look at these sets and The Rule of Five Cross Numbers.
High Five: The Rule of Five Cross Numbers
Each of the final two sets along the south column share one mathematical property: The sum of four consecutive cross numbers is equal to the fifth consecutive cross number times three. This is the Rule of Five Cross Numbers where the Cross Numbers in question represent the five digits of a hand. The sum of four consecutive cross values is represented by the four fingers and is equal to the thumb - or fifth cross value - times three. In fact, the sum of the length of all four fingers from tip to knuckle matches the length of one's entire thumb times three.
Also try this. Use your hands to separate the 10 digits into two groups, the low five and the high five. First count the low five then count only the four fingers of the high five, "6, 7, 8, and 9." Now add these four numbers together and divide by three - the result is 10 which satisfies the Rule of Five Cross Numbers, As an expression of a successful alignment we might imagine the satisfying of the rule of five - the appearance of the goose eggs - as being celebrated with a "thumbs up" or a "high five". The sets demonstrate how shape is maintained and copied as expansion unfolds uniformly along a central column. The model itself may shed light on how quantum science influences everything - from our language to our biology.
An Important Organizing Principle
The location where the Rule of Five Cross Numbers applies among the cross values is marked by start and end values along the prime column. Off-column matches of the rule are not shown.
(11 + 12 + 13 + 15 = 17 x 3) South Column
(23 + 26 + 29 + 33 = 37 x 3) North Column
The final two sets leading to the Finial Circle are the first contiguous sets of the column having starting points of 29 and 47. This contiguity will continue forever beyond the Finial Circle if and only if one condition is met which we will discuss shortly. At this stage, it is worth noting how proportion is maintained as expansion drives the process of individuation.
(29 + 33 + 37 + 42 = 47 x 3) South Column
(47 + 54 + 61 + 69 = 77 x 3) South Column
This rule of the Cross Numbers is also called the Rule of Five Executives and it highlights points of equivalence directed in a path along the south column as she crosses through the Finial Circle. At 47 the sum of the preceding 4 cross values is equal to the value of 47 repeated 3 times. That's the first point of equivalence or "goose egg". And the same holds true for 77 since 47 + 54 + 61 + 69 = 77 x 3 which brings us to the second goose egg after which a new spiral emerges with a value of 1 at location 125.
Additional observations: The values of the new spiral along the south column do not replace the south column values from 125 onward; they exist on a different plane. However the regenerating or motion part of spiral 935 always takes this path. Therefore 124 times the number of generations passed is added to each spiral value. The spiral values which emerge South at location 125 follow the Golden Egg Sequence 1, 2, 4... Most interestingly, subtracting the terms of the Golden Egg Sequence (starting with 1) from the south column values beyond the Finial Circle (from 125) results in a sequence which satisfies the Rule of Five Executives thereafter to infinity. When I first observed this, it was a thrilling moment. I had already known about the "goose eggs" which gave rise to the Golden Egg Sequence; I had known about the smooth prime ring at the Finial Circle (Cntrl F "distribution") broken only by a highly composite opening at 77 / 125. The evidence pointing to the birth of a new Golden Egg Sequence was gathering momentum.
The center of the new spiral emerges from above the finial circle's 77th interval as interval 125 comes very close to full circle. Under certain circumstances as when rotations come together at this location a spin is produced and a new spiral emerges. One wave motion of The Spiral of Life requires four generations. Here what I am describing is a basic, ironed-out representation of a cross-section. The kinds of forms that she can take range from the plate to the cone to the cylinder to the Mexican hat and as far as one's imagination can conceive. It is a symbolic representation which nevertheless highlights - by means of measure or sequence - mathematical and physical symmetries. What kind of physical reality does it describe? Rotation.
The Spiral of Life gives rise to another ray of origin (and a new spiral) starting from the South column's 125 and this process repeats ad infinitum creating a gyration or wave motion observed in cross-section. In number theory these progressive terms along the axes are called Lucas numbers and are similar to the Fibonacci Sequence with the exception being that a set amount is also added to the previous term. In the illustration, this value is '1'; however, since Spiral-935 is a dynamic model, values other than one will also be explored particularly as the point of origin ebbs and wanes. Spiral 935 is a significant breakthrough in our understanding of life's origin and sheds light on the occurence of tantalizing patterns among the prime numbers. Within this model, symmetry after symmetry ties intimately with the properties of the integers themselves to reveal not only an architecture of all natural numbers but an exceptional understanding of the concepts of zero and one. I invite you to bookmark this page as I will be adding and editing it regularly. The organization of this essay is somewhat spiral. I have placed newer information up top and pushed back previous findings. So you may find areas which repeat or that have been changed.
Vertically, there are eleven numbers between 77 and 99 on the above column and ten are prime. The ten prime numbers I refer to as cross primes. The remaining number, 9 is a cross prime square. The ten cross primes and 1 square form pairs opposite each other on the column with the exception of center 3 at the core of the spiral's engine.. The first cross pair are 5, 9 followed by 11, 13, and 17, 23 and 29, 37; finishing with 47, 61. Numbers 77 and 99 I refer to as column finials. 77 is a biprime and 99 is a square prime repdigit which mark the end of the sequence of cross primes. When the column pairs are set as fractions or divisors are being multiplied the finial pair prove themselves to be a very useful set.
Horizontally, there exists an eleventh cross prime on Spiral 935's crossbar between finials 88 and 112. It is number 7 and lies just to the left of column center 3. The crossbar has her own centered number - 4 - which one may visualize as being very close to column center 3. Seven is the only prime number on the crossbar. For a more complete understanding of how Spiral-935 evolved please explore Spiral-31 and other images within this photoset.
The engine driving Spiral-935 needs priming and an energy source To get it started we simply add the value of 2 to the centered 3 to give it a total value of 5. Then we put the formula to work.
The formula is x + y + 1 = z.
x, y, and z are 3 numbers on a column segment ascending in value with x equal to or nearest to center 3.
Spiral-935 takes its shape from a centered 3-number column segment having values of 9, 3, and 5 from top to bottom - notated as 9-3-5. Starting at the center number 3, we can calculate the next number set (z) in the column using the formula since we are given the initial values for x and y in both directions, namely, 3, 9 moving upward and 3(+2), 5 moving downward.. .
moving down the column from center:
(2) +3 + 5 + 1 = 11
5 + 11 + 1 = 17
11 + 17 + 1 = 29
17 + 29 + 1 = 47
29 + 47+ 1 = 77
moving up the column from center:
3 + 9 + 1 = 13
9 + 13 + 1 = 23
13 + 23 + 1 = 37
23 + 37 + 1 = 61
37 + 61 + 1 = 99
how to create Spiral 935 from zero
It is also possible to create Spiral 935 using the formula by starting with '0' for values x and y. This method establishes values for Spiral 935's horizontal axis, or crossbar starting from center and extending right when spiraling counterclockwise, the preferred viewpoint for reasons explained at the end of this essay; (The illustration above spirals clockwise.) Using the formula with starting values of x and y at zero, the progressive values attained are 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, etc. I refer to this series created by the spiral formula, as the Golden Egg sequence as it begins with two "goose eggs" one for x, the other for y.
0 + 0 (+1) = 1
0 + 1 (+1) = 2
1 + 2 (+1) = 4
2 + 4 (+1) = 7
Zero Origin relates to the model for "priming" the spiral by adding '2' to the centered 3. The spiral owes her start to a force of attraction between two objects which combine regularly with three to reach a point of equivalence (with five) allowing the spiral to form. The zero/zero start of x and y is thus reflected in the priming formula of (3+2) + 5 + 1 = 11 at column center. The concept of equivalence lies at the heart of Zero. The Fibonacci sequence starts with two equivalent numbers, 1 and 1. In Spiral 935 many Fibonacci sequences are found including the double eleven sequence beginning with 11 and 11
The Golden Key
Starting from center ANY three consecutive numbers on Spiral 935's column or crossbar may be reduced to one by subtracting the smallest two numbers from the largest. The symmetry of Spiral 935 reaches to the core and through it. Following the formula in reverse reveals other sequences such as the Golden Key sequence linked to the North Column.
From Cross to The Golden Key
South : 11 - 05 - 01 = 05 ------05 + 05 + 01 = 11-----05 + 11 + 01 = 17
+East : 10 - 04 - 01 = 05 ----->04 + 05 + 01 = 10-----05 + 10 + 01 = 16-----10 + 16 + 01 = 27,
North : 09 - 03 - 01 = 05 ----->03 + 05 + 01 = 09 ---- 05 + 09 + 01 = 15----09 + 15 + 01 = 25<
+West : 07 - 04 - 01 = 02 ----- 04 + 02 + 01 = 07-----02 + 07 + 01 = 10
South : 05 - 03 - 01 = 01------- 03 + 01 + 01 = 05-----01 + 05 + 01 = 07
The formula creating the column and crossbar loops through the core to create new sequences intersecting but not on the cross. Associated with the North Column and moving from 9-3-5-9- to 15 the Golden Key sequence forms the triangular shape of a key before straightening and dividing the Finial Circle at her golden mean. The sequence of the Golden Key is
9, 3, 5, 9, 15. 25, 41, 67, 109, 177, 287, 465, 753, 1219...
At 15 the key sequence makes her final appearance on the cross and at 25 diverges, taking a course close to but moving gradually away from the crossbar. At 109 the hidden sequence intersects the Finial Circle at 111 degrees, dividing the spiral arm exactly at 16 / 26ths.
125 - 109 = 16
125 - 099 = 26
-------------->16 / 26
This is the precise ratio (0.615384 or 8/13) both crossbar finials reach in relation to their larger cross number. (88/143 and 112/182) This fraction may be expressed as a ratio of the sum of the first two cross numbers + 1 divided by the sum of the first three cross numbers + 1. An interesting relationship with between the golden finial fractions of the crossbar and column has been observed. Subtracting the crossbar finial's 8/13 from the south column finial's 77/125 (.616) and multiplying by 1,000 yields 8/13. .616 divided by crossbar finials 88 and 112 yields exactly .007 and .0055, respectively. These are important Golden Relationships.
(.616 - (8/13)) x 1,000 = 8/13
.616 / 88 = .007
.616 / 112 = .0055
.616 = (29 x 08) / 100 x .05 + 1/2 (similar to a formula found on Professor Steven J. Finch's page on the determination of critical value)
Astounding Symmetry: Crossbar Squares and the Golden Ratio .616
There is one set of 5 crossbar numbers between Finial 88 and Cross Center. We'll call this set A. And another set of 5 crossbar numbers between Finial 112 and Cross Center we'll call set B. If we take the sum of the square values of each set and add to each set the column value 99 located above and directly between each set, then the ratio between set A and B will equal the South Column Finial Ratio of 77/125 or .616.
Set A: 7^2 + 12^2 + 20^2 + 33^2 + 54^2 + 99 = 4697
Set B: 10^2 + 15^2 + 26^2 + 42^2 + 69^2 + 99 = 7625
4697 / 7625 = 77 / 125 = .616
Not only does the hidden Golden Key of Spiral-935 form the shape of a key, it is the master key linking Spiral 935 to the Fibonacci sequence. From 25 onward, one need only subtract the hidden key numbers from the nearest crossbar number to arrive at the Fibonacci sequence.
Crossbar values - 15, 26, 42, 69, 112, 182, 295, 478...
The Golden Key - 15, 25, 41, 67, 109, 177, 287, 465...
Fibonacci values - 00, 01, 01, 02, 003, 005, 008, 013...
Golden Key Sequence Essentials: Dividing the Spiral in Halves, Thirds, and The Golden Mean
1.) The Golden Key divides the Finial Circle at 111 /180 degrees, the Golden Mean.
2.) The Golden Key and the Golden Egg sequences divide Spiral-93 in half. Golden Egg 54, for example, divides 109 into two whole integers separated by 1: 54 and 55.
3.) The Golden Key divides the spiral rotation into thirds. At the finial circle the rotation is 48 along spiral values (125 - 77 = 48).
125 - Golden Key 109 = 16
16 / 48 = 1 / 3
The Lost Fibonacci Sequence
Comparing opposite sides of the Spiral we find an inverse relationship between Fibonacci and Golden Key formulas. They swap roles. Where the Fibonacci sequence defined the "altitude" lifting off from the crossbar that role belongs to the spiral values, or 'golden egg' values of crossbar 88. Instead of adding the golden egg sequence to the opposite crossbar we subtract it creating a descending ray starting at cross bar values 20 and 33 where the initial values are 0 and 0; then we begin descent with value 1 reducing 54 to 53, The sequence of the Golden Egg is the same spiral formula - only starting with x and y at Zero. Subtracting the Golden Egg sequence from the Crossbar values reveals the Lost Fibonacci Sequence.
++88 Crossbar: 20, 33, 54, 88. 143, 232, 376 ,609 ...
++Golden Egg: 00, 00, 01, 02, 004, 007, 012, 020 ...
Lost Fibonacci: 20, 33, 53, 86, 139, 225, 364, 589 ...
The Lost Fibonacci Sequence is as much a part of the Fibonacci Sequence as the North Column is a part of The Golden Key Sequence: they are one path having two parts which come together at the core and connect at the cross. To recover the Lost Fibonacci Sequence we backtrack through the Fibonacci sequence using the same rule that applies to cross values between 3 and 5 at the cross center . We add 2 to 3 going out from center and subtract 2 from 3 going in. So 8 - 5 = 3 (-2) = 1. After we've subtracted 2 we have crossed the threshold and may begin adding again with '1' as the new core number: 5 + 1 = 6: 6 + 1 = 7; 7 + 6 = 13; and so on.
The Lost Fibonacci Sequence is:
1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101..
Fibonacci sequences and Golden sequences are paired. Where the Golden Key divides one side of the spiral at her golden mean, Lost Fibonacci divides the other. Where Fibonacci defines the "ascent" of the Golden Key on one side, a Golden sequence will define the "descent" on the other. It demonstrates how corresponding formulas may achieve an effect dependent on their location. When placed at opposite locations their roles may reverse.
Valentine's Day, Mega-Fibonacci, and The Tressarian Twins Sequence
On Valentine's Day 2013 Tressa Montalvo gave birth in Houston, Texas to two sets of identical twins - Ace, Blaine, Cash, and Dylan. The odds of such a birth are said to be one in several million...and to occur on Valentine's Day must make the odds astronomically small.
For awhile I've been debating what to call the two sets of twin Fibonacci sequences formed by the cross. Quadronacci or Tetranacci might be nice but they are in current use for sequences having four terms added.. The sequence formed by the cross divides every full rotation into two halves - left and right - one slightly larger than the other however with each half having two equal parts. This unique sequence is the direct product of the cross - representing the organizing forces of attraction.
Due to the timing of contemplation I first considered naming it the Beligyre Sequence, an anagram of Lee Rigby, the UK police officer slain by a follower of an evil man whose fantasy to torture and murder his questioning aunt and uncle made its way to nauseating posterity in Surah 111. However, wishing for a more jubilant association, I decided to name the twin set of integers the Tressarian Sequence since it alludes to a double set of twin Fibonacci sequences. I also refer to it as the Tressacci Twin or TT sequence for short recognizing that in shape TT resembles Pi and in name - a sports car from Audi. However if one is willing to part with $700,000 or more then a sports car can be yours that will go from 0 to 60mph in Pi (3.14) seconds - the Ferrari Enzo. Only 400 were made so they are very hard to find.. But every now and then they are abandoned - particularly in countries with stiff debt laws like Dubai.
Starting with center 3 and moving to crossbar center 4 establishes the first digit - 1. Moving to South Column 5 offers the second digit - 1. From 5 to Crossbar 7 we arrive at 2 and another 2 brings us to the North Column's 9. From that point forward we continue along the spiral in same manner, making note of the count from one cross value to the next with each one-quarter rotation. The Tressarian Sequence is:
1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 7, 7, 8, 8, 11, 11, 13, 13, 16, 16, 21, 21, 29, 29, 34, 34, 47, 47, 55, 55, 76, 76, 89, 89...
The Tressarian Sequence will help explain how the Fibonacci Sequence works in the growth patterns we see in nature. When we observe the sequence in petals, pinecones, or even in the bracts of pineapples we may infer a pattern of twin sequences at work - one for each half of the petal. Once the concept of a twin sequence emerges based on natural forces it is possible to fathom a multiple of twin sequences at work in the creation of a single cone. And that's just the beginning. Spiral-935 has also demonstrated how the Fibonacci Sequence not only appears in spiral shapes but also as a rate of elevation marking successive points of equal division. The implications are that the twin Fibonacci Sequences found imbedded within the Tressarian Sequence may open the door to the discovery of a variety of other growth patterns that have previously been thought unrelated to Fibonacci.
Column and Crossbar Center
Four represents the centered number in Spiral 935's crossbar as three represents the centered number in the cross column. It is likely the first centered number was two and the center shifted as more numbers gravitated. Spiral-935 differs from other spiral representations which typically use "0" as both the point of origin and centered number of the spiral. My purpose is to describe a spiral of numbers in motion, dynamic, and living. The crossbar values from center to right are: 4, 10, 15, 26, 42, 69, 112, 182, 295, 478, 774, 1253, etc. By lining the columns of values side by side we may extrapolate the vertical column numbers and cross primes by the following method:
(4 + 7 - 10) x 2 + 1 = 3
(7 + 12 - 15) x 2 + 1 = 9
(12 + 20 - 26) x 2 + 1 = 13
(20 + 33 - 42) x 2 + 1 = 23
(33 + 54 - 69) x 2 + 1 = 37
(54 + 88 - 112) x 2 + 1 = 61
(88 + 143 - 182) x 2 + 1 = 99
(143 + 232 - 295) x 2 + 1 = 161
(232 + 376 - 478) x 2 + 1 = 261
circle versus spiral values at the finial circle The illustration represents a hypothetical tightening of the belt at 77 with the foreshortening of the Finial Circle from 125 to 121. With the discovery of equivalent fractions at the crossbar finials along spiral values I find it useful to consider a compression/expansion model of the belt at the Finial Circle in keeping with spiral values. UPDATE: Continue reading through "evening the odds" for a more complete explanation of the Finial Circle's creation and properties.
circle values (77, 88, 99, 110, 121)
(for measuring circle properties between column or crossbar pairs)
121 - 99 = 22 (the count of numbers at 180 degrees of this spire)
22 + 99 = 121 (adding the same amount to form a 360 degree circle)
121 - 77 = 44 (the total count of numbers forming the circle's circumference)
spiral values (77, 88, 99, 112, 125)
125 - 99 = 26
26 + 99 = 125
125 - 77 = 48 (the total count of numbers forming the spiral's finial circumference
evening the odds at the finial circle
Prime symmetry found between the even-odds of the finial circle. First observed in Spiral-935's column, the symmetry of prime numbers occurs at the finial circle in her sum values. Odd numbers may be thought of as the sum of two numbers with a difference of 1 between them. If the lowest of these two numbers is an even number then the inspected number is called an even-odd number. If the lowest number is odd, it's an odd-odd number. From 77 to 125 there are 13 even-odds but only the middle 11 - centered precisely around 101 - share a characteristic which solves the mystery of the finial circle. Between 9 squared and 11 squared lie the eleven even-odds: 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, and 121. Five of the eleven are prime numbers outright and contain no primes in their pairings. The remaining six are each paired with prime numbers. Considering the count in this way (important when translating mathematics to chemistry and biology) brings the even-odd count of prime numbers between 9 squared and 11 squared to 11.
If a circle can be created from a spiral one would anticipate the point at which the circle is formed to have weaker bonds which can unlock and join with an adjacent arc to form a circle. The weaker bonds may appear not only in the divisible even numbers (which, in many cases, may have strong bonds) but in the important even-odds. Of course not all even-odds have necessarily weak bonds; however 77 and 125 - nonprimes which are both the sum of consecutive nonprime integers - stand out in their neighborhood.
the Holy Grail
The numbers below illustrate the framework around which the creation of a circle from a spiral occurs around the stronger bonds between nine squared (81) and eleven squared (121). By viewing the Finial Circle through the lens of her even-odds it reveals a smooth distribution of prime numbers:
77 = 38 + 39
81 = 40 + 41prime
85 = 42 + 43prime
89 = 44 + 45 (prime sum)
93 = 46 + 47prime
97 = 48 + 49 (prime sum)
101 = 50 + 51 (prime sum)
105 = 52 + 53prime
109 = 54 + 55 (prime sum)
113 = 56 + 57 (prime sum)
117 = 58 + 59prime
121 = 60 + 61prime
125 = 62 + 63
The above represents a close-up of Spiral-935's Finial Circle as seen through the even-odds. 77 and 125 are triple composites. They are actually tetra composites if we consider that each of their digit sums is also composite. In stark contrast, every even-odd number between 77 and 125 either 1.) has a prime part in one of her two parts separated by one (we may call these partial primes) ; or 2.) is a prime number outright. It is worth noting that none of the prime number even-odds along the Finial Circle has a prime part - each one being comprised of two composite parts separated by one. This property gives the Finial Circle her "smoothness" in terms of the distribution of primes.
The sum value is 1111. (11 * 101) for the even-odds. (81 - 121)
The sum value is 1212 (12 * 101) for the odd-odds.
The sum value is 1313 (13 * 101) for the even-odds (77 - 125)
The sum of all primes is 813 (271 * 3)
The sum of all 6 non-primes is 298 (81-121)
The sum of all 8 non-primes is 500 (77 - 125)
What makes the even-odds and odd-odds relevant?
Symmetric patterns are created around them throughout the spiral making adding, dividing, and multiplying much easier - key to making replicas and creating life. The Golden Key Sequence, for example, not only divides the spiral along her Golden Mean but also serves to divide the entire spiral length in nearly equal portions (like a coiled tape measure) - from point of origin 1 (near spiral center 3) to her mid-way point (along the Golden Egg Sequence) and ending at her Golden Key (a ray rising just north of the opposite crossbar).
Odd numbers appear every third number on the crossbar extending through the core. A similar pattern exists along the column (comprised exclusively of odd numbers) with odd-odd numbers occurring at every third value from North and through the Golden Key Sequence.) The crossbar center is flanked by odd-odd 7 and odd-even 10. Extending outward odd numbers occur at every third interval, alternating between even-odds and odd-odds.
To refresh, the Golden Egg Sequence is the crossbar from Center to 88 and beyond (4, 7, 12, 20, 33, 54, 88, 143, etc.) The Golden Key originates from the North but loops through the center and forms a ray, lifting off from the opposite crossbar at a rate of elevation equal to the ascending Fibonacci Sequence. It is located on the opposite side of the spiral towards 110/112.
Dividing the length of Spiral 935 is accomplished by dividing the Golden Key values into two whole integers separated by one. For example Golden Key value 67 divides to 33 (Golden Egg) and 34. Golden Key 109 divides to 54 (Golden Egg) and 55. This organization gives Spiral 935 flexibility and deeply ties the Golden Mean to the ability to grow spiraling structures into nearly equal parts.
Most of us are accustomed to counting in ten increments of ten numbers. However, from the start we can see numbers are easily divided between odd and evens. And taking it one step further, between even-odds, odd-evens, odd-odds and even-evens (such as 1, 2, 3, and 4 respectively.) Even-odds start at 1 and may be thought of as the primordial numbers - meaning they reflect back to the original '1' ; they are always the first number - of each successive four-number grouping. The first 43 even-odds are:
001, 005, 009, 013, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97
101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169...
There are several possible ways of counting the sum of all numbers at the Finial Circle yet whichever method we pick - one number - a factor of every combination of finial sums - stands out - and that number is 41.
sum of even-odds -81-121 ---1111
sum of odd-evens- 81-121 ---1000
sum of odd-odds--- 81-121----1010
sum of even-evens-81-121----1020
Total ---------------------------------4141----(41 * 101)
41 and prime-hunting
Her symmetric properties make 41 a major player in the world of prime-hunting. There are several formulas involving 41 perhaps the least complicated of which is that if you start with any two integers totaling 41; squaring one of them and adding to it the value of the other integer,will most likely result in a prime number - or else a multiple of five (4+1).
To commemorate the latest discoveries at the finial circle, I went prime hunting using the above formula with a twist. Instead of two integers summing to 41 I used two integers summing to 4141 (41 x 101). Only 7 of 22 were prime using only the above formula. However I was able to predict primality with 100% accuracy by considering the number of prime numbers that appeared after dividing the last two-digits of the two integers (un-squared) into either two equal parts (odd-evens or even-evens) or two parts separated by 1 (odd-odds or even-odds) If 2 or 3 prime numbers appeared In these parts, the result was prime but If less than 2 prime numbers appeared, the result was not prime.
Choosing integers for use in the above formula with sum values of the form 41 x 10101, or 41 x 1010101... has shown positive results in terms of prime prediction. 271 - a prime factor of 813 (271 * 3) - has demonstrated even greater reliability. 813, as you may recall, is the sum total of the prime even-odd numbers and the nonprime even-odd' numbers' prime offspring at the finial circle. Engaging in his recreational exercise allowed me to see that 271 x 41 = 11111, The sum of all even odds at the finial circle is 1111. An early Christmas present. Thank you for being there to observe the mysteries of the finial circle come to light.
3.1415, 23, 41, and 8,888
Since this area ties in with the Finial Circle I thought to mention another aspect which ties into today's calendar day - Pi Day or 3.14. The Finial Circle is formed around an arc of primes and prime parts between 9 and 11 squared. The place where the circle starts and ends is in an area with a higher degree of compositeness compared with the other numbers at the finial circle. Notably 77 and 125 mark the composite area where the circle forms. If we sum the four numbers from 77 to 80 (1 prior to nine-squared) we arrive at 314, the first three digits of Pi and - due to the compositeness of the location - what I call the "mouth" of the Finial Circle. For a hungry circle, Pi is the perfect order! Happy Pi Day, everybody!
Since we have reached a milestone in 8,888 views I wish to explore some properties of this fascinating number in relation to Spiral 935. Pi quite nicely divides 8888 in the following cross number multiples of Pi.
8888 / 23 Pi = 123.00 = 3 x 41
8888 / 69 Pi = 41.00
No other multiples of PI - cross or otherwise - dvides 8888 so precisely as do the pi multiples of 23. In a related property of Spiral 935 so far it appears that prime column pairings (prime numbers at opposite sides of the column) occur at multiples of 23 of the column height. However testing primality for column pairs at height multiples of 23 requires greater computing power than my laptop offers.
Factors of 8888 include 88 and 101 - both located on the Finial Circle.
11 x 808 = 8888
88 x 101 = 8888
8888 / 70* Pi = 40.41 While 70 is one-off from cross number 69 the resulting fraction is nonetheless pattern-forming; and we see a similar re-occurrence in cross number 112.
8888 / 112 Pi = 25.26
About 8888 and 41. I recommend using the search function on this page and noting all the occurrences of 41....in the even-odds for example...or at the finial circle it's in the Finial Circle's upper-bound breakdown of 9 squared, (40 and 41) 81's prime 41. (Compare to the upper-bound breakdown of Finial Circle's 11 squared...121's prime 61.)
When multiplied by 77/125 (.616), 8888 yields ((11+11)^2 x 1414 ) / 125
Pi The Golden Ratio, zero and the core of spiral-935
The circumference of the Finial Circle is 48 along spiral value 112. The foreshortening of the Finial Circe to 44 occurs when ending the spiral formula by adding a half arc equal to the preceding 22-count arc between 77 and 99. 44 divided by pi yields 14 (which is exactly equal to the column height of 14 between spiral numbers 100-124.) The Finial Circle is the only location along the entire column where Pi equals the circumference divided by the diameter when the diameter is defined by the column count . At the core the diameter is compressed and pulsing (not static) and beyond the finials it is rapidly expanding. For the count of column numbers to be equal to the diameter well beyond the finial circle would require Spiral-935's numbers to be moving very fast (creating a helix) and for their speed to be increasing. Also if the numbers represent objects which are very tiny relative to the space between this may alter our perception of the spiral's shape.
The Finial Circle itself may be thought of as the "Zero" of Spiral-935. Zero, in this sense, does not mean the absence of anything but rather reaching a point of equivalence. For example at the finials, the Golden Ratio of .61 (and 1.61) is fully reached. Now imagine for a moment we divided every number in Spiral 935 by 100. Everything would essentially be the same - albeit smaller - but notice how the finial circle as It crosses the column would exist between column numbers .61 and 1.61 - phi and PHI, the two ratios of the Divine Proportion.
Golden convergence begins at 26/42 on the crossbar and 29/47 on the column. However it fails to fully converge until reaching the finials. (I define the point of golden convergence to be when the first two digits of the mean appear in decimal form .61 at both ends of crossbar and column.) As a rule, all additive sequences resolve to the Golden Ratio except one. The question of convergence then becomes a matter of how and when not if. That is why the location of total golden convergence between column numbers 61 and 161 coupled with Pi at the finials (44/pi = column height 14) presents a memorable confluence of form and event. The only explanation I may provide for this is the unique center of Spiral 935. Among the competing ratios at the core, 3/5 stands out as the centerpiece and at .60 is as close to the Divine Proportion as can be formed by single digits. Indeed, the whole of Spiral 935, with its Finial Circle and vertical column resembles the very symbol of Phi.
Subtracting the ratios at the two crossbar finials 88/143 and 112/182 leaves a remainder of zero as both are represented by the repeat decimal .615384. Unique symmetry at the finials can also be found in square calculations. Each set of three square numbers lie on the cross (within both column and crossbar) and end at the finials.
4.4121 x 42^2 + 69^2 = 112^2
4.4233 x 29^2 + 47^2 = 77^2
4.4334 x 33^2 + 54^2 = 88^2
4.4411 x 37^2 + 61^2 = 99^2
Whole numbers 3, 4, and 5 (3+2) comprise the spiral center. Since the decimal values above all fall within the range of these core numbers I thought to investigate what the core would look like as an initial cube value in the formula. To my surprise, the decimal values converge to a whole number. Spiral 935's inner core may be represented (in formula) by the following three decimal fractions before reaching 5 (the cube root of 5 cubed) at the completion of the spiral's first fifteen digits. ( 5^3 + 10^2 = 15^2) 5 emerges at the center as the relation completes at 15, a location where - in perfect symmetry - the Fibonacci sequence begins at zero.
4.44796^3 + 9^2 = 13^2 (column values from center north)
4.56290^3 + 7^2 = 12^2 (cross bar values from center west)
4.57885^3 + 5^2 = 11^2 (column values from center south)
5.00000^3 + 10^2=15^2 (crossbar values from center east)
When the value of 5 is reached near the column center it creates a point of equivalency tipped by the addition of 1 to reach 11, the first column number beyond the inner core. (3+2) + 5 + 1 = 11. This may symbolically be represented as two open hands with the 3 longer fingers of each hand curving inward to meet the tips of the shorter fingers to become 5. As the compressed fingertips of both hands meet it expands into a ball of space and energy between them before turning into the symbol for '11' as the hands compress, palms together.
88 and the piano numbers
The piano numbers appear from 03 to 23 and are of two basic kinds. Those which lie on the cross form the notes of the cross scales while those numbers between the cross (against the field) form the notes of the field scales. Thru 10 all crossbar numbers are counted as field numbers except 7.
The cross scales include numbers 11, 12, 13, 15, 17, and 20 [ A, A#, B, C#, D#, F# ]
A five-tone cross scale may be comprised of numbers 13, 15, 17, 20, transitional [ 23 (A) ]
The field scales include numbers 14, 16, 18, 19, 21, and 22 [ C, D, E, F, G, (and G#) ]
Most scales can be broadly categorized into these two basic divisions. One interesting feature of both cross and field scales is how they resolve respectively to B Major, C Major, E Major, or alternately, to the chromatic scale. The cross and field scales divide the 88 keys of the piano in the most elegant way possible. Different nuances are imparted when descending versus ascending so be sure to try the scales in both directions. If there exists a more beautiful way than the cross and field scales to bifurcate the 88 piano keys then I am eager to hear it.
Alternating between the cross and field scales brings to mind Dizzy Gillespie and Frank Paparellli's "A Night In Tunisia" (lyrics below) given the song's use of half-step up and half-step down chord changes. Written in Texas, it was originally entitled, Interlude after Dizzy noticed the melody he laid down lie at the crossroad of Asiatic and Western scales. It was written during a break from making a short film. Using the bottom of a trash bin as a makeshift table, stool or drum perhaps Dizzy wrote it at a time when the lights of the heavens were just starting to make their appearance.
lyrics to 'A Night In Tunisia'
The moon is the same moon above you
Aglow with its cool evening light
But shining at night, in Tunisia
Never does it shine so bright
The stars are aglow in the heavens
But only the wise understand
That shining at night in Tunisia
They guide you through the desert sand
Words fail, to tell a tale
Too exotic to be told
Each nights a deeper night
In a world, ages old
The cares of the day seem to vanish
The ending of day brings release
Each wonderful night in Tunisia
Where the nights are filled with peace
There are thousands of other songs matching the cross and field scales. The reason I've brought attention to this one is that Dizzy himself recognized that there was something special about this song that had raised it from "the vicissitude of the contingent" to its exalted place "in the realm of metaphysics."
Below are a few examples of some highly matching scales.
cross scales: (group one) B Major; A# Harmonic Minor; A Prometheus Neopolitan (group two ) D major, E major, A major, E melodic minor, B natural minor and A major pentatonic
field scales: C Major; F dim Lydian; D Blues; F Kumoi; D Locrian 2; C Mixolydian flat 6
Piano numbers are so-named because the sum of the six-note cross scale numbers is 88 -- a common number of keys on a piano; Interestingly, the sum of the five-note cross scale is also 88. The sum of the field numbers is 110. Subtracting the G# (22, a highly transitional tone) from the field scales, yields a remainder of 88.
Eleven through twenty-two (11 - 22) correspond to the 12 tones of the chromatic scale. In the illustration the spiral formula breaks down to allow for a tightening circular ring between 77 and 99 (as opposed to a spiral ring around 125). The result is that 110 appears as a finial at one end of the crossbar (instead of 112) and 88 appears at the other, reflecting the two sums of the piano series represented horizontally, both cross and field.
defining attributes and symmetry of the 9-3-5 column)
13 - 9 = sum of 3 and 5 divided by 2
11 - 5 = sum of 3 and 9 divided by 2
37 - 23 = sum of 11 and 17 divided by 2
29-17 = sum of 9 and 13 divided by 2
99 - 61 = sum of 29 and 47 divided by 2
77 - 47 = sum of 23 and 37 divided by 2
9 - 3 - 1 = 5
5 + 3 + 1 = 9
reaching for the stars
The column of Spiral 9-3-5 reaches indefinitely upwards and downwards still forming pairs at each end only one pair of which are known prime numbers and the others being odd numbers that can be factored into prime numbers. The occurrence in terms of location and timing of these divisors form patterns that not only are very useful in terms of prediction but also interesting in terms of how they originate. The column numbers can get quite large so to make it easier to put them into categories and discuss them I've given them names in addition to the column height value I have assigned them. The first pair of non-primes appear at a column height of 13 and are 99 and 77. The pair of divisor sets are 3,3,11 and 7,11; 99 and 77 are called trizor 11 and bizor 11 respectively. Of course, 1 is also a divisor, but for the sake of efficiency, I've decided not to reflect that fact in the names since it is well-understood that 1 is a divisor of all integers. Therefore bizor is a number having two divisors (excluding '1") A trizor is a number having three divisors (exluding 1). Other terms are quadzor, pentzor, hexzor and so on. A number following these terms may either be the lowest divisor of that number or the common divisor when describing a set of numbers. For brevity, numbers such as 77 and 99 may both be referred to as zor-11's as they share a common divisor in 11.
other attributes of the cross primes and finials
93 = The square root of (112^2 - 88^2 + 99^2 - 77^2 - 23).
93 = The sum of every other cross column prime (starting on 37 and ending on 29 which includes center cross prime 3. The remaining column numbers total 162 or 9^2*2 or 161 plus 1. Eliminating the square 9 results in 153 or 3*51; matching nicely with 3*31 or 93. The sum of prime column numbers is thus 3*31 + 3*51 = 6*41. Subtracting the sum of the South column's alternating primes, 93, from the sum of the North Column's alternating primes, 153, yields 60 which is equal to the repeating cycle of 60 digits on the column in the one's place. So while not evident at first glance, intervals of 10 turn up very significantly from calculations Involving Spiral-935' prime column. Subtracting the lower alternating prime sum 93 from the higher prime sum of 153 plus square 9 yields 69 or 23*3 - equal to the final cross value of Spiral-935 prior to reaching the first finial, 77, at the South Column.
93 = The sum of the first three cross primes from the bottom, 47, 29, and 17.
93 = The centered number of the sole trinity of consecutive composite odds - 91, 93, 95 - (bounded by prime numbers 89 and 97.)
48 = The count of integers at the Finial Circle (77 - 124)
76 = The count of integers up to the Finial Circle (1 - 76)
`124 = The total count of the primary integers of Spiral 93 comprising one generation.
41 = The sum of the first five column numbers - (3, 5, 9, 11, and 13.)
31 = The sum of four consecutive cross numbers beginning with 5. ( 5, 7, 9, 10)
42 = The sum of five consecutive cross numbers beginning with 5. (5, 7, 9, 10, 11)
28 = The sum of the first 5 cross numbers - (3, 4, 5, 7, 9)
19 = The sum of the first 4 cross numbers - (3, 4, 5, 7)
121 = The sum of the first three cross primes from the top: 61, 37, and 23.
The cross primes (and square) 9, 3, 5, 11, 17 appear to express that the sum of 9, 3, and 5 is equal ( | | ) to 17 which is a true statement. Apparently God has a sense of humor.
255 = The sum of all cross numbers including 9 on the column. In mathematics, 255 is the tenth perfect Totient number the first two Totients being cross prime 3 and cross square 9.
85/255 = 1/3 The numerator, 85, is the sum of all five sexy prime quintuplets: 5, 11, 17, 23, and 29. The denominator, 255, is the sum of all 11 cross numbers on the column (10 primes and one square).
256 = 2^8 The sum of all cross column numbers inside the finial circle including 1. (2 and 4 are a part of the cross bar; 1 and 3 belong to the column.) 1, 2, 3 and 4 form a stem at spiral center which is an extension of the knot at 3 and 4. As a power of 8, 256 is the first zenzizenzizenzic number greater than 1.
priming the center to achieve the spiral effect.
Priming the center occurs when moving from center 3 to the next column number, 5. We add 2 to center 3 for a total value of 5. Priming and an adequate energy supply are key in allowing spirals to form their shape. From nine to thirteen the numbers appear to rotate with greater distance between them as if being propelled at a higher speed. It is conceivable that by mirroring the rotation of spiral galaxies using numbers it could yield results which reflect the properties of spiral galaxies whose rotation speeds are higher near the center. Some argue the black holes thought to be at the center of spiral galaxies do not exist but are better classified as neutron stars having a magnetic pole with a force of debatable consequence. It has also been proposed that spiral galaxies form from the inside out and begin with a large compressed core which fragments over time. It's helpful to think of the generated numbers of this spiral as intervals marking moments in time like a clock. The prime column produces Fibonacci-like sequences of numbers that mirror real life spirals. The advantage of these cross primes is they describe a sequence that has a beginning, a middle, an ending and can replicate so long as there is an energy source.
sign of the cross
Going up from center 3 we do not need to add 2 to center 3 because the center has already been "primed" and has added the '2' on its journey south to 5 before heading north to 9. The value of an engine car may be '3' but the added energy output is consumed as it passes through each car and doesn't skip cars. At 77 and 99 Spiral-935 reaches a column height of 13 numbers allowing for the formation of a circular frame around the 10 primes and 1 squared number. I was amazed that after completing its prime column with a pair of twin-digit numbers, the shape of Spiral-935 could either continue spiraling to column number 125 (5 cubed) or alternately - by duplicating her previous half-spire count - converge into a circle at 121 (with 77), at exactly 11 squared! 77 and 99 appear to act as finials for the column marking the end of prime progression in harmony with crossbar finials 88 and 112 (or 110 for the circle) If a strobe were to flash in turns at every multiple of 11 (11, 22, 33, 44....) one may observe how closely the pattern resembles the tradition of the sign of the cross.
Merging the spiral with the circle yields a circumference of 44. Continuing hypothetically with the spiral formula (without circle convergence) will reveal other primes and semi-primes but the next pair of prime numbers (one at top and one at the bottom of the column) will occur at a column height of 23 numbers and at a height of 31 numbers we find a prime number on top and a semi-prime below, bizor-31 with a property unique to all column numbers known at this time of correspondence.
Outside the Finial Circle, prime column pairs are scarce. At a column height of 23 the prime pairs are 863 and 1,109. Another prime pair is located at a column height of 69 (23*3). Within the bounds of the Finials, 23 is also noteworthy for being the last of only two cross primes which are the average of two other cross primes: 11 (5 and 17) and 23 (17 and 29). 11 and 23 are the co-anchors for sexy prmes 5, 11, 17, 23 and 29. Sexy primes are prime numbers separated by 6. They are named after the Latin word for six which is 'sex'. (Nos 1 - 11 in Latin are unum, duae or duo, tres, quattuor, quinque, sex, septem, octo, novem, decem, and undecim.) Remarkably, the five, sexy prime quintuplets are also cross primes given all five of them reside on the cross column. They comprise 50% of all cross primes (excluding cross square 9) and their sum (85) comprises 33.33% of the sum of all column numbers between the fnials (255).
At a height of 31 we find the only semi-prime whose divisor 31 matches the column height its circle forms with column numbers 5,921 and 7,607. No perfect prime pair beyond height 23 has ever been found. (Important Update: At a column height of 69 only the second prime column pair beyond h