Mathematical mystery of ancient Babylonian clay tablet solved

Posted: August 24, 2017 by oldbrew in History, Maths, research

Some Pythagorean triples [credit: Cmglee / Wikipedia]


Could Babylonian base-60 maths be about to make a comeback? The tablet has been dated to between 1822 and 1762 BC and is based on Pythagorean triples, as Phys.org reports. It uses ‘a novel kind of trigonometry based on ratios, not angles and circles’.

UNSW Sydney scientists have discovered the purpose of a famous 3700-year old Babylonian clay tablet, revealing it is the world’s oldest and most accurate trigonometric table, possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.

The new research shows the Babylonians beat the Greeks to the invention of trigonometry – the study of triangles – by more than 1000 years, and reveals an ancient mathematical sophistication that had been hidden until now.

Known as Plimpton 322, the small tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, academic, diplomat and antiquities dealer Edgar Banks, the person on whom the fictional character Indiana Jones was based.


It has four columns and 15 rows of numbers written on it in the cuneiform script of the time using a base 60, or sexagesimal, system.

“Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realised it contains a special pattern of numbers called Pythagorean triples,” says Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

“The huge mystery, until now, was its purpose – why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet.

“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.

“The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new,” he says.

The new study by Dr Mansfield and UNSW Associate Professor Norman Wildberger is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

A trigonometric table allows you to use one known ratio of the sides of a right-angle triangle to determine the other two unknown ratios.

Continued here [includes short video].
– – –
Access to science paper via the Wikipedia link (Note 7) here

Comments
  1. E.M.Smith says:

    We still use 60 oriented math in things as diverse as time and navigation. 60 minutes, 360 degrees (or 6 x 60 ) in a circle.

    The biggest advantage is ready divisibility by many factors…
    1, 2 , 3, 4, 5, 6, 10, 12, 15, 20, 60 if I’ve done it right… makes fractions easy to work and both time and angles get lots of dividing (fractional parts).

    We don’t need base 60 numbers to use 60 based systems, though…

    [reply] and 30

  2. oldmanK says:

    Nice!

    Tallbloke tweeted “Mathematical mystery of ancient Babylonian clay tablet solved”. Not fully, not yet. According to Thorkild Jacobsen 4th millennium Mesopotamia is ‘conspicuous by it absence in the archeological record’. That is pre 3200bce.

    EM Smith says “We don’t need base 60 numbers to use 60 based systems, though…” Sure. Decimal, base 10 is more easy, Base 20 for large numbers – pre Maya-; base 16 (hex) for the four fingered hand – Homer S- would have been an advantage. So why 60? (the average season length in days?) 🙂

  3. oldbrew says:

    One of the Pythagorean triples is 39, 80, 89 (89² – 80² = 39²)

    In solar system synodic period terms, this is > 99.8% true:
    39 Saturn-Uranus = 80 Hale (solar) = 89 Jupiter-Saturn

    That period also = ~60 Saturn orbits.

  4. Thanks, Rog. Great find and great article. Have shared it to Facebook.

  5. I’ve often said that Pythagoras was standing on some mighty strong shoulders with his so-called inventions in the field of mathematics, and that a lot of what has been attributed to him could be found, by any scholar prepared to dig more deeply, in earlier Indian and Babylonian cultures. Even his famous Pythagorean theorum can be attested earlier and without which the Indians, thousands of years before he lived, would not have been able to build their fire altars.

  6. oldbrew says:

    In the conclusion to their study the authors say:

    P322 is historically and mathematically significant because it is both the first trigonometric table and also the only trigonometric table that is precise. Irrational numbers and their approximations are seen as essential to classical metrical geometry, but here we have shown they are not actually necessary for trigonometry. If the dice of history had fallen a different way, and the deep mathematical understanding of the scribe who created P322 not been lost, then very possibly ratio-based trigonometry would have developed alongside our angle-based approach.

    This new interpretation of P322 significantly elevates the status of Babylonian mathematics, and the vast number of untranslated tablets are likely to contain many more surprises waiting to be found. The discovery of trigonometry is attributed to the ancient Greeks, but this needs to be reconsidered in light of the much earlier, computationally simpler and more precise Babylonian style of exact sexagesimal trigonometry. In addition to being historically significant, P322 also brings the founding assumptions of our own mathematical culture into perspective. Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today.

  7. oldbrew says:

    60 = 3*4*5.

    The first Pythagorean triple is 3,4,5 i.e. 3²+4² = 5².
    – – –
    The properties of a primitive Pythagorean triple (a, b, c) with a < b < c include:
    Exactly one of a, b is divisible by 3.
    Exactly one of a, b is divisible by 4.
    Exactly one of a, b, c is divisible by 5.
    The largest number that always divides abc is 60.
    [bold added]

    http://en.wikipedia.org/wiki/Pythagorean_triple#General_properties

  8. oldmanK says:

    Season length in days are between ~87 to ~93 days. If a scale for a measuring system for the seasons, such as a quadrant for eg. equinox to solstice, is required the best number is 90 (divide by 3x3x2x5 ). That gives a circle of 360 divisions. The circle (circumference ) easily divides by its radius into six parts of 60.

    But why that????. 2×5 would have been an easy start; the fingers on two hands. 2x5x2 for the pre-Maya; they added the toes?. We use 2x2x2x2 for digital work. It seems that what was, primarily, familiar, and was also convenient, was used.

  9. “Babylonians beat the Greeks to the invention of trigonometry”

    No earthbound ancient people “invented” trigonometry and geometry. It was all inherited from the “gods”. It is all present in the Great Design of the ‘gods’ (uncovered and thoroughly verified by my research 20 years ago), of the surface of the Earth, its motions in space, and the entire solar system. it was all designed, and the design imposed roughly 20,000 to 10,000 years ago. Only shards of the ancient knowledge were retained by the earthbound people of earliest known history (Babylonians, etc.).

  10. oldbrew says:

    oldmanK : ‘The circle (circumference ) easily divides by its radius into six parts of 60. But why that????’

    Close to the number of days in a year perhaps (6*60, +5.25). That would have been easy to observe. An orbit is very nearly circular.

  11. JB says:

    I never took the Greeks to have invented anything, whether art, mathematics, philosophy, politics, etc.. Just variations on a theme like everyone else was doing around them. When Solon was told by the Egyptian priest their culture had “forgotten” their past due to the catastrophe, he wasn’t just referring to their cultural history. I don’t believe this is “a rare example of the ancient world teaching us something new.” It’s another example in a long, discontinuous thread of modern man’s recovery of what was forgotten like the Greeks did.Take for example one of the ancient quarry cuttings that indicate a cutting wheel >6M in diameter and 1/8″ wide, with a cutting speed in one of the hardest stones known twice the rate we can produce. Ancient man not only had a technology we can’t reproduce, they had a measuring system and precision we barely can reproduce.

    The 60/360 measurement system of the Ancients has been well documented, and matches the same approach used in the geometry of the megalithic pyramids. Whenever I do not have access to precision measuring tools I find myself resorting to ratios. Carpenters use them regularly. To me this isn’t genius, it just using the ol’ noggin. The advent of the table of ratios was just the next logical step if a person is engaged in a large building project.

    Mechanization by the Industrial Revolution has seduced Man into thinking he’s a lot smarter than anyone before him, despite the evidence to the contrary.

  12. oldmanK says:

    oldbrew : to start from the end > ‘Close to the number of days in a year perhaps (6*60, +5.25)’. That can only be known when observing against some datum. And one may add, that datum is/has to be for some clarity, on the horizon. Next, for one whole year one adds the season days. But they are never 90 per season. They are either more, or less, and never add up to 360.

    Pragmatism says use a scale with a number the is convenient for comparison, easily divisble. Ergo, 90 per season.

    (Probably by now several know what I’m driving at. Behind some custom or practice there is a concept, and proof of its onetime existence is some remnant of where it was employed. Here is a probable one, and from a similar other I have been able to forecast solstice day and time, so it works. That it is all coincidence is impossible; intention is very evident. Link: https://melitamegalithic.wordpress.com/2017/04/17/melitamegalithic/ )

    Now to ‘ ‘The circle (circumference ) easily divides by its radius into six parts of 60. But why that????’ Sorry, that was a mental jump on my part. Whoever conceived the above perceived also why that happens. And some 4-5kyears ago the earth orbit was nearer to circular – didn’t need to bother with ellipses. Plus they did not have near 2000yrs of misleading bias. Another thing, most of us who fiddled with dividers/compasses have drawn the six petalled rose within the circle with same radius; learned how to bisect and then add to it. Dead easy once the concept is grasped.

  13. oldbrew says:

    From the paper: ‘all the values in P322 are exact – and this is important because it distinguishes P322 from all modern day trigonometric tables.’

    Some people point out that 60 can be counted on two hands, using
    4*3 knuckles of fingers on one hand, and all 5 digits on the other:
    12 * 5 = 60.
    – – –
    ‘Babylonian records of observations of heavenly events date back to 1,600 BCE. The reason for adopting their arithmetic system is probably because 60 has many divisors, and their decision to adopt 360 days as the length of the year and 360 [degrees] in a circle was based on their existing mathematics and the convenience that the sun moves through the sky relative to fixed stars at about 1 degree each day.

    From about 700 BCE the Babylonians began to develop a mathematical theory of astronomy, but the equally divided 12-constellation zodiac appears later about 500 BCE to correspond to their year of 12 months of 30 days each. Their base 60 fraction system which we still use today (degrees / hours, minutes and seconds) was much easier to calculate with than the fractions used in Egypt or Greece, and remained the main calculation tool for astronomers until after the 16th century, when decimal notation began to take over.’

    http://nrich.maths.org/6070
    – – –
    ‘The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree.’

    http://en.wikipedia.org/wiki/Babylonian_calendar

  14. oldmanK says:

    oldbrew: some comments to your post.

    para1; Those a simple handy tables — in use to this day in many crafts. I used such in carpentry for making windows, and the shape of my plot of ground.

    para2: quite so, and one may find more of such. The difference lies in the possibility/probability that one is driven to develop/use a particular scale (such as 90) in his devised system. The ‘degree’ in arc measurement does not seem to have existed independently before.

    Para 3: this extract assumes the early Babylonians to already have had the ‘360 degree circle’ concept. It refers to dates of 1600BCE. The calendar in my link dates to prior 3200BCE. Knowledge diffusion had been going on in the Med for more than that (as shown in agrilore going back to 5500BCE)

    re Para 4: I am afraid this is a subject where I have little staying power and quickly get lost. But I think it is more complicated than I like it to be. Vide here: http://www.dioi.org/vols/w91.pdf at page 30.

    We read of the Babylonians, but there were others before, and from whom they were descended. That is more interesting when it comes to origins, and sources of lore and ‘science’ (and language). For example this from last para ” plus an intercalary month inserted as needed by decree.” show where cultic obfuscation had already set in. Nature (agriculture) does not observe and follow cult practices.

  15. Paul Vaughan says:

    Above OB has already noted exactly what I was going to note ((3*5)*(4)=60; 5^2=25).

    “fingers & toes, fingers & toes, 40 things we share,
    41 if you include the fact that we don’t care”
    — The Tragically Hip

    3 articulation points for each finger (1 tip + 2 knuckles = 3 points)
    times 5 fingers per hand and 5 toes per foot
    gives 4 times 15 points per foot or hand
    matches 4 columns of 15

    MSM takes note:
    http://www.ctvnews.ca/sci-tech/babylonians-not-greeks-may-have-been-first-to-study-trigonometry-study-1.3560013

    It’s a beautiful reminder.
    It’s not enough to be principled when we can be more simply principled.
    Unnecessary complexity of “expertise” reduces naturally.

    As/if time permits I’ll share connections with φ, Φ, & √5.

  16. oldbrew says:

    To get the party started…
    16*63*65 = 21*13*4*60
    39*80*89 = 89*13*4*60

    13*4*60 is common to both of these Pythagorean triples.
    Therefore their multiplication ratio is 21:89 = 1:φ³

    http://en.wikipedia.org/wiki/Golden_ratio#Geometry

  17. Chaeremon says:

    60 and 360 belong to a unique and surprisingly small finite series “highly composite numbers that are half of the next highly composite number”, http://oeis.org/A072938 , f(k) such that f(k+1) = 2*f(k), numbers m such that d(m)>=d(k) for 0<k<2m and d(n) being the number of divisors function.

  18. oldmanK says:

    Chaeremon; very interesting. This again poses the question of whether the choice of 360 division to the circle was mathematical flair, or a matter of necessity. The latter being 90 divisions to a quadrant to approach a divisible average of days per season.

    The compass still uses 360 deg to a circle, but specific necessities also added a scale of 6283mils/rad (or rounded to 6400 or 6000) for more accurate uses http://www.compassdude.com/compass-units.php

    Quote from link: “There are 2 PI radians in a circle. PI is a constant of approximately 3.1416. That is 2 * 3.1416, or 6.283 radians. Divide each radian into 1000 mil-radians and you see there are 6283 mil-radians in a circle. Mil-radians are called mils for short.
    17.78 mils equal 1 degree.”.

    In digital electronics binary is a necessity, augmented by a base 16 for coding. We inherited the Arabic 0-9 digits for base 10, but the Romans used base 10 with an unholy method of expressing a number (purposely intended to confuse – I bet). My point, necessity ruled not flair.

  19. Paul Vaughan says:

    OB, a suggestion:
    Correct * to +
    above where you wrote
    “The first Pythagorean triple is 3,4,5 i.e. 3²*4² = 5².”
    intending
    “The first Pythagorean triple is 3,4,5 i.e. 3²+4² = 5².”

    “Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.” — https://en.wikipedia.org/wiki/Pythagorean_triple#Fibonacci_numbers_in_Pythagorean_triples

  20. oldbrew says:

    Thanks PV – corrected.

    Every first odd number > 1 makes a P triple with two other consecutive numbers that sum to its square.
    3,4,5 (4 + 5 = 3²)
    5,12,13
    7,24,25
    9,40,41
    etc.

  21. oldbrew says:

    Math Warehouse says:

    The Babylonians came up with a general formula for creating Pythagorean triplets.
    This formula represents triplets for any U and V:

    Hypotenuse U² + V²
    Larger leg U² − V²
    Smaller leg 2UV

    As an example if you make U = 2 and V = 1, you will get the 3,4,5 triplets
    4 + 1 = 5, 4 − 1 = 3, 2(2)(1) = 4

    Hints for creating unique triplets using the Babylonian formula:
    U and V should not have any common factors
    U and V should not both be ODD
    U and V should not both be EVEN

    Therefore, to create a unique triplet, choose an even and an odd number for U and V
    and make sure that the numbers do not have common factors

    http://www.mathwarehouse.com/geometry/triangles/pythagorea-theorem/pythagorean-triplets.php
    – – –
    With this formula the smaller leg is always a multiple of 2, so if the result has 3 even numbers, divide them all by 2.
    For the case where the smaller leg is an odd number, the smaller and larger legs swap formula e.g. 7,24,25:
    4² + 3² = 5² = 25
    4² – 3² = 7
    2 * (4*3) = 24

  22. Paul Vaughan says:

    oldmanK constructively provoked:
    “the Romans used base 10 with an unholy method of expressing a number (purposely intended to confuse – I bet)”

    The components of the simplest Pythagorean Triple (3,4,5) are integer partitions of a golden binomial expansion of a Conway Triangle hypotenuse:

    5 = (√5)^2
    = (Φ+φ)^2
    = (Φ+φ)(Φ+φ)
    = ΦΦ+Φφ+φΦ+φφ
    = ΦΦ+φφ+Φφ+φΦ
    = (ΦΦ+φφ)+Φφ+φΦ
    = (3)+1+1

    3 = (ΦΦ+φφ)
    4 = (ΦΦ+φφ)+Φφ = (ΦΦ+φφ)+φΦ
    5 = (ΦΦ+φφ)+Φφ+φΦ

    √5 = Φ+φ (where Φ=1/φ) is a golden partition of a Conway Triangle hypotenuse by a diameter 1 circle:

    5^2 = 4^2 + 3^2
    = 25 = 16 + 9 arises from Recursive Pareto Principled Division of Unity.

    A property of Pareto Principled Division of Unity is fractal similarity:

    Red (1/5) and blue (4/5) are similar (in the sense of the formal definition of similar triangles — in this case similar Conway Triangles) to white (1=5/5=1/5+4/5).

    Hierarchical application of the Pareto Principle is exploited to efficiently achieve
    4/5 + (4/5)(1/5) = 20/25 + 4/25 = (80+16)% = 24/25 = 96% perfection with

    1/5 + (1/5)(4/5) = 5/25 + 4/25 = 9/25 = (3^2)/(5^2) = 36% of the effort smartly invested and
    1 – 9/25 = 25/25 – 9/25 = 16/25 = (4^2)/(5^2) = 64% of the effort averted.

    The most primitive Pythagorean Triple summarizes the partitioning of effort into invested and averted.

    Anywhere there’s spin there’s division. Wherever symmetry results from division, there’s fractal geometry and recursive boundary conditions.

  23. oldmanK says:

    A bit of info – or gossip. A little delving into the Babylonian mind (which i should have done earlier, I admit). Babylonian is a descendant of earlier Akkadian and related to Ugaritic. The latter was a old/earlier interest of mine for several reason, one being that my native tongue is Semitic, and exposed to old peasant/rural Maltese, and which made certain Ugaritic words/expression familiar. (In one instance, which happened to relate to earlier research, reading specific bits of cuneiform directly made more sense than via Hebrew [as academia insists] – but I’m no linguist, just fascinated). Ugaritic numbers are much the same as Maltese.

    Ugaritic has numbers 1 to 10 (pretty same pronunciation). Same for 11 and 12. Same for 20. Same for the ‘hundred’ and for ‘thousand’. The concept in speech, of the number system is base 10. It is expressed so up to number 59 in roman fashion. https://en.wikipedia.org/wiki/Sexagesimal

    So the question remains. Why then jump to sexagesimal/base60 ? (the Mayan calendar has something of a similar mix but base20, yet my opinion here is they also tried to approach the number of day in the year, in one case 20×18 ).

  24. oldbrew says:

    Re: ‘5^2 = 4^2 + 3^2
    = 25 = 16 + 9 arises from Recursive Pareto Principled Division of Unity.’
    – – –
    In the Babylonian formula, if U = 2 and V = 1:
    (2² + 1²)² = (2(2 * 1))² + (2² – 1²)²

    https://tallbloke.wordpress.com/2017/08/24/mathematical-mystery-of-ancient-babylonian-clay-tablet-solved/comment-page-1/#comment-129936

    Re Pareto: obviously 5 is 1/5th (20%) of 5². Dividing the square of every greater Fibonacci number by 5 will return a number close to another F number e.g:
    8² / 5 = 12.8 (F = 13)
    13² / 5 = 33.8 (F = 34)
    etc.

  25. Paul Vaughan says:

    Maybe Babylonians were smart investors with historic humor.

    80-20 = 60
    96-36 = 60

    60 is the Pareto difference between performance and investment.

  26. oldbrew says:

    PV – re ‘√5 = Φ+φ (where Φ=1/φ) is a golden partition of a Conway Triangle hypotenuse by a diameter 1 circle’

    In terms of the Babylonian ‘U and V formula’:

    Hypotenuse U² + V²
    Larger leg U² − V²
    Smaller leg 2UV
    means
    U = √Φ and V= √φ (in a Conway Triangle)

    U² + V² = √5
    (larger and smaller legs get reversed)
    2UV = 2
    U² − V² = 1

    Conway Triangle = 1,2,√5

  27. oldbrew says:

    PV quoted:
    “Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.”
    – – –
    In such triples, the sum of the two largest numbers (sides) is the square of a Fibonacci number which is also a multiple of the other number.

    3,4,5: 4+5 = 3² (1*3 = 3)
    5,12,13: 12+13 = 5² (1*5 = 5)
    16,30,34: 30+34 = 8² (2*8 = 16)
    39,80,89: 80+89 = 13² (3*13 = 39)
    105,208,233: 208+233= 21² (5*21 = 105)
    etc.

    The multiple in brackets also follows a Fibonacci progression: 1,1,2,3,5 etc.
    – – –
    the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

    The sum of any two consecutive ‘short legs’ in the list equals a bypassed Fibonacci number:
    3 + 5 = 8, 5 + 16 + 21, 16 + 39 = 55, etc.

    The sum of any two consecutive ‘middle legs’ in the list equals double the same bypassed Fibonacci number.

    The sum of any two consecutive hypotenuse values in the list equals a Lucas number:
    5 + 13 = 18, 13 + 34 = 47, 34 + 89 = 123, etc.
    [this is true of any two alternate Fibonacci numbers]

    The area of any triangle in the list (short * middle/2) = the product of four consecutive Fibonacci numbers.
    – – –
    In any Pythagorean triple, two of the three numbers will sum to the square of a whole number(n1).
    The difference between the same two numbers will also be the square of a whole number(n2).
    n1 * n2 will be the third number in the triple.

    Example 1: 3,4,5
    5 + 4 = 3²
    5 – 4 = 1²
    3 * 1 = 3

    Example 2: 39,80,89
    89 + 80 = 13²
    89 – 80 = 3²
    13 * 3 = 39

  28. Paul Vaughan says:

    The imperfection remaining after recursive Pareto Principled division of unity:
    (1/5)^2 = 1/25 = 4%
    for a difference of 64-4 = 60 (see above)

    The difference can be calculated directly:
    (4-1)/5 = 3/5 = 60%.

    72° in trig reps of φ are rooted in recursive Pareto Principled division of unity:

    72 = 2*100*(0.36)
    360/5 = 2*100*(3/5)^2

    “The term was coined by Ramanujan (1915). The first 15 superior highly composite numbers […] are also the first 15 colossally abundant numbers […]”

    “Highly-composite numbers are labelled in bold and superior highly-composite numbers are starred.”

    For future reference note well from the table of primorial factorizations:
    11*210=2310

  29. dadgervais says:

    Not of any significance but, in High School (some 50 years ago), I derived the following:

    For all u,v such that gcd(u,v) = 1, and v odd,

    A = 2*u*v + 2*u*u
    B = 2*u*v + v*v
    C = 2*u*v + 2*u*u + v*v

    are a Primitive Pythagorean Triple (PPT). All PPTs are of this form.

    I doubt that I was the first (or even among the first thousand) to find this.

    –dadgervais

    p.s. Farey Sequences are an efficient way of generating co-prime number pairs. Each Farey term generates one or two unique PPTs (two iff both numerator and denominator are odd).

  30. oldbrew says:

    Re The Babylonian general formula for creating Pythagorean triplets.
    Hypotenuse U² + V²
    Larger leg U² − V²
    Smaller leg 2UV
    – – –
    For any P triplet: (U² + V²) + 2UV = the square of an odd number

  31. Paul Vaughan says:

    Internal adjustment of the unstable major western fault need not hinge on external powers when it’s feasible to rebalance internal divisions for global stability.

    2 Valleys of Stability in 1 Farey Tree:

    √5 = φ+Φ is a difference of squares:
    https://tallbloke.wordpress.com/2017/09/03/why-phi-the-rainbow-angle/#comment-130126

    A sharp leader with respect for the utility of fractal (scale) symmetry will take naturally principled responsibility for adopting the uncostly valley of stability, which demands only internal rebalance.

    forthcoming: second-order Ramanujanian matrix model of solar system’s current configuration