Mathematics News

Why early humans preferred the Golden Ratio

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https://www.youtube.com/watch?v=4VrcO6JaMrM

Further to early humans preferred the Golden Ratio (1.618) too?: Repeated use of the numbers 2, Pi and Phi and the relationship between them could not have been chance.

I, O’Leary for News, wish to thank bornagain for consistently helpful notes in the comments over the years, and to draw attention to his notes on the Golden Ratio in particular:

The golden ratio is found throughout nature and has often been referred to as ‘the fingerprint of God’ (vid ):

From the shape of some galaxies to quantum mechanics, the golden ratio is found throughout nature.

Do We Live in a “Golden Ratio” Universe? (Evolution News & Views December 2, 2014)
*The curl of an elephant tusk
*The shape of a kudu’s horn
*Hurricane spirals
*The distribution of planets in the solar system
*A biological species constant, T
*The spiral structure of the cochlea ear-bone in a fossil hominin
*The logarithmic spirals of galaxies
*The structure of DNA
*The growth of many plants (phyllotaxis)
*The Periodic Table of the Elements
*Spiral shells of certain mollusks, like snails
*Spiral shells of living and extinct ammonites
*Stress patterns in nanomaterials
*The stability of atomic nuclides
*The topology of space-time

What is sculpting the Milky Way into its beautiful spiral shape? In the following article it is claimed that ‘the topology of space-time’ itself is based on the golden ratio and, as such, could be very instrumental in explaining why the Milky Way forms such a beautiful spiral shape (2014):

We suggest that there is a strong case that this so-called ‘Golden Ratio’ (1.61803…) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time…

As well, the golden ratio is found on the quantum scale:

Golden ratio discovered in quantum world: Hidden symmetry observed for the first time in solid state matter (January 7, 2010)

Excerpt: For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.” Radu Coldea is convinced that this is no coincidence. “It reflects a beautiful property of the quantum system — a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material,” he explains.

‘Fibonacci quasiparticle’ could form basis of future quantum computers (Dec. 15, 2014) Excerpt: “The Fibonacci anyon is a non-Abelian anyon whose quantum dimension is the golden ratio (1.617…), and is the simplest anyon capable of performing universal quantum computation,” Vaezi explained. “Using the fusion rule of Fibonacci anyons, it can be shown that the degeneracy of the ground state in the presence of n well-separated Fibonacci anyons on a sphere is the nth number in the Fibonacci sequence.”

Notes: In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons… the Nth Number is some arbitrary positive integer in a sequence.

DNA also has the beauty of the golden ratio embedded within it:

DNA spiral as a Golden Section

Excerpt: The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.
DNA in the cell appears as a double-stranded helix referred to as B-DNA.This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms.

… a cross-sectional view from the top of the DNA double helix forms a decagon:

A decagon is in essence two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon.
The ratio of the diagonal of a pentagon to its side is Phi to 1. So, no matter which way you look at it, even in its smallest element, DNA, and life, is constructed using phi and the golden section!

Cross section of DNA compared to the Rose window at York Minster (the largest Gothic cathedral in northern Europe)

The golden ratio is not only beautiful to look at but it even sounds beautiful when put to music.

One doesn’t hear much about this sort of thing outside of compulsory math classes. Probably because our culture is much more attuned to forgettable books on why everything is a cosmic accident

Reader-contributors like BA77 make a great deal of difference to a shoestring operation like ours. So again, thanks.

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20 Replies to “Why early humans preferred the Golden Ratio

  1. 1
    bornagain says:

    Thanks News. My pleasure. I’m glad that, however small, I can sometimes be of help to you and others.

    Of related note:

    ‘The argument from beauty’ is a Theistic argument:

    Aesthetic Arguments for the Existence of God:
    Excerpt: Beauty,,, can be appreciated only by the mind. This would be impossible, if this `idea’ of beauty were not found in the mind in a more perfect form.
    http://www.quodlibet.net/artic.....etic.shtml

    ENV has an article that was, though technical, humorous in detailing the futile attempts of two materialists who tried to reduce the subjective ‘sense of beauty’ to mere material mechanism.,,

    Beauty Evades the Clutches of Materialism – March 27, 2013
    http://www.evolutionnews.org/2.....70321.html

    Of related interest to theoretical mathematics that are fruitful to the progress of science, it is said that the best mathematical theories, that are later confirmed empirically to be true, were born out of the mathematician’s ‘sense of beauty’.
    http://www.uncommondescent.com.....ent-537503

    Music:

    Brooke Fraser- “C S Lewis Song”
    http://www.godtube.com/watch/?v=DL6LPLNX

    Creatures are not born with desires unless satisfaction for these desires exists. A baby feels hunger; well, there is such a thing as food. A duckling wants to swim; well, there is such a thing as water. Men feel sexual desire; well, there is such a thing as sex. If I find in myself a desire which no experience in this world can satisfy, the most probable explanation is that I was made for another world.
    C.S. Lewis (Mere Christianity, Bk. III, chap. 10, “Hope”)

  2. 2
  3. 3
    groovamos says:

    Wait a sec. When I saw quantum mechanics mentioned above I was immediately suspicious.

    All of the dimensionless constants of nature are irrational numbers. They are pi, root pi, e, and gamma (lower case). And arguably root 2 since it is so common. Help me out if I forgot one.

    As irrational numbers, all of these can be calculated to any number of places by Taylor’s series. And so apparently can phi (lower case). So what is the difference? The four that I mentioned are basic to higher mathematics including differential equations. The occurrence of phi depends upon an arithmetic construction.

    Why am I suspicious? I lived in Austin in the ’80’s. In case that isn’t enough of a hint, we’re talking New Age culture on steroids. Yeah it was fun, kinda like hippie days with shorter hair, spirituality, study groups, paid seminars, yoga, camp outs, retreats, nudity and easy sex thrown in for good measure. Oh that’s right we had that last one in the ’70’s

    I’m not one to eschew quantum mechanics as a pathway into some serious mysteries and and all that good stuff. But pretty well grounded in the STEM culture too I’ve grown really down in the last two decades on sales pitches into the mysteries. And quantum mechanics is at the top of the toolbox for shyster people in the new age culture, Nassim Haramein being the worst. In other words the most successful. I have tried to read papers from this guy and he has no clue regarding higher mathematics, the papers are devoid of calculus: http://resonance.is/explore/nassim-haramein/

    He may have hired someone by now to get around this last issue. Oh and his papers make no sense, almost forgot.

    OK so searching around, the most likely piece to make some sense regarding the topic at hand to non-specialists like me on this is here: http://www.sciencedaily.com/re.....143909.htm

    The phi constant shows up as a proposed ratio for two measurements. That’s it. Can anyone else show me a derived theoretical application for phi? Proposing a ratio for two measurements don’t cut it with yours truly.

  4. 4
    wd400 says:

    This is just numerology. Many of the examples are simply wrong (the DNA measurements for example) others seem to claim any logarithmic spiral is a golden spiral (nautiluses, for example, have logarithmic spirals, but are not very close to a golden spiral).n I’m also keen to hear about these “living ammonites”!

  5. 5
    George Edwards says:

    I wonder if a maple key falling to the ground obeys the golden ratio.

  6. 6
    bornagain says:

    Of note:

    wd400 says these measurements of DNA are wrong.

    Excerpt: The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.

    and yet even wiki says 34 and 22 which is in pretty close agreement:

    The structure of DNA is non-static,[8] all species comprises two helical chains each coiled round the same axis, and each with a pitch of 34 ångströms (3.4 nanometres) and a radius of 10 ångströms (1.0 nanometre).[9] According to another study, when measured in a particular solution, the DNA chain measured 22 to 26 ångströms wide
    https://en.wikipedia.org/wiki/DNA#Properties

    at to grooves:

    DNA in the cell appears as a double-stranded helix referred to as B-DNA.This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms.

    wiki says:

    One groove, the major groove, is 22 Å wide and the other, the minor groove, is 12 Å wide.
    https://en.wikipedia.org/wiki/DNA#Grooves

    again in pretty close agreement.

    But then again anyone who can deny that DNA is designed just after looking at a cross section of DNA, as wd400 can deny it is designed, is not really interested in truth are they?

    cross section of dna
    https://www.google.com/search?q=cross+section+of+dna&biw=1600&bih=745&tbm=isch&tbo=u&source=univ&sa=X&sqi=2&ved=0ahUKEwjck6n8nevJAhXKWh4KHQuyAR4QsAQIGw

  7. 7
    wd400 says:

    “pretty close” doesn’t really cut though, does it? Are we really meant to be astounded that’s it’s possible to find some measurements that give a ratio within ~10% of a low real number?

  8. 8
    ppolish says:

    Wd400, your “just numerology” sure seems to be guiding biology in many situations. What else besides “numerology” guides biology?

  9. 9
    wd400 says:

    Ppolish, I’m not sure you read my comment. It’s numerology to go looking for these patterns with the sloppiness described in BA’s comment. The majority of the examples are simply not true.

  10. 10
    bornagain says:

    wd400 seeing as you pride yourself in population genetics, which is the reigning king of ‘sloppy’ numerology masquerading as hard science, it is the height of hypocrisy for you to criticize the ‘rough fit’ of the golden ratio to so many different beautiful structures in the universe. A ‘rough fit’ which, because of the inherent beauty in it, rightly inspires people to wonder if there is somehow an inherent connection. Myself, I hold the inherent connection to be because of God’s craftsmanship. i.e. ‘the fingerprint of God!’

    “In materialism all elements behave the same. It is mysterious to think of them as spread out and automatically united. For something to be a whole, it has to have an additional object, say, a soul or a mind. “Matter” refers to one way of perceiving things, and elementary particles are a lower form of mind. Mind is separate from matter.”
    Kurt Gödel – Hao Wang’s supplemental biography of Gödel, A Logical Journey, MIT Press, 1996. [9.4.12]

    “So you think of physics in search of a “Grand Unified Theory of Everything”, Why should we even think there is such a thing? Why should we think there is some ultimate level of resolution? Right? It is part, it is a consequence of believing in some kind of design. Right? And there is some sense in which that however multifarious and diverse the phenomena of nature are, they are ultimately unified by the minimal set of laws and principles possible. In so far as science continues to operate with that assumption, there is a presupposition of design that is motivating the scientific process. Because it would be perfectly easy,, to stop the pursuit of science at much lower levels. You know understand a certain range of phenomena in a way that is appropriate to deal with that phenomena and just stop there and not go any deeper or any farther.”,,, You see, there is a sense in which there is design at the ultimate level, the ultimate teleology you might say, which provides the ultimate closure,,”
    Professor Steve Fuller discusses intelligent design in Cambridge – Video – quoted at the 17:34 minute mark
    http://www.uncommondescent.com.....nd-others/

    A few notes on the ‘sloppy’ numerology inherent in wd400’s craft of deception, i.e. population genetics:

    Ann Gauger on genetic drift – August 2012
    Excerpt: The idea that evolution is driven by drift has led to a way of retrospectively estimating past genetic lineages. Called coalescent theory, it is based on one very simple assumption — that the vast majority of mutations are neutral and have no effect on an organism’s survival. (For a review go here.) According to this theory, actual genetic history is presumed not to matter. Our genomes are full of randomly accumulating neutral changes. When generating a genealogy for those changes, their order of appearance doesn’t matter. Trees can be drawn and mutations assigned to them without regard to an evolutionary sequence of genotypes, since genotypes don’t matter.
    http://www.uncommondescent.com.....tic-drift/

    Majestic Ascent: Berlinski on Darwin on Trial – David Berlinski – November 2011
    Excerpt: The publication in 1983 of Motoo Kimura’s The Neutral Theory of Molecular Evolution consolidated ideas that Kimura had introduced in the late 1960s. On the molecular level, evolution is entirely stochastic, and if it proceeds at all, it proceeds by drift along a leaves-and-current model. Kimura’s theories left the emergence of complex biological structures an enigma, but they played an important role in the local economy of belief. They allowed biologists to affirm that they welcomed responsible criticism. “A critique of neo-Darwinism,” the Dutch biologist Gert Korthof boasted, “can be incorporated into neo-Darwinism if there is evidence and a good theory, which contributes to the progress of science.”
    By this standard, if the Archangel Gabriel were to accept personal responsibility for the Cambrian explosion, his views would be widely described as neo-Darwinian.
    http://www.evolutionnews.org/2.....53171.html

    (With the adoption of the ‘neutral theory’ of evolution by prominent Darwinists, and the casting aside of Natural Selection as a major player in evolution),,,
    “One wonders what would have become of evolution had Darwin originally claimed that it was simply the accumulation of random, neutral variations that generated all of the deeply complex, organized, interdependent structures we find in biology? Would we even know his name today?
    What exactly is Darwin really famous for now? Advancing a really popular, disproven idea (of Natural Selection), along the lines of Luminiferous Aether?
    Without the erroneous but powerful meme of “survival of the fittest” to act as an opiate for the Victorian intelligentsia and as a rationale for 20th century fascism, how might history have proceeded under the influence of the less vitriolic maxim, “Survival of the Happenstance”?”
    – William J Murray
    http://www.uncommondescent.com.....ent-510124

    Thou Shalt Not Put Evolutionary Theory to a Test – Douglas Axe – July 18, 2012
    Excerpt: “For example, McBride criticizes me for not mentioning genetic drift in my discussion of human origins, apparently without realizing that the result of Durrett and Schmidt rules drift out. Each and every specific genetic change needed to produce humans from apes would have to have conferred a significant selective advantage in order for humans to have appeared in the available time (i.e. the mutations cannot be ‘neutral’). Any aspect of the transition that requires two or more mutations to act in combination in order to increase fitness would take way too long (greater than 100 million years).
    My challenge to McBride, and everyone else who believes the evolutionary story of human origins, is not to provide the list of mutations that (they think) did the trick, but rather a list of mutations that (actually) can do it. Otherwise they’re in the position of insisting that something is a scientific fact without having the faintest idea how it even could be.”
    Doug Axe PhD.
    http://www.evolutionnews.org/2.....62351.html

    etc.. etc…

  11. 11
    daveS says:

    How do the skeptics explain this?

  12. 12
    ppolish says:

    Golden Ratio in Nature often guides the ability to recreate & grow. Important stuff:)

    http://io9.gizmodo.com/5985588.....-in-nature

    Tree branching often guided by golden ratio. Not the “Darwin Tree” diagram in Origin of Species however. Bogus tree right there lol.

  13. 13
    groovamos says:

    wd400: Ppolish, I’m not sure you read my comment. It’s numerology to go looking for these patterns with the sloppiness described in BA’s comment. The majority of the examples are simply not true.

    I’m siding with WD here. When you have pi and e which show up in all types of RIGOROUS studies of nature, including the propagation of electromagnetic waves in free space, this attraction to phi stuff is a kind of babes in toyland phenomenon. Since the spectacular relevance of the dimensionless constants to analysis is out of the grasp of the untrained, we get this attraction to geometrical patterns which show up in art, and seem to show up in the patterns of living things, without any analytic treatment. There is no need for rigor in this babes in toyland thing.

    The people reading here not trained in science probably don’t know what I mean by “analysis”. The people untrained in mathematics on this thread probably have no idea why the area of a disk is EXACTLY pi(r^2) or how to prove it. Or how to prove the area under a unipolar half cycle sinusoid of unity amplitude and unity period is EXACTLY 2/pi and what relevance this has to science and engineering. Or why all kinds of phenomena of decay (e.g. how in time does the temperature of your house drop after the heat is turned off) rely on e in analysis. Or e.g. the waveform decay of a photon.

    In other words the proposed relevance of phi to the physical sciences doesn’t even reach the level of heuristic. It’s all speculation peppered with gaga.

  14. 14
    Bob O'H says:

    daveS @ 11 – I don’t know about the existence of God but I think that’s a pretty definitive proof of the existence of some sort of Higher Power with a warped sense of humour.

  15. 15
    bornagain says:

    groovamos, my claim is/was that the golden ratio shows up in very diverse places that have no possible physical relation to each other other than God’s craftsmanship.

    To think that there should be some underlying unifying mathematical principle besides God, as you (and others) do, to explain why all these diverse golden ratio patterns show up in such diverse, and unrelated, places, (patterns which we can see with our own eyes by the way), is to dramatically miss the point that I personally was trying to make. Namely, that God is the ONLY unifying explanation for these beautiful patterns.

    I guess a mathematician would miss that specific ‘fingerprint of God’ point! 🙂

    All Things Bright and Beautiful
    ~Cecil Frances Alexander

    All things bright and beautiful,
    All creatures great and small,
    All things wise and wonderful,
    The Lord God made them all.

    Each little flower that opens,
    Each little bird that sings,
    He made their glowing colors,
    He made their tiny wings.
    http://www.storyit.com/Classic.....things.htm

    The Artists – The Artists is a short film about two rival painters who fail to see the bigger picture. – video
    http://vimeo.com/33670490

  16. 16
    groovamos says:

    OK BA I’m going to quote the quote from News under the topic of “mathematics”, but when I try to open the link I get a blank page:

    We suggest that there is a strong case that this so-called ‘Golden Ratio’ (1.61803…) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time…

    So you see this has nothing to do with your perceived harmony of the patterns nature, which itself I am not refuting. This quote is saying something of a “strong case” related to mathematics.

    “Strong case(s)” in mathematics doesn’t cut it. Either the constant phi shows up in the progression of analysis or it doesn’t. I want to see it on a page.

    There are a few people posting on this board who have training in the hard sciences and math, but I have to admit, that it gets a little embarrassing to me when the ID proponents who do not come up with stuff that can be justifiably ridiculed by materialists. “News” posting under the tagline “mathematics” is always going to get my attention for good reasons, in other words red flags and this thread is case in point.

    OK let me say this: when you see the word “analysis” or the phrase “functional analysis” as used by mathematicians, it is understood to be the endeavor to solve problems via the treatment of functions using the tools of higher mathmatics (calculus, differential equations) with additional tools from linear algebra, statistics, etc.

    If you don’t know what I mean by “function” I’m not going to even bother. Suffice to say I am not aware of phi appearing in any functional analysis, and would be very interested if you or anyone else can point me to it if it exists. Please, no sarcasm here, I would like to see it.

  17. 17
    bornagain says:

    groovamos

    as to:

    “So you see this has nothing to do with your perceived harmony of the patterns nature, which itself I am not refuting.”

    Would you even try to refute that there is a overarching harmony to nature?

    “Our monotheistic traditions reinforce the assumption that the universe is at root a unity, that is not governed by different legislation in different places.”
    John D. Barrow

    Stephen Hawking’s “God-Haunted” Quest – December 24, 2014
    Excerpt: Why in the world would a scientist blithely assume that there is or is even likely to be one unifying rational form to all things, unless he assumed that there is a singular, overarching intelligence that has placed it there? Why shouldn’t the world be chaotic, utterly random, meaningless? Why should one presume that something as orderly and rational as an equation would describe the universe’s structure?
    I would argue that the only finally reasonable ground for that assumption is the belief in an intelligent Creator, who has already thought into the world the very mathematics that the patient scientist discovers.
    http://www.evolutionnews.org/2.....92351.html

    as to:

    “This quote is saying something of a “strong case” related to mathematics.”

    That specific quote you take issue with was taken from this 2014 paper.

    Number theory and the unity of science – 2014
    http://www.sajs.co.za/sites/de....._SciCo.pdf

    If you do like their claim, take it up with them. They are the ones who made the claim in the first place are they not? Not I. I’ve made it clear several times now that I think the diverse patterns are a ‘fingerprint of God’ i.e. a mark of His craftsmanship.

    Moreover, contrary to what many mathematicians take for granted, I personally don’t think there ever will be a purely mathematical ‘theory of everything’.

    Even Hawking himself agrees

    “Note that despite the incontestability of Euclid’s postulates in mathematics, (ref. on cite), Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel (ref. on cite), halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”. Thus, based on the position that an equation cannot prove itself, the constructs are based on assumptions some of which will be unprovable.”
    Cf., Stephen Hawking & Leonard Miodinow, The Grand Design (2010) @ 15-6
    https://books.google.com/books?id=7MzOBAAAQBAJ&pg=PA536#v=onepage&q&f=false

    In fact, the belief that there should be a unification between General Relativity and Quantum Mechanics does not follow from the math, but is a belief that is born out of Theistic presuppositions. And since that belief is a thoroughly Theistic presupposition, I hold that Christ offers the correct, empirically backed, solution for the much sought after ‘theory of everything’

    Christ’s Unification of General Relativity and Quantum Mechanics:
    https://docs.google.com/document/d/1Gb75eVQyt3wU0Pwcl5nes4N2axenxQKZa4pYxSSeGzk/edit

  18. 18
    groovamos says:

    Correction: functional analysis is a branch of analysis, I’m not a mathematician and this bubbled up later in my thoughts, the two terms are not interchangeable.

    So BA another quote from the piece as you linked: “A convincing case for assuming a cosmic character of the Golden Ratio can be made based on…”

    So we’re back to my argument. A “convincing case” is not rigor, it is an interesting thing to think about as opposed to having applicability. The constant phi, an interesting thing with interesting properties, does not show up in analysis as far as I know. I am ready to be shown otherwise. Until I am, the constant is to me and interesting thing with no applicability. Show me the money.

  19. 19
    bornagain says:

    groovamos, this seems to be quite the bug for you. Again, I am not claiming that general relativity and quantum mechanics can be linked mathematically via phi. They are the ones suggesting that there could be a deep link because of the fact that the golden ratio shows in so many different places. In fact, in my last post I specifically stated that I firmly believe, via Godel, that mathematicians will never find a ‘mathematical’ theory of everything. And that Christ is the correct solution for the much sought after ‘theory of everything’.

    If you are having such an issue with these people pondering the cosmic significance of the golden ratio, since it shows up in so many places different places, I suggest you write them and vent your frustration on them.

    Moreover, it seems they are far from the only ones who have ever wondered what the connection could possibly be for the golden ratio showing up in so many different places. Wikipedia reveals quite a history, with even Penrose chipping in

    Roger Penrose (b. 1931) discovered in 1974 the Penrose tiling, a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[23] This in turn led to new discoveries about quasicrystals.
    https://en.wikipedia.org/wiki/Golden_ratio#Timeline

  20. 20
    bornagain says:

    Here is a neat quote that is of related interest:

    “People take it for granted that the physical world is both ordered and intelligible. The underlying order in nature-the laws of physics-are simply accepted as given, as brute facts. Nobody asks where they came from; at least they do not do so in polite company. However, even the most atheistic scientist accepts as an act of faith that the universe is not absurd, that there is a rational basis to physical existence manifested as law-like order in nature that is at least partly comprehensible to us. So science can proceed only if the scientist adopts an essentially theological worldview.”
    —Paul Davies (cited in, The Historic Alliance of Christianity and Science)

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