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I was really looking forward to this programme but was very much disappointed. Too much time was spent on the “golden section”. Of course it is fundamental to the Fibonacci, and all Lucas series, but too much was made of it. There is so much of interest in the Fibonacci series that was skated over to make room for flights of fancy over the aspect ratios of picture frames and credit cards (not to mention ratios of body part lengths) that I despaired. Anyone wishing to know about the Fibonacci series would have got so much more about the subject from Ian Stewart’s Royal Institution Christmas Lectures in 1997. Maybe that was what the programme lacked, Ian Stewart.
R. Pope/ Fibonacci /question: human dna helix prop
Did I hear years ago on radio 4 that the human dna helix proportion is the same ideal 1:6 proportion? (may have even been on this program)I have always remembered because I think they also spoke about pulse rhythm too. It was an explanation of why humans respond to this proportion in art and music.Is this correct?(about 7 years ago)
Tom - phi
1-(phi-1)=phi squared.
Robert Greenfield
Why did no one bring up the subject of of the Creator of the Fibonacci sequence? Darwinists do not get their own way with this one!
Tam Giles, Leonardo and Golden Section
Leonardo's first employer was Lorenzo de Medici who also attracted the most advanced group of intellectuals to his court. All were versed in Plato, Pythagoras, Euclid, the relatively new number system and all new ideas. They were hoping to reconcile these with Catholic theology. From contemporary writers such as Vasari we know that Leonardo was highly revered. '...having a divine and marvellous intellect...' (quote Vasari) Even if we hadn't Leonardo's writings it would be unlikely that such a man, living among this ferment of ideas, could possibly be ignorant of the Golden Section/Divine Proportion/.618 ratio. But from his own writings: 'If you say ..that the non-mechanical sciences are mental, I would reply that painting is mental and that like music and gometry, which deals with the proportions (ratios)of continuous quantities, and arithmetic, with those of discontinuous quantities, painting deals with all the continuous quantities and with the qualities of proportions of shade to light and through perspective, with distance ' (from leonardo's treatise on painting)This must be read in the light of the then general view that painting and sculpture, being produced by manipulation of tools and matter, was ranked with manual labour as opposed to mental. This seemed to be also shared by Jacky Stedall who doubted that Leonardo consciously used the G.S. for determining his compositions without knowing what was in his mind. The belief that the artist's work was on a 'lower' level than that of poets and musicians (who were accorded the same rank as philosophers, mathematicians, etc) was a thorn in Leonardo's flesh and he returns again and again to this argument.
John Oliver - Fibonacci Sequence
Absolutely compulsive listening, even if my head was spinning slightly between junctions 18 and 19 of the M4. Brought back many (and some happy) memories of why I should never have studied A-Level Maths, although I am now having a fine old time trying to work out if the rate at which I failed said subject and gained others lends itself to a Fibonacci Sequence. I was so impressed that everyone on the programme had managed to get through the discussion without once mentioning Dan Brown...until someone blew it at 10.43am.
Kevan Martin - Fibonacci
Great program - If thats what MB can do with a hangover he should have one every week! It might be mentioned that one of the Liber Abaci's greatest achievements was to introduce Europe (ever fearful of the void) to zero. Medieval mathematics in Europe had been held back by a lack of zero and although Fibonacci called it a sign rather than a number, his knowledge of Muslim mathematics gave him Arabic numerals and their ease of use for commerce meant that Italian merchants adopted Arabic counting and it led to double entry bookkeeping (although Florence banned Arabic numerals in 1299, because they were so easily changed - adding zeros or changing 0 to 6, for example).
Michael Duffy; Fibonacci sequence
Agreed: an interesting programme, yet again. A general point, however:Has Melvyn ever discussed the idea of transferring the programme to TV? It seems to me that it would be the kind of uplifting programme that BBC TV should aspire to (and does) produce. Also the mathematics and science programmes would benefit greatly from a visual aid or two. Clearly more preparation would be needed but I suspect the contributors do not turn up 'cold' anyway. Melvyn: tell is what you think in your next newsletter.
Anthony Constable - golden rectangle
Excellent programme - Fibonacci is always fascinating and Melvyn's team was harmonious and erudite on the topic. But I have to say that the golden rectangle is not quite the appealing shape it is sometimes claimed to be. Perhaps this is the reason why it is so hard to find examples of the golden rectangle in everyday rectangular objects. The ratio 1.61803 rarely occurs in book dimensions. It is absent in standard paper sizes including A4. My credit cards do not come close enough (1.583) to be called genuinely 'golden'. Before the days of universal A4, I once measured a whole batch of scientific and medical journals and only found an ancient copy of The Lancet to be an almost perfect Golden Rectangle. My bookshelves are teeming with rectangular tomes and I haven't yet found one with a 1.618 ratio. If it is so good for architecture, why not books? Is the answer simply that, after all, it is an inconvenient and unattractive shape!!
Dr David Barnett - Fibonacci sequence
A trivium about the golden ratio. This number is the most strongly irrational of the irrational numbers in the sense that its continued fraction representation is a is an infinite sequence of 1s. As a result, approximation by ration of pairs of integers requires surprisingly large numbers to achieve any given accuracy.
John Brock : The Fibonacci sequence
Thanks for sending the newsletter in a larger typeface.However, I don't find the rabbit reproduction anology very helpful in illustrating the Fibonacci sequence. I find it easie to think of a game of leapfrog.A junps over B and takes whatever B has but, A keeps whatever they had before they jumped. At this point an invisible benefactor gives person B back whatever they had before they were jumped on. Then B jumps over A and takes whatever they now have. Again the benefactor gives A whatever they lost. The process might continue ad infinitum. But:the benefactor has to keep intervening at each stage replacing whatever was taken from the one who has been jumped on. And pretty soon neither A nor B feels much better off than the other, just encumbered.Further; the benefactor has to keep making ever larger inputs to feed this system. Its easy to see the implicit lmits and why the growth rate might settle around 1.6 anyway.
Andrew Edwards - Fibonacci
This was thoroughly enjoyable and Marcus du Sautoy's enthusiasm was totally infectious and even managed to interest my teenage son. Thanks so much!
Alan Lloyd - Fibinacci Numbers
There was a comment that many complex items are touched by the simple rules of a Fibonacci number.This "complexity arising from simple rules" is further explored by Stephen Wolfram in his book & annual conference "A New Kind of Science". Science has usually averred that complex activities _must_ arise from complex rules. Stephen Wolfram, your guests, and experiences in nature emphasise the possibility of complex behaviour arising from simple rules.Alan Lloyd
Robert Gore-Fibonacci and Music/Architecture
I would have liked a little more emphasis on how growing plants reveal the Fibonacci numbers and thus how we imbibe their proportions without being aware of it. I hope there is a follow up program on related topics and perhaps mentioning the Greek method of looking for harmony in architecture by measuring length height and breadth and then taking the longest as the dominant note, trying to play them as a chord, and listening to see if a harmony or dissonnence occurred. Some, we call them artists, who may be utterly non mathematical seem to be aware of the proportions of nature and to give their creations beauty with really knowing why. Perhaps It is time to revive this knowledge and studies in our art schools, to draw it back consciously into all creativity, including industrial design, to achieve a more rational basis for judgements about beauty and ugliness.
Dr Joan Thackray - Fibonacci and Golden Section
It was a pity that there was not enough time in the programme yesterday to do more than briefly mention the use of Golden Section (GS) in music. A reference to Roy Howat's meticulously researched book Debussy in Proportion (CUP, 1983) would have been of great interest to your listeners. Dr Howat’s pioneering work in this area has considerably expanded analytical approaches to Debussy’s music even though scholarly opinion still remains divided about the latter’s interest in GS proportions as a means of structuring his music. That Debussy was secretive about compositional matters is not in doubt – Erik Satie could certainly have explained many mysteries we now have little hope of solving. A few enigmatic remarks and indirect observations about ‘shape’ and ‘proportion’ can be found in Debussy’s correspondence, but most have tenuous associations, and only one – this in a letter to his publisher, Jacques Durand, in August 1903 – could reasonably be taken as evidence of his interest in the matter. Returning corrected proofs to Durand, Debussy requests the insertion of a 'missing' bar (a repeat of the previous bar) in the piece ‘Jardins sous la pluie’, and closes his letter thus: ‘elle est nécessaire, quant au nombre; le divin nombre’. (Lettres de Claude Debussy à son éditeur. Durand, 1927). The evidence presented so persuasively by Dr. Howat – and which readers may choose to accept or reject – is, of course, all there in the music.
Terry Collins
In response to Norman and Ruth on Fibonacci:the equiangular spiral which is related to the Fibonacci sequence and which was mentioned by the panellists occurs as the shape of the spiral arms of galaxies. For this reason, the signal chosen by SETI (Search for Extra-terrestrial Intelligence) contains the Fibonacci numbers, giving them a cosmic relevance. If the human race ever receives a reply from space, we may have to develop a language based on the Fibonacci sequence!
David Doff - phi
A great programme! Not often noticed is the connection between phi and the square roots. If we look at a unit triangle (each side length =1), unit square, unit pentagon, unit hexagon ... the lengths of the diagonals come out in the sequence: 1, root 2, phi, root3 ...
Tom Milner-Gulland - Fibonacci Sequence
Numbers only exist because of the human urge to use logical functions - addition, multiplication etc. as a form of duplication (and, reciprocally, division). An original entity, a one, becomes a two, which becomes more, and yet more by way of the idea of function, which in the real world translates as dynamic process. So when we analyse harmonies and visual proportions and obtain from them numerical ratios, we are identifing with processes, and the beauty of the manifest forms resides in their avoidance of the superfluous; ultimately - and in Darwinistic fashion - the unnecessary members of the replication process. Interfering beats in musical chords make for dissonance, as by Helmholtz's psychoacoustic theory of consonance and dissonance; and in maths, certain numbers or functions become essential while others are composite.To add to the literature in the Subject Research page, see H.E. Huntley's The Divine Proportion (Dover, 1970).
MIF at Pathway Initiatives - Fibonacci Sequence
Of course, Melvyn was right on the case when he raised the question, Is there something Platonic, even divine, about these numbers? For Plato, the realm of ideal, archetypal, potential forms, between the physical and the spiritual and known then as the Aether, was a living reality - only ignored and denied later in the 'descent into matter' of human consciousness.Marcus danced off into talk of musical inspiration etc - essentially an implicit but unexpressed affirmation of Melvyn's thought. The mystery of the Golden Ratio, Fibonacci sequence, prime numbers, pi and more have all been resolved in the 2006 'free to download' online published paper Threshold Mathematics at www.aetheraware.org/, should anyone care to check it out.
Chris Palmer - maths stuff
Blimey!! Quite an effort listening to this! I made myself listen as I have a daughter doing A level maths (at Brighton 6th form college) The most interesting bit for me was how maths applies to everyday life, and other subject areas. Does Melvyn understand all he talks about every week?! What a brain he must have!
Chris Whittaker. In defence of the architects of a This was an accessible and entertaining introduction to a fascinating topic but I am perplexed that no one challenged Jackie’s suggestion that the builders of the Parthenon may have been unaware of the mathematics underlying their work. This undoubted error was compounded by the lack of any reference in the programme to the mathematical basis of Classical and Neo-Classical architecture!Vitruvius Ten books of Architecture were written and published between c.30-20 B.C. and clearly indicate the mathematical design of even very modest buildings. The edition by (Rowland and Howe 1999) allows us to see the diagrams illustrating this in detail.Andrea Palladio in Renaissance Italy took Vitruvius’s work forward. (Palladio 1965) is a large format version and again clearly shows the importance of mathematic ratios in the proportions of buildings. Palladio’s work has had an immense influence on subsequent Georgian/ Neo-Classical architecture. Good introductions are (Tavernor 1991) and anything by the late John Summerson e.g. (Summerson 1963).There’s certainly enough for another programme on the golden section and architecture! Given the fundamental aesthetic role of the golden section, it would be interesting to explore the relationship between it demise in modern architecture and the feelings of alienation, expressed by many, towards modern/ postmodern buildings.Yours, Chris WhittakerReferencesPalladio, A. (1965). The Four Books of Architecture. New York, Dove Publications, Inc.Rowland, I. D. and T. N. Howe (1999). Vitruvius Ten Books on Architecture. Cambridge, Cambridge University Press.Summerson, J. (1963). The Classical Language of Architecture. London, Thames &Hudson.Tavernor, R. (1991). Palladio and Palladianism. London, Thames and Hudson.
Anna on Fibonacci
I think that you were angling for some sort of semi mystical/aesthetic reason - anyway something more mysterious than "just" dry old Mathematics to explain why/how humans react in such a positive way to the perfect dimensions of the Golden Ratio whether it is in looking at a classically proportioned building or hearing a beautifully balanced Symphony; nevertheless, I think it is a fascinating question i.e. the Human response to a theoretical mathematical formula. I think it all comes down to the scale of the human body in relation to our environment. We take in so many relative shapes and sizes everyday without thinking but we are completely conditioned to living within a certain scale whether one is surrounded by trees and Nature or whether one is in the urban jungle.Trees are the size they are naturally governed by natural laws and buildings are the size they are a) as an unconscious replication of the natural scale and B)to be manageable for human beings of average height to negotiate and build. This goes beyong aesthetics ,which up to a point, are subjective. The pleasure one gets from contemplating a beautiful Georgian house of Classical proportions (according to Fibonacci s ratio) is as restful and reassuring and non threatening as looking at a spreading Oak tree. It completely corresponds to an unconscious sense of scale,size and proportion that has been around us since birth. I also think everyone has ,according to their own height and size, a particular scale of this Golden Ratio with which they feel most comfortable even if it varies as much as a centimetre from another persons and we subconsciously try and surround ourselves with this perfect size for us whenever we can at home or work.
Norman and Ruth The Fibonacci Sequence
Great topic to look at, but what a pity that the really BIG questions which arise from the prevalence of the Fibonacci Sequence and the Golden Number in most of nature, were largely ignored. Could we have a follow up with questions such as; why this patterning might be so prevalent, and what might it be in the human psyche that finds architecture, painting, music etc so attractive when it is contructed on the Golden Number framework ? and is the Golden Number found only in planet Earth or is it found within the Universe ? etc, etc
Dr Janet Rutherford - Fibonacci Series
I was disappointed that your panalists seemed unaware of the Pythagorean roots of the series, particularly with relation to music. It was indeed his noticing of the relation between the lengths of the strings of an instrument and the production of harmonies that set in motion studies into the golden section and hence the golden rectangle. No mention was made either of the difference between organic and synthetic molecules, etc etc. The question of Leonardo's use of the Fibonacci series, as well as various geometric shapes and concepts, is almost endlessly fascinating. (Though it has been demonstrated that his man in a circle fudges the length of the legs and arms). The obsession with the golden mean's connection with the human body stems from Vitruvius, whose mistranslation of his Greek sources (he confused volume and area), have much to do with the different aesthetic legacies of the golden mean in Greek east and Latin west. I have recently written an article, 'Pythagoras, Byzantium, and the Holiness of Beauty' (Irish Theological Quarterly, vol 71, Nos 3&4, 302-319) about the presence of logarithmic spirals in Rublev's icons, and their Pythagorean associations.If I have a criticism of the programme itself, it would be that it might have been helpful to have one historian of ancient philosophy on the panel. As I say, I know that the programme was on the Fibonacci series, but this is bound to lead onto phyllotaxis and golden proportions. Long before Galileo discovered that the rules of nature are mathematics, Pythagoras had stated: Things are numbers. There are mathematicians who are aware of this; I came upon one on the internet, who claims to be a neo-Pythagorean (which makes being a neo-Neoplatonist seem tame!)To conclude, I would like to say how much I enjoy this programme. Recently, I particularly liked the one on Wordsworth's 'Prelude'; I could have listened to another half hour of it.With thanks for your attention, Janet Rutherford.
James Baring - Phi and Fibonnaci
Phabulous. What a happy and enthusiastic discussion it was too. Nothing cheers people up more than sharing the corner of a veil being lifted. The logic of numbers gives rise to geometry and that is how the journey starts, how order comes from chaos and why it is neither meaningless nor pointless.
Bridget Herbert - Fibonacci sequence
What a pity Melvyn Bragg did not pursue his line of questioning, alluding to the possibility of some kind of "Divine Plan" related to the Fibonacci sequence. The Fibonacci sequence, the Fibonacci spiral, the Golden Section AND the Platonic Solids are, in my opinion, the basis of Creation itself. "It is a sequence about growth", it was said... It is the BEGINNING of all growth, it could be argued.It would have been a wonderful adjunct to hear the views of Dr JJ Hurtak, or Drunvalo Melchizedek, with their esoteric take on the same subject.
Edwina re Fibonacci series
Perhaps the natural propensity towards being attracted to the golden mean propertion maybe to do with the shape of our eyes and the shape of our field of vision?
Fibonacci numbers
What a cracking programme - after a hesitant start the contributors took flight - not always achieved on these programmes despite Melvyn's best efforts. Not just technically brilliant, great entertainment too. Made me recall some of the best of open lectures at uni, listening to the great minds of the time - intellectual nectar.Now, Melvyn, let's have Richard Dawkins and Steve Jones back on air - a Christmas cracker.Keep it up!Peter Gibbs
tony porter/fibonacci
Before discussing 15th century aesthetics, could you please refer to the paintings of Piero della Francesca, a close friend of Pascioli, and the scholarship of B.A.R.(Sam)Carter,who used to teach perspective at the Slade. there is a BBC television documentary programme that reconstructs Carters exposition of Pieros "flagellation" which demonstrates clearly the sophistication of Piero's mathematics.
MIF at Pathway Initiatives - Fibonacci Series (nex
This may be yesterday's news by the time someone reads it but...The mystery of the Fibonacci Numbers Series has been resolved in 2006 in an online publication called Threshold Mathematics available at www.aetheraware.org/ along with other mathematical enigmas such as the Prime numbers, the Riemann Hypothesis and the Golden Ratio/Mean.
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