Golden numbers
by Carl Wieland and
Russell Grigg
What on earth do rabbits, the Parthenon, mathematics, sunflowers, art and pinecones
have to do with each other? They are all interconnected in a fascinating way, giving
evidence of a beautiful, not yet fully understood patterning in the world.
Italian mathematician Leonardo Fibonacci (rhymes with ‘Archie’), also
known later as Leonardo Pisano (Leonardo of Pisa, c. 1170–1240) theorized
about the rate of multiplication of breeding pairs of rabbits beginning with one
pair. He reckoned that the way in which the numbers of pairs would increase followed
a mathematical progression in which each number after the first two was the sum
of the two preceding numbers. That is, 0, 1, 1,2,3,5,8,13,21, etc. (3+5=8, 5+8=13,
8+13=21, and so on). This has become known as the Fibonacci series.
Fibonacci numbers can be seen in botany. The arrangement of the whorls on a pine
cone (above right) and the petals of a sunflower (above left) follows a sequence
of Fibonacci numbers.
If you look at the seeds in the head of a sunflower or daisy, you will see that
they are arranged in two sets of spirals, one set running clockwise, the other anticlockwise.
Count the number of spirals going in one direction, and the number going in the
other. You will find that these are always two numbers which are next to each other
in the Fibonacci series (e.g. 8 and 13). A similar arrangement is found in the way
pine cones are constructed, in snail-shell spirals, animal horns and in the arrangement
of leaf buds on a stem.1
Computer modelling2 has
apparently shown that the way in which a group of circles of varying sizes is most
efficiently packed is in a series of spirals that have this Fibonacci patterning—but
no one yet seems to know why.3
Pleasing to artists
The so-called Golden Section (or Golden Ratio), known to most artists and architects,
is also related to Fibonacci patterns. Most people, if asked to choose from a series
of rectangles the one most pleasing to the eye, will choose one in which the ratio
of the two sides (that is, the larger side divided by the smaller) is approximately
1.62.4 In other words, the
long side is 1.62 times the length of the shorter. A rectangle framing the front
of the famous ancient Greek building, the Parthenon (below), has sides which follow
this ‘Golden Ratio’. This proportion is widely found in art and architecture.
The Parthenon in Athens. The ratio of height to length at its front is approximately
1:1.62 — following the ‘Golden Ratio’.
Statistical experiments have shown that ‘people involuntarily give preference
to proportions that approximate to the Golden Section.’5 This Golden Ratio (1.62, or 1.618 to four significant
figures) seems to be naturally pleasing to the human eye. Authoritative works on
art and architecture make bold claims in alleging, for example, that ‘the
Golden Section is aesthetically superior to all other proportions’, which
claim is said to be ‘supported by an immense quantity of data, collected from
both nature and the arts …’.6
When we take the Fibonacci series (ignoring the zero), dividing each number by the
one before it gives: 1, 2, 1.5, 1.6, 1.625, 1.615, 1.619, 1.617, 1.619, 1.617, 1.618
and so on ad infinitum. After the first few, the numbers keep hovering around 1.618.
To three significant figures, they stay precisely on this Golden Ratio of 1.62 indefinitely.
No one yet seems to know why dividing these Fibonacci numbers should give proportions
which happen to be pleasing to the eye.
Returning to living things, we also see that when you count the spirals on a sunflower
hub one way, then the other way, dividing the larger number by the smaller gives
this same Golden Ratio.
Unexplained link-ups
Why should there be all these fascinating and unexplained linkups between things
which are mathematically beautiful and things which are beautiful to the human eye?
And why do these in turn link up to number patterns found in living things?
A mathematician, when interviewed on television in relation to some of these matters
said:
‘I personally believe there is some greater deity that’s organized it.
Everything is too cleverly organized, as far as I’m personally concerned,
to have just happened by happenchance. Whether you say all this was constructed
by God, or whether you believe in some other way of doing it, I’m not quite
sure, but yes, I think there is some power behind it all, but what it is I have
no idea.’7
Unfortunately, our young people are being indoctrinated in humanist/evolutionary
fallacies which try to deny the logical conclusion of intelligent design. For example,
it is commonly claimed that nature (chance) invented man’s mind, which invented
mathematics.8 How then is
it that we find the same mathematical patterns in nature as in that which appeals
to our sense of beauty?
Surely it is more logical to conclude that the connections exist because nature,
mathematics and the human mind, with its subtle sense of beauty, have one supreme
link — they are all the created products of God, the Master Designer.
How to draw a golden rectangle. First draw a square ABCD. Then, find the
midpoint M of side AB. Then, use a compass to extend AB
to a point E, so that ME=MC. Rectangle AEFD
is a golden rectangle. To divide AB according to the golden section, use
a compass to find a point G on AE so that EF=EG.
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Readers’ commentsPeter M., South Africa, 20 October 2010
I have been doing a lot of flower photography lately and had been struck by the mathematic a geometric shapes in seed and petal arrangements. So this article, for me, is very timely. Anyone, not a fool, can see an artist, a mathematician, an engineer, a micro-biologist, an information specialist, a geneticist, a genius, with heart and a superb sense of taste, and many, many other inexhaustible qualities, not to mention omnipotent powers has designed and manufactured these living jewels.
Adolfo E., Canada, 20 October 2010
I am sure that in the hypothetical case that SETI (Search for Extra-Terrestrial Intelligence) were to receive a radio signal from outer space corresponding to the Fibonacci series, evolutionists would immediately claim it as “proof” that intelligent life also evolved in other parts of the universe. The fact that they have so many examples of the Fibonacci series in nature and don’t attribute them to intelligence is telling about their bias against a Creator. |
Related articles
Further reading
References
- Encyclopædia Britannica 7:279,
1992. Apparently, Fibonacci numbers also feature in the genealogy of descent of
the male bee, but no details are provided. Return to text.
- This was stated without detail on a Quantum television
program, screened by Australian Broadcasting Commission, November 13, 1991.
Return to text.
- New Scientist, April 18, 1992, p. 18. Also Physical
Review Letters 68:2098. French physicists have built a physical
model which seems to show that such ‘Fibonacci spiralling’ is a result
of the system’s keeping the energy required for the growth of its parts (for
example, the seeds) to a minimum. Return to text.
- Dividing any line (AB) by a point (C) such that AB/AC = AC/BC
will ensure that these fractions equal the golden ratio, no matter how long the
line. Return to text.
- The Oxford Companion to Art, Ed. Harold Osbome, First
Edition, Oxford University Press, Oxford, 1978, p.489. Return to
text.
- Ibid. p.488. This claim could still be so, even if
it should be claimed that the Parthenon proportions mentioned were deliberately
chosen because of Greek fascination for numbers and geometry. Leonardo da Vinci
was fascinated by this Golden Section, or ‘divine proportion’ as it
was also called, particularly in relation to the proportions of the human body.
See also The Geometry of Art and Life by Matila Ghyka, and The Divine Proportion
by H.E. Huntley, both available in Dover editions. Return to text.
- The speaker was Dr Michael Gore of the National Science and
Technology Centre, Canberra, Australia (Ref. 2). Return to text.
- See James Nickel, ‘Why Does Mathematics Work?’,
Journal of Creation
4:147–157, 1990. Return to text.
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