Mathematical Expression of Ordered Elegance
Dear Friend of Creation,
Flowers blooming all around. Leaves budding from the trees and birds singing in the breeze. The sights and sounds of spring remind us again of the wonderful refreshing beauty of God’s creation!
But what makes for beauty? What is at the root of that harmonious order which makes things pleasant to look at and listen to? Philosophers have wondered about this for ages and have discovered that nature is full of a very pleasant beauty and harmony tied together by a mathematical order.
The ancient Greek mathematician Euclid, founder of geometry, proposed what later became known as the “Golden Mean”, or the “Golden Ratio”, and still later as the “Divine Proportion”. Draw any line and there will be one unique point dividing that line as in the figure to the left where the ratio a+b : a = a : b. That ratio is 1.6180339887…., an important irrational number (continuing infinitely) which is given its own Greek letter, Phi. Its inverse ratio – the “Golden Section” – just happens to be 0.6180339887…. , the exact same number minus the 1. If you draw a large “Golden Rectangle” of length and width in the Golden Ratio and then mark out the square within that rectangle formed from the Golden Ratio point, you will have another Golden Rectangle which can be further subdivided in like manner until you are left with just two small equal squares. If you then draw a smooth curve joining the corners of each square within the series of Golden Rectangles, you will generate a logarithmic spiral originating at the inside corner of the two small equal squares which opens up each quarter turn by a factor of Phi, 1.618…, a “Golden Spiral”, as in the figure on the right. The sides and the areas of each square formed from the Golden Ratios within the Golden Rectangles are in proportion to each other according to the famous Fibonacci Series of numbers, named for mathematician, Leonardo Fibonacci. The series, beginning from the origin, is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377… That is, beginning from the point of origin, the point and then each square in the figure to the right could be given a precise value of 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.
The same series is also formed by adding the first two numbers to get the next such that every number in the series is simply the sum of the preceding two: 0+1=1, 1+1=2, 1+2=3, … 13+21 = 34, etc. The series happens to be intimately related to the Golden Ratio as above in another way. The ratio of any two adjacent numbers in the Fibonacci Series converges to the Golden Mean: 8/5 = 1.6, 34/21 = 1.62…, 89/55 = 1.618…, and 377/233 = 1.6180… The further out in the series you go, the more significant figures will be added to the value of Phi. Nearly infinite adjacent numbers in the Fibonacci Series will produce ratios which will calculate Phi to a nearly infinite number of significant figures! Why would a series of numbers formed from the addition of adjacent numbers work out in such a way as to give Euclid’s Golden Ratio? – except that a great Creator-of-All set it up to work that way!
In fact, the Golden Ratio and related number series is found throughout nature and occurs as a recurrent theme for design and beauty. The number of petals on a flower are most often a Fibonacci Series number. Galaxies form in a logarithmic spiral approximating the Golden Spiral. The DNA molecule, the program for life, measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral, a Golden Ratio good to three significant figures. Golden Spirals and portions of spirals, curves opening up by a factor approaching the Golden Mean, are seen throughout creation. A human ear, for example, is essentially a Golden Spiral.
Human proportions in particular seem “coincidentally” based on the Golden Ratio, resultant Golden Spiral (curves) and the related Fibonacci Series. The ratio of head height to width, outside corners of the eyes to mouth width, mouth width to nose width, chin to eyebrows vs eyebrows to the top of the head are all essentially Golden Ratios. And though there are exceptions, we seem to appreciate beauty in people the more they fit the “Divine Proportion”, as the Golden Ratio came to be called in the Renaissance. Leonardo Da Vinci’s Mona Lisa is a case in point.
Indeed, the Golden Ratio appears to be part of God’s signature on His handiwork, especially in mankind. The human hand itself is a particularly good example – 2 hands each with 5 fingers, from the wrist outward there are 3 knuckles for each finger and 2 for the thumb, with the bone length leading up to each knuckle in a Divine Proportion to the bone length leading from that knuckle. Furthermore, hand length to hand width is also an approximate Divine Proportion. Thus, there may well be deeper meaning to the biblical phrase “holy hands”.
Interestingly, you can add any two sequential numbers together and develop a Fibonacci-type series where all numbers in the series are the sum of the two preceding numbers, and their ratio will also converge to the Golden Ratio! For example, starting with 5 and 6 we would get 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500, 809,… The ratio of 118/73 = 1.616. The ratio of 500/309 = 1.6181… No matter what two numbers we begin with, the ratio of two successive numbers – each one the sum of the two immediately previous and similarly summed numbers – in ALL of these Fibonacci-type sequences always approaches a special value: the Golden Mean, 1.6180339887... and this seems to be the secret behind such series!
There apparently is a special harmonizing and beautifying principle tucked into the addition of two numbers to form the next one. And there may be an underlying spiritual truth. I am reminded here of God’s signature design for a man and a woman and the procreation of children … “the two shall become one flesh…” (Genesis 2:24, Matthew 19:4-6). The institution of marriage and the family is of primary importance in human life, and from it flows much beauty, harmony and order as we comply with the intent and ideal of God’s design.
The amazing beauty and intricacies of Creation should indeed remind us of God. “For the invisible things of Him from the creation of the world are clearly seen…” (Romans 1:20). Even the power and the marvel of mathematics applied to our physical world is astonishing and should remind us of our Maker. That’s because the living Word (Logos – the expression of thought) made all things (John 1:1-3). As the wonderful Easter hymn puts it, “Thine is the glory, risen conqu’ring Son”.
At Creation Moments we pray that you may truly rejoice in our awesome Creator this spring as we are all reminded of the new life that He gives through His miraculous and saving power. And we ask that you open your hearts once again to give what you can to help us spread the Word of that creative power! God bless you!
Yours in Christ,
P.S. This month we’re excited to send you The Hand of God for a donation of any amount! This fascinating book about Fibonacci Numbers and the Divine Proportion is illustrated throughout with full-color photos and diagrams. It covers everything I wrote about in this letter plus a whole lot more. To get the book – or to make a tax-deductible donation of any amount – call 1-800-422-4253 during regular office hours or click here.