The Golden Proportion: composite double-page spread

pp.52-53

The structure of this double opening reflects that of the Golden Proportion: on each single page two golden rectangles are placed on top of each other which are divided into squares and the golden reciprocal 1/⦶. There are further sub-divisions.

The only exception is at the top centre where a smaller Golden Rectangle lies vertically joining the two pages. This structure  is very flexible and can be divided in different ways according to the needs of the information. The border pattern defines the overall arrangement.

In designing this double opening, I was mindful of the fact that to hold together such a variety of different drawings ( as on pp. 44-45), I needed to create a defined border. I was reminded of  certain Islamic patterns, so chose a structure that would contain all the elements I had in mind. Firstly, I drew in the outline of the borders, but leaving out the little stars, and the decoration within the outside circles; these were added later, when I could could determine more accurately their scale and weight.

Most of the various drawings were sketched in on draft paper, and were shuffled around between the two pages.  The more important data was considered first and arranged, leaving out the written text and the more decorative details as the placing and appearance of these gave more scope for adjustment as the pages were completed.

On p. 53, the large 10 pointed star was drawn first, with its central triangles filled in with blue. This gives an ‘eye-hold’ to the page, which helps to settle the other elements. The other areas on both pages were filled in more or less in sequence, and the text completed later; and the more decorative elements were added afterwards, balancing the eye-flow and overall unity. Finally the decoration of the borders and of the roundels at the corners were finished.

A ten-fold star doubles up the five-fold star of the Golden Proportion ⏀. Here it is displayed with smaller satellite stars round its perimeter. In the outer left-hand side is revealed how the Pentagram and the Pentagon are realised within a double circle. 

In the column below it, the drawings show how the construction of the Golden Spiral evolves from a Triangle and a Square; further down, there is a grid that are displayed the Fibonacci Numbers: 1,2,3,5,8.

Lastly, this square displays much of the information on p.50. Know the part, know the whole; each cut creates a three-in-one bond of units; each part relates to all other parts and to the whole.

Moving across, in the image of the 5 circles, if the radius of any one of the circles is 1 then the line OX is⏀.

To the right, this square contains a quotation of the American astronomer and scientist Carl Sagan.’ We are star stuff contemplating the stars’.

The other quotation is from Shakespeare; ‘ Beauty doth of itself persuade the eyes of men without an orator’. In the square on the lowest level, surrounding the five-pointed star is an extract from Timaeus by Plato: ‘Two things cannot be rightly put together without a third, there must be some bond of union between them. And the fairest bonds that which makes the most complete fusion of itself and the things which it combines’.

In the last square, a drawing of how the five-star shape is repeated outwards, and so echoing the Golden Proportion - the Divine Proportion as it is sometimes called - in every part. Around the edge is a quotation from Fra Luca Pacioli,  ‘The Golden Proportion is always similar to itself’. Pacioli was an Italian mathematician and Franciscan friar. He lived 1447 - 1517.

The section joining the two pages show the 10 pointed star emerging from the doubles circles. Below this is a drawing describing how intersections of a large square can yield smaller squares and Golden rectangles;  a double square, and its diagonal: the square root of 5.

p.53 

On the top row of drawings on the left, the image shows an example of a spiral formed by a Fibonacci number of elements - these could be stamens or seed heads  - as in a clematis head or a dandelion ‘puff’. This same formation is also found for instance in a sunflower head or in the common daisy, where the whirls grow in both directions, the numbers being always two consecutive Fibonacci e.g. 21/34 or 55/89. Taking a sunflower seed head, and observing how the seeds are packed, it seems that this particular formation is the one that favours the maximum amount of space and light for each seed.

In the next square to the right, the diagram shows a cross-section of what is called the Primordia: the growing point of plants, where it can be seen that the points appear in a spiral pattern from the centre at an angle at 137.5 degrees - the golden angle - each ‘blip’ on the drawing appear at 137.5 degrees further on the stem. Thus 360 degrees /222.5 = 1.6179…. very near the Golden number of 618…And  222.5 + 137.5=360 degrees. ( My drawing on the left is rather misleading as the numbers should be swapped around).

The last square on the top and the bottom rows displays two five-fold figures, which generate further self-generating  and self -similar Golden relationships of different sizes.

On the next row down, the drawing shows how several diagrams fit together: the square from which the Golden rectangle is created; and when two of these are placed- either side of the square- that creates a semi-circle. 

The right-hand square shows two 5-pointed stars dancing together. The quotation is from Plotinus: ‘All things are full of signs, and it is a wise man who can learn about one thing from another’.

Moving downwards, the chart in this area shows the Fibonacci Series : each successive number arising from adding the two previous numbers together. 0,1,1, 2, 3,5,8,13, 21,34 and so on. The Golden Number I.6180339.… emerges as a proportion between 2 consecutive numbers: e.g.55/34. However the lower numbers in the sequence are not  at all accurate, and range alternately above and below the line: e.g 5/3 = 1.6666…... But higher up the sequence: 233/144 = 1.618055…. the numbers get closer to the Golden number, but will never be exact. 

Below this chart, is a diagram of Pascal’s Triangle. Blaise Pascal was a 17th century French mathematician; though the pattern named after him is believed to be far older. In the arithmetical sequence on the left, the sequence is developed by squaring - counting the boxes horizontally 2, 2x2, 2x2x2 ….thus 2,4,8 etc. Each number is the sum of the numbers in the boxes. Also each number is the sum of the two numbers above it.

The Fibonacci sequence is found by taking the numbers diagonally downwards. 

In the next square to the right, the Fibonacci sequence is found in growth patterns of many kinds - here displayed in the growth patterns of rabbits. An empty circle represents an immature pair and a starred circle a mature pair. Mature pairs stay alive and breed a new pair. Immature pairs mature. Working horizontally, the Fibonacci sequence emerges: 1,1 2,3,5,8….

The square diagonally above this entitled Sun, Moon, Earth depicts the geometric plan of a megalithic Stone Circle -  a flatter circle of type A according to Alexander Thom’s catalogue.

Alexander Thom was a Scottish engineer ( 1894  - 1985). He spent decades studying and mapping the neolithic stone circles all over the British Isles. He discovered that many of them were not circles at all but sometimes egg-shaped or flattened. But whatever shape, they responded to uncanny geometric precision, long before the age of Pythagorus.

He concluded that the stone rings were astronomical observatories, recording the significant passages of time taken by the heavenly bodies as they moved across the sky. He concluded that the megalithic builders must have had sophisticated knowledge of geometry and astronomy, as well as being outstanding observers of the night skies over a considerable span of time; quite apart from the astonishing work involved in building these immense structures. 

In his studies, covering many decades, Thom discovered what he called the Megalithic yard - 2.72 feet, which seemed to be the unit of measurement used in these monuments throughout the British Isles. At the time of publishing some of his theories were hotly debated, though in recent times mostly agreed upon.

The information for these two pages was gleaned from many books, but mostly from The Wooden Books series, designed and published by John Martineau.