Sundials and the Golden Ratio

A Letter from Michael Guest

When I read your article (by Billy Gawn) about dowsing for Golden ratio in the September 98 EEG Newsletter, I was involved in a problem relating to the construction of sundials.  As you probably know astronomy has been one of my life-long interests but it is only latterly that I have become fascinated by sundials.  My problem had been on the back burner a little while, displaced by other pressures.  It revolved around the construction of the dial plate showing the hours indicated by the shadow of the sun.  Although simple in principle, the trigonometry can get quite tricky, especially when it’s much longer ago than one cares to remember that one did it at school. I was realty struggling to understand the formulae given in the reference book I was using.

When the dial lies horizontally on the Earth’s surface, it makes an angle with a line of sight parallel with the plane of the equator.  The circle is then seen as an ellipse. The amount of flattening of the ellipse is related to the latitude of the place.

In order to calculate the correct ellipse, one needs to understand how the various elements of the ellipse are interrelated.  The amount of flattening is controlled by the difference between the major and minor axes. The earth revolves round the sun in an elliptical orbit, which is nearly circular so these axis are almost the same length. The sun occupies a special point on the major axis called the focus. There is a second, vacant focus at the same distance the other side of the centre. I needed to know how to generate the formula for the focal positions, as they were the keys to drawing the ellipse of a sundial. Frustratingly, my principal reference book omitted to mention it and I found it impossible to derive from the other elements.

Golden ellipse of the Sun

I became obsessed with the problem of assembling all the relevant factors so that I could construct a comprehensive computer programme, which would define any ellipse at any latitude. From Norton’s Star Atlas I got the orbital elements and some definitions, from the Handbook of the British Astronomical Association all the solar and planetary details for testing, from Pau1 Carless’ unpublished book Geometry and Nature a lot about geometry and the generation of shapes and roots.  To cope with the trigonometry I had to buy a book on A-Level Maths. Even so, certain things eluded me and I finally chased them down through Lancaster Brown’s Megaliths and Man to Alexander Thom’s Megalithic Sites in Britain. When I came to put it all on computer I found the formulae given by my sundial book didn’t work: the author hadn’t mentioned that sines and cosines have to be calculated in radians: fortunately my computer manual came to my help. Finally I pressed the button and out came the answers. They were wrong! But after an adjustment they came right. Phew!

As you know, I am very interested in the Golden Ratio that increases by 1. 618 between adjacent terms in the Fibonacci sequence, which I already had on computer. I had also drawn a sheet of interesting Fibonacci rectangles with sides of 1.618:1. You’re Golden Triangle reminded me that there is also another Golden Rectangle with a ratio of 1.618: 1.272, which is the square root of 1.618.  Its proportions are much more ‘restful’ than the more elongated 1.618: 1 rectangle. Both shapes have been widely used by artists from the renaissance onwards which a visit to any great art gallery will demonstrate.

Golden Ellipse inside Golden Rectangle

While playing around with these shapes, it occurred to me to construct an ellipse using the same ratios for the major and minor axes.  This produces an ellipse boxed into a Golden rectangle.  I first constructed an ellipse using 1.618 for the semi-major axis and 1.272 (the square root) for the semi-minor axis. The result is to me a beautiful and serene shape, very satisfying.  I constructed a second using the ratio of 1.618 to l, the more elongated and flattened ellipse, which is much more suggestive of action than repose.  Both these ellipses incorporate within their geometry the Golden Triangles which you dowsed so interestingly.

However, now comes the interesting bit. You recall that my original purpose was to specify all the elements of an ellipse for any latitude on the earth’s surface. In effect I was defining how a circle on the ground would look if viewed from space parallel with the earth’s equatorial plane. I now had a computer programme, which would show the effects at the whole range of latitudes.

It so happened that the latitude exampled by the author of my sundial book was 51° 48′. When I set the computer programme parameters accordingly, it produced the appropriate ellipse dimensions. Imagine my amazement when I realised they almost exactly with the dimensions of the Golden Ratio ellipse, 1.618: l. By adjusting the latitude very slightly, I produced an exact match at  51.8263°.

Now, my own latitude here at Walsall is not very much greater than this but I thought it worth looking on the map to see what places of interest might correspond. What better than to check for circles on the ground which would project as Golden Ellipses when seen from space. On an O.S. map I carefully measured the latitude of Avebury: it was 51.43°; then Stonehenge at 51.46°. But the nearest was the Rollright Stones at 51.97°.  What a strange coincidence that these famous circles should fall so close to the Golden Ratio latitude, and so close that you would not see the difference on the enclosed diagram.

But more was to come. When I set the Golden Ellipse using the square root as the length of the semi-minor axis, i.e.1.618 to1.272, I found its corresponding latitude was the reciprocal of the other one, 38.1737°. Together they add to 90°. Obviously a circle viewed at low latitude on the earth’s surface would make a less elongated ellipse but inside all this there must be some lovely maths at work.

So I looked through a world atlas to see what interesting places lay on the 38.1737° parallel. I did not have far to look: Athena is just below it by a few miles! It runs straight across ancient Greece! And what had I been busting mv head about for the last few days: Pythagorean triangles and Euclidean geometry! !

I cannot claim that these findings amount to anything more than coincidence but what a wonderful reward for such an effort.  My thanks to you for nudging my mind on to this Golden trail at a critical moment in my thinking.

© 1998 Michael Guest & BSD EEG