Some quite heavy-on-the-eye geometry made to look pretty here. I got a little confused when I read about the Golden Spiral – it made me think there was one, and only one, Golden Spiral but, in fact, it is the construction which decides if something is a Golden Spiral or not – and not the angle it’s drawn with. Usually a Golden Spiral drawn with a 90˚ angle is said to be the Golden Spiral – the one you see inscribed within a Golden Rectangle. A Golden Spiral can actually be any spiral drawn using similar segments of a circle which are reduced by Phi and arranged as shown in the first image below.
The blue triangle is similar to the others, but smaller than the magenta triangle by the ratio 1:1.618. You can guess how the green relates. The second grouping in the first image here shows the important characteristic of Phi – it is the only number capable of creating a sequence of numbers that relate to one another by addition and multiplication – in 3, 5, 8, 13, 5+3=8 and 5x1.618=8 as 8x1.618=13.
Phi can be found in a sequence in which any number is the sum of the 2 preceding ones. I tried creating a sequence of numbers whereby any number is the sum of the 3 preceding ones by choosing any three numbers to begin with and working through the numbers. It doesn’t matter what numbers you begin with because the more you add numbers to the sequence, the more stable the ratio will become and the closer you’ll get to finding the special number. I got 1:1.839 as the ratio for a sequence relating to every 3 numbers. 1:1.926 was reached by looking at a sequence using the preceding 4 numbers. If you look at the second and third image below, the groups to the right show jagged stars marking the failings of these ratios to act in a similar way to Phi.
The fourth and fifth images show Golden Spirals in steps of 10˚ up to 360˚. Some significant spirals occur at 36˚, 60˚, 72˚, 90˚, 108˚, 144˚ and 180˚ – 60˚ giving a hexagon, and 72˚, a pentagon, for instance. It’s interesting how wonky they get after 180˚ – perhaps it’s not necessary to go beyond that point though, because you can no longer join up the points of the curves with straight lines.
The sixth and last images demonstrate the pattern of end points in golden spirals – all ending on the circumferences of circles (in pink/dark pink) which are reduced by Phi from the largest. This is expected, really, but in the sixth image, where I’ve reflected all of the spirals up to 180˚, we find them producing proportionally smaller circles by their interactions. All this can be anticipated with an understanding before hand, but seeing it at work is something else.