Suppose a circle is divided by two radii and the two arcs a and b are in the golden ratio:
b / a = ( a + b ) / b = φ ≈ 1.618
Then the smaller angle formed by the radii is called the golden angle. It is equal to about 137.5° or 2.4 radians. We will denote the golden angle by γ. Its exact value, as a fraction of a complete circle, is ( 3 – √5 ) / 2 ≈ 0.382 cycles.
Nature and Art
The golden angle is found in many contexts in nature. In phyllotaxis, it is the angle separating successive petals or florets in many flowers.
The golden rectangle is a rectangle whose width and height are in the ratio of the golden number φ ≈ 1.618. Allegedly, it has great aesthetic appeal and a great deal – of both sense and nonsense – has been written about the occurrence of the golden number in art.
Moments of Synchrony
As the hands of a clock turn, the angle between the minute and hour hands varies continuously from zero to 180° and back to zero in a period slightly longer than an hour. More precisely, the hands come together every 12/11 hours. It is convenient to measure angles in cycles, or units of 360°, and time in hours and decimal fractions of an hour. Then the angular speed of the minute hand is 1 cph (cycle per hour) and that of the hour hand is 1/12 cph. So the relative speed of the hands is (1 – 1/12) = 11/12 cph. Thus, the minute hand catches up to, or laps, the hour hand after a period of 12/11 hours.
We define moments of synchrony to be those times when the two hands overlap. Clearly, midnight and midday are two such moments. There are 11 moments of synchrony in each 12 hour period. They occur at times:
TS ( N ) = 12 ( N –1 ) / 11 , N = 1, 2, 3, … , 11
So TS( 1 ) is at midnight or midday, TS ( 2 ) is at 12/11 hr or 1h 5m 27.27s and so on.
Golden Moments
Between every two moments of synchrony there are two golden moments, that is, times when the angle between the hands of the clock equals the golden angle γ. Since the relative angular speed is 11/12 cph, the time t after synchrony of the first golden moment is such that
( 11 / 12 ) t = γ or t = ( 12 / 11 ) γ
This implies an interval of 0.416727 hours or almost exactly 25 minutes. The next golden moment is when
( 11 / 12 ) t = 1 – γ or t = ( 12 / 11 ) 0.618 = 0.674
which is at about 40m 27s after synchrony. Further pairs of golden moments occur at intervals of about one hour five and a half minutes.
The first two golden moments after midnight. there are 44 such moments every day.
In total, there are 44 golden moments in a day. Do they have any aesthetic or psychological significant? Presumably not, but they are mathematically interesting.
Exercise: You may wish to construct a complete table of golden moments so that, at various times of the day, you can “savour the moment”.
Acknowledgement: The clock faces are drawn using Mathematica code downloaded from the blog of Christopher Carlson.
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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/
ThatsMaths 