90 Degrees of Separation:
The Golden Rectangle & Logarithmic Spiral
Part IV
©Lorena Loo
The Golden Rectangle & Logarithmic Spiral
Part IV
©Lorena Loo
In this next diagram, the constant a is the same and the growth factor G is the same but the value of gamma is varies.
Mathematically, anytime there is a geometric progression in powers of phi (the golden ratio) the Fibonacci relationship will always appear. This is a
consequence of the property of phi that . So for a geometric progression where the nth term is Tn we have:
For those familar with the Flower of Life books by Drunvalo Melchizedek, the rectangular spiral is referred to by him as the male spiral and the logarithmic spiral as the female spiral.
consequence of the property of phi that . So for a geometric progression where the nth term is Tn we have:
For those familar with the Flower of Life books by Drunvalo Melchizedek, the rectangular spiral is referred to by him as the male spiral and the logarithmic spiral as the female spiral.
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
It turns out that is directly a function of the growth factor G and the angle and hence b.
Another way of writing the polar equation for a logarithmic spiral is therefore:
The constancy of this angle for a logarithmic spiral is the reason why this type of spiral also is referred to as an equiangular spiral.
In the case of the logarithmic spiral determined by the golden rectangle, is 72.96760887º.
The value of alpha then is another way of characterizing the shape of a log spiral which is determined by the combined value of the growth factor G and the angle . This means there are actually an infinite number of sets of values
for G and that produce the exact same spiral. For example, a growth factor
of and angle of 45º is also a golden log spiral of the golden rectangle. It
is more convenient to use a growth factor of phi and an angle of 90º because of how it readily correlates to a rectangular spiral with the identical growth factor and also to the golden rectangle.
Another way of writing the polar equation for a logarithmic spiral is therefore:
The constancy of this angle for a logarithmic spiral is the reason why this type of spiral also is referred to as an equiangular spiral.
In the case of the logarithmic spiral determined by the golden rectangle, is 72.96760887º.
The value of alpha then is another way of characterizing the shape of a log spiral which is determined by the combined value of the growth factor G and the angle . This means there are actually an infinite number of sets of values
for G and that produce the exact same spiral. For example, a growth factor
of and angle of 45º is also a golden log spiral of the golden rectangle. It
is more convenient to use a growth factor of phi and an angle of 90º because of how it readily correlates to a rectangular spiral with the identical growth factor and also to the golden rectangle.
The values of the growth factor and angle gamma collectively determine the
shape of the spiral. i.e. The value of determines how our spiral looks.
If you draw a line from the pole to any point on a log spiral and then construct the tangent to the spiral at the same point, the angle determined by the radial vector and the tangent to that point will be the same for any point on the spiral. We will call that angle (alpha). The diagram below shows the constancy of this angle for a few different points on the spiral.
shape of the spiral. i.e. The value of determines how our spiral looks.
If you draw a line from the pole to any point on a log spiral and then construct the tangent to the spiral at the same point, the angle determined by the radial vector and the tangent to that point will be the same for any point on the spiral. We will call that angle (alpha). The diagram below shows the constancy of this angle for a few different points on the spiral.
Another reason for looking at the golden mean log spiral in terms of the 90º angle and growth factor of phi relates to the Fibonacci numbers. For a rectangular spiral of growth factor phi, successive arms of the spiral grow by a factor of phi as the spiral winds outwards. If we take any three successive arms of this spiral, it turns out that the longest arm is equal in length to the sum of the previous two arms of the spiral. For example,
L5 = L4 + L3, L4 = L3 + L2. More generally, L(n+1) = Ln + L(n-1). i.e. The length of the (n+1) spiral arm equals the sum of the lengths of the nth and (n-1) spiral arms. This is precisely a Fibonacci relationship.
outwards. So the same relationship holds for any three successive radial distances of the vertices of such a spiral from the pole. i.e. R(n+1) = Rn + R(n-1). Once again we find a Fibonacci relationship. The defining relationship of the Fibonacci numbers is thus inherently found in terms of length in the golden rectangular spiral.
We also know that the radial distance of successive vertices from the pole of the phi rectangular spiral also increase by a factor of phi as the spiral winds
Now for the golden log spiral, successive points on the spiral separated by a 90º rotation increase in their radial distances from the pole by a factor of phi. This too means that any three successive of these radial distances separated by a 90º rotation will also obey the Fibonacci relationship. It seems then in terms of a 90º rotation and the golden spiral, whether rectangular or logarithmic, phi and Fibonacci go hand and hand.
Continue to Part V
© Copyright Lorena Loo
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