90 Degrees of Separation:
The Golden Rectangle & Logarithmic Spiral
Part III
©Lorena Loo
The Golden Rectangle & Logarithmic Spiral
Part III
©Lorena Loo
The equations for the two rectangular spiral diagonals (thin orange lines) are:
Solving for x and y in these two equations yields the intersection point of
(Author Note: Subtract (4) from (3) to solve for x, then substitute this value of x into (4) to obtain the value for y.)
The intersection points of the two sets of diagonals are one and the same point! Therefore their intersection points coincide. Also since the two diagonals in each set are orthogonal to one another, this means the angle between any one of the four diagonals and its neighboring diagonal on either side is 45º.
This means we need only construct the two consecutive diagonals of the golden rectangle in order to find the pole for its rectangular spiral. That pole will also be the pole for the golden spiral, the spiral determined by the golden mean rectangle.
A Rotation & Length
In Part I, we looked at an equation to locate the vertices of the rectangular spiral so the equation was limited to just those points rather than all the points on the rectangular spiral. Now we are going to examine the equation for a logarithmic spiral. Any point on a spiral can be expressed as both a distance from its origin or pole and a rotation about that pole. That allows us to use polar co-ordinates for a spiral.
An example of a polar co-ordinate system is in the diagram below where angular rotation in a clockwise direction is a negative angle, counterclockwise a positive angle. The two violet colored axes intersect at the pole (origin) at 90º. Three gold colored points are illustrated to show points located at a radial distance of 6 from the origin and a rotation of -45º, radial distance 5 and rotation of 30º and radial distance 10 and rotation 135º.
Solving for x and y in these two equations yields the intersection point of
(Author Note: Subtract (4) from (3) to solve for x, then substitute this value of x into (4) to obtain the value for y.)
The intersection points of the two sets of diagonals are one and the same point! Therefore their intersection points coincide. Also since the two diagonals in each set are orthogonal to one another, this means the angle between any one of the four diagonals and its neighboring diagonal on either side is 45º.
This means we need only construct the two consecutive diagonals of the golden rectangle in order to find the pole for its rectangular spiral. That pole will also be the pole for the golden spiral, the spiral determined by the golden mean rectangle.
A Rotation & Length
In Part I, we looked at an equation to locate the vertices of the rectangular spiral so the equation was limited to just those points rather than all the points on the rectangular spiral. Now we are going to examine the equation for a logarithmic spiral. Any point on a spiral can be expressed as both a distance from its origin or pole and a rotation about that pole. That allows us to use polar co-ordinates for a spiral.
An example of a polar co-ordinate system is in the diagram below where angular rotation in a clockwise direction is a negative angle, counterclockwise a positive angle. The two violet colored axes intersect at the pole (origin) at 90º. Three gold colored points are illustrated to show points located at a radial distance of 6 from the origin and a rotation of -45º, radial distance 5 and rotation of 30º and radial distance 10 and rotation 135º.
The growth factor used in the three above spirals is phi, the golden mean, and so we can call these spirals golden mean spirals. Logarithmic spirals with a
value of (90º) for can always be fitted to a rectangular spiral with the
same growth factor so that the corners or vertices of the rectangular spiral lie on the log spiral. We already know that a rectangular spiral with growth
factor is just contained within a golden rectangle. As mentioned in the article Spirals, a logarithmic spiral defined by a golden rectangle can be closely approximated by drawing quarter circles of a series of whirling squares. The diagram below shows how close that ruler and compass construction comes to a true log spiral. Here the green spiral is the true logarithmic spiral for the golden rectangle while the red spiral is the ruler and compass quarter circles construction. The rectangular spiral within the golden rectangle is indicated in purple.
value of (90º) for can always be fitted to a rectangular spiral with the
same growth factor so that the corners or vertices of the rectangular spiral lie on the log spiral. We already know that a rectangular spiral with growth
factor is just contained within a golden rectangle. As mentioned in the article Spirals, a logarithmic spiral defined by a golden rectangle can be closely approximated by drawing quarter circles of a series of whirling squares. The diagram below shows how close that ruler and compass construction comes to a true log spiral. Here the green spiral is the true logarithmic spiral for the golden rectangle while the red spiral is the ruler and compass quarter circles construction. The rectangular spiral within the golden rectangle is indicated in purple.
© Copyright Lorena Loo
The approximated quarter circles spiral remains entirely within the golden rectangle. But the true log spiral actually extends a little beyond the borders of the golden rectangle on three sides as indicated by the green arrows in the diagram above. If you start at the bottom left corner of the golden rectangle and follow the log spiral (in green) as it approaches the top border of the rectangle, it begins to extend beyond the rectangle within about 7º of reaching vertical. When it reaches vertical, it has gone through a rotation of 90º and the point on the spiral is then exactly on the border of the rectangle. That point coincides with a vertex of the rectangular spiral. After that the spiral remains within the golden rectangle until once again it is within about 7º of coinciding with the next border of the rectangle. Again it extends a little beyond the side of the rectangle until it coincides again with the rectangle's border at another completed 90º rotation which is at next vertex of the rectangular spiral. Follow the log spiral through another 90º rotation and you find exactly the same thing.
A Spiral Named Logarithmic
So why is a logarithmic spiral called a logarithmic spiral? The answer lies in the more conventional way of expressing the equation for a log spiral. Instead of writing it as:
Mathematicians like to use the polar form:
Here a and b are constants value and e is the irrational number 2.718281828459045…. and the base of natural logarithms. This number, which goes by the fifth letter of the English alphabet, is extremely important in mathematics. Virtually all if not all important formulas in statistics involve e. The number e is present when we flip a coin, in calculating biological growth such as bacterial growth and even in determining our bank interest. As e appears in growth formulas, it is only "natural" we use it in equations for "growth" spirals.
Since e is also the base of natural logarithms, we can take the natural logarithm of both sides of the above equation to also express it in logarithmic form as:
The log spiral is so-named because it can be expressed in the form of a logarithmic equation.
Now suppose we have a spiral with growth factor G and gamma as the angle between two points on the spiral whose radial distance from the pole differ by a factor of G. Using the polar equation with e, we have:
Solving for b we obtain . Our polar equation for the log spiral can
now be expressed in terms of e and G and .
We saw before how the constant a in the equation affects the size of the spiral but does not determine its shape. It is the growth factor G and the angle which determine the shape of the spiral. The diagram below illustrates four log spirals with the same value of the constant a and angle but with different values of the growth factor G.
A Spiral Named Logarithmic
So why is a logarithmic spiral called a logarithmic spiral? The answer lies in the more conventional way of expressing the equation for a log spiral. Instead of writing it as:
Mathematicians like to use the polar form:
Here a and b are constants value and e is the irrational number 2.718281828459045…. and the base of natural logarithms. This number, which goes by the fifth letter of the English alphabet, is extremely important in mathematics. Virtually all if not all important formulas in statistics involve e. The number e is present when we flip a coin, in calculating biological growth such as bacterial growth and even in determining our bank interest. As e appears in growth formulas, it is only "natural" we use it in equations for "growth" spirals.
Since e is also the base of natural logarithms, we can take the natural logarithm of both sides of the above equation to also express it in logarithmic form as:
The log spiral is so-named because it can be expressed in the form of a logarithmic equation.
Now suppose we have a spiral with growth factor G and gamma as the angle between two points on the spiral whose radial distance from the pole differ by a factor of G. Using the polar equation with e, we have:
Solving for b we obtain . Our polar equation for the log spiral can
now be expressed in terms of e and G and .
We saw before how the constant a in the equation affects the size of the spiral but does not determine its shape. It is the growth factor G and the angle which determine the shape of the spiral. The diagram below illustrates four log spirals with the same value of the constant a and angle but with different values of the growth factor G.
We can generalize the equation for the vertices of a rectangular spiral from Part I to that of a general log spiral as follows:
Here r is the radial distance from the pole (origin) and theta the angular rotation about the pole. G is the growth factor for the spiral, gamma is the angle of separation between two points on the spiral whose distance from the pole differs by a factor of G, the growth spiral. Both theta and gamma must be either in degrees or radians but here we will use the convention of radians. Also, a in our equation is any constant value..
Let's take an example. For a spiral where any two points separated by a (90º) rotation differ in distance from the pole by a factor of G, the growth spiral, the equation becomes:
In the illustration below, this equation is plotted for three different values of a for the same range of theta (in reality the spirals continue forever both inwardly and outwardly). The three spirals maintain the same shape or curve, only their sizes differ.
