90 Degrees of Separation:
The Golden Rectangle & Logarithmic Spiral
Part I
©Lorena Loo
The Golden Rectangle & Logarithmic Spiral
Part I
©Lorena Loo
Much misinformation and misunderstanding in both published print sources and on the web abound about the golden rectangle and the log spiral. As I also found material covering this topic lacking in properly developing a solid understanding of the log spiral and golden rectangle, I decided to write my own article on the subject.
The Rectangular Spiral
Our starting point is the rectangular spiral. But what exactly is a rectangular spiral? One picture is worth a thousand words. Below are animations of the same rectangular spiral: as clockwise spiralling inwards and as counterclockwise spiralling outwards.
The Rectangular Spiral
Our starting point is the rectangular spiral. But what exactly is a rectangular spiral? One picture is worth a thousand words. Below are animations of the same rectangular spiral: as clockwise spiralling inwards and as counterclockwise spiralling outwards.
The thin blue lines are two successive "diagonals" of the rectangular spiral whose intersection determines the pole or center of the spiral. Using the Cartesian co-ordinate system, we can determine the equations of the two diagonals using the general form:
y = mx + b, where m is the slope of the line and b is the y intercept.
Here are the equations:
y = mx + b, where m is the slope of the line and b is the y intercept.
Here are the equations:
© Copyright Lorena Loo
The diagonals of any two consecutive of these spiralling rectangles intersect at a point which is the pole of the spiral. The pole is the center of the spiral, that point which is the spiral's ultimate destination if it were to spiral infinitely inwards.
The following are properties of the rectangular spiral:
1. The aforementioned two diagonals intersect at 90º. i.e. Orthogonal to one another.
2. Every second vertex or corner of the spiral lies on the same diagonal.
3. The growth factor of the spiral is also the constant multiplicative factor by which the distance of each successive vertex (or corner) of the spiral increases from the pole as we spiral outwards. Or decreases as we spiral inwards.
We now prove the three aforementioned properties using basic math principles.
Below we have drawn the rectangular spiral with growth factor G as indicated by three successive arms of the spiral having lengths G2, G and 1. The blue colored ordered pairs are the x,y co-ordinates in the Cartesian plane of four consecutive vertices of the spiral with the origin (0,0) as indicated.
The following are properties of the rectangular spiral:
1. The aforementioned two diagonals intersect at 90º. i.e. Orthogonal to one another.
2. Every second vertex or corner of the spiral lies on the same diagonal.
3. The growth factor of the spiral is also the constant multiplicative factor by which the distance of each successive vertex (or corner) of the spiral increases from the pole as we spiral outwards. Or decreases as we spiral inwards.
We now prove the three aforementioned properties using basic math principles.
Below we have drawn the rectangular spiral with growth factor G as indicated by three successive arms of the spiral having lengths G2, G and 1. The blue colored ordered pairs are the x,y co-ordinates in the Cartesian plane of four consecutive vertices of the spiral with the origin (0,0) as indicated.
Recall that we can think of the rectangular spiral as a series of whirling half rectangles. The starting rectangle has a height to width ratio of 1/G. The next rectangle in the sequence has a height to width ratio of G. Every rectangle in the sequence will either have a height to width ratio of one of those two ratios. In fact, these ratios alternate. i.e. Odd numbered rectangles in the sequence have the height to width ratio of 1/G, even numbered ones have height to width ratio of G. But this height to width ratio is exactly the same as the slope of the line joining opposite vertices of the given rectangle. In other words, it is the slope of the rectangle's diagonal. We need only be concerned with whether the diagonal slants to the right or left of vertical. Right of vertical means a positive value for the slope, left a negative value for the slope.
Let's look at the line joining vertices 3 and 5. It happens to be the diagonal of the third whirling rectangle in the sequence and so has a slope of -1/G. Two lines which have exactly the same slope and are known to have at least one point in common must coincide. Since the slopes of diagonal 1 and 3 and that of 3 and 5 are identical and the two diagonals share the same point (vertex 3) in common, they are coincident. i.e. Vertex 5 must lie on the diagonal of 1 and 3.
Following the same line of reasoning, we know that diagonal 5-7 coincides with diagonal 3-5 which coincides with diagonal 1-3. So vertex 7 must also lie along diagonal 1-3. We need not proceed any further to now realize that all odd numbered vertices of a rectangular spiral lie on the same diagonal.
To prove the same for the even numbered vertices, we make use of property 1 which we have already proven. So we know that diagonal 1-3 is perpendicular (orthogonal) to diagonal 2-4. Also as the diagonals joining any two consecutive even numbered vertices has slope G, they all are orthogonal to diagonal 1-3. Since diagonal 4-6 has the same slope as diagonal 2-4 and they have vertex 4 in common, they must be coincident. i.e. Vertex 6 lies along the same diagonal as 2 and 4. Similarly all the other even numbered vertices of the spiral lie along this very same diagonal joining vertices 2 and 4.
Property 2 has now been demonstrated.
Let's move on to the final property. Now consider the first two outer consecutive arms of the spiral. We label the vertices as in the diagram below. As this is a rectangular spiral, it follows that .
Let's look at the line joining vertices 3 and 5. It happens to be the diagonal of the third whirling rectangle in the sequence and so has a slope of -1/G. Two lines which have exactly the same slope and are known to have at least one point in common must coincide. Since the slopes of diagonal 1 and 3 and that of 3 and 5 are identical and the two diagonals share the same point (vertex 3) in common, they are coincident. i.e. Vertex 5 must lie on the diagonal of 1 and 3.
Following the same line of reasoning, we know that diagonal 5-7 coincides with diagonal 3-5 which coincides with diagonal 1-3. So vertex 7 must also lie along diagonal 1-3. We need not proceed any further to now realize that all odd numbered vertices of a rectangular spiral lie on the same diagonal.
To prove the same for the even numbered vertices, we make use of property 1 which we have already proven. So we know that diagonal 1-3 is perpendicular (orthogonal) to diagonal 2-4. Also as the diagonals joining any two consecutive even numbered vertices has slope G, they all are orthogonal to diagonal 1-3. Since diagonal 4-6 has the same slope as diagonal 2-4 and they have vertex 4 in common, they must be coincident. i.e. Vertex 6 lies along the same diagonal as 2 and 4. Similarly all the other even numbered vertices of the spiral lie along this very same diagonal joining vertices 2 and 4.
Property 2 has now been demonstrated.
Let's move on to the final property. Now consider the first two outer consecutive arms of the spiral. We label the vertices as in the diagram below. As this is a rectangular spiral, it follows that .
As we have proven orthogonality, because point D is the pole and intersection of the two diagonals, then .
From the Theorem of Mean Proportionals, we know that:
From the Theorem of Mean Proportionals, we know that:
So b = Ga = G2c.
As B, A, C are the first three consecutively ordered vertices of the rectangular spiral and b, a, c are their respective distances from the pole (D), we have shown that these distances decrease by growth factor G as we spiral inward, increase by the growth factor G as we spiral outward.
As B, A, C are the first three consecutively ordered vertices of the rectangular spiral and b, a, c are their respective distances from the pole (D), we have shown that these distances decrease by growth factor G as we spiral inward, increase by the growth factor G as we spiral outward.
We see this type of spiral consists of straight line "arms" where the angle between any two consecutive arms is 90º. Each successive arm of the spiral is the same multiplicative factor smaller (if spiralling inwards) or larger (if spiralling outwards) than its immediately preceding arm. We call this constant multiplicative factor the growth factor, G, of the spiral where G is greater than 1. For a growth factor of 1.5, each successive arm of the inwardly winding rectangular spiral is 1.5 times shorter than its preceding one; for the outwardly winding rectangular spiral, each successive arm is 1.5 times longer than its preceding one.
In a rectangle, the angle at each of the corners or vertices is 90º. Any two successive arms of this spiral then can be viewed as half a rectangle. So you can think of a rectangular spiral as consisting of whirling half rectangles as depicted below where the thick lines represent the arms of the spiral.
In a rectangle, the angle at each of the corners or vertices is 90º. Any two successive arms of this spiral then can be viewed as half a rectangle. So you can think of a rectangular spiral as consisting of whirling half rectangles as depicted below where the thick lines represent the arms of the spiral.
As the product of the slopes of these two diagonals is G(-(1/G)) = -1, this proves the orthogonality of the two diagonals at their intersection which is statement 1.
Let's move on to the second property. Below we have labelled the first 8 vertices of our rectangular spiral, beginning with 1 at the outermost vertex and increasing in order as we spiral inwards. The blue lines are the two diagonals connecting vertices 1 and 3, 2 and 4. We need to show that all odd numbered vertices lie on the same diagonal that joins 1 and 3 and all even numbered ones lie on the same diagonal that joins 2 and 4.
Let's move on to the second property. Below we have labelled the first 8 vertices of our rectangular spiral, beginning with 1 at the outermost vertex and increasing in order as we spiral inwards. The blue lines are the two diagonals connecting vertices 1 and 3, 2 and 4. We need to show that all odd numbered vertices lie on the same diagonal that joins 1 and 3 and all even numbered ones lie on the same diagonal that joins 2 and 4.
Also, triangles ABC, ABD and ACD are all similar right angle triangles.
So
. Therefore b = Ga.
But also
and hence a = Gc.
