Dynamic Symmetry Lesson: Angular Spiral Construction

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Spirals were very important to the Greeks as evidenced in their Ionic columns. The proportions were calculated carefully, resulting in classical beauty known for this period. The proportions used are at the heart of Dynamic Symmetry, using inner divisions that are proportional to the original rectangle. Diagonals and Reciprocals aid in the construction of spirals. We begin with angular spirals formed from line segments between a main diagonal and one of its reciprocals. 

The well-known Golden Rectangle would lead to a “Golden” Angular Spiral described elsewhere. Note though that a 5 by 6 rectangle is chosen here as an example, since it has more turns than a "wider" rectangle/panel and in fact is closer to the Greek Ideal spiral for the Ionic Column. When constructing this 5 by 6 shape, realize that the instructions will hold for any rectangle. Following this set, we will list a few angular spirals constructed for popular panel proportions.

Note in particular how each rectangle is divided initially into a smaller vertical rectangle with the same proportion as the original (on the right in these cases) plus a leftover rectangle; this leftover rectangle is known as a gnomon to this smaller rectangle.

This demonstration is for a spiral on the upper right in each case. Exercise: after viewing this set of constructions, you may construct the angular spirals for each proportion in the lower left, for example.

1. Begin with a rectangle with dimensions 5x6. Draw a main diagonal (Baroque) with one Reciprocal perpendicular to the diagonal from the lower right corner to the top edge. Draw a vertical from this point on the top edge to the bottom edge. This forms a smaller vertical rectangle with the same proportions as the original rectangle. 


5 to 6 Rectangle

  1. Begin to draw line segments, alternating vertical and horizontal segments, joining a segment to the closest point on one of the diagonals: the baroque diagonal of the original rectangle or the diagonal of the first constructed reciprocal rectangle (alternating between these two diagonal lines). Note that these line segments will be circling the principal point, the polar point at the intersection of the two diagonals. 




  1. Continuing... 





  1. Any three consecutive (red) line segments can be visualized as part of a 4 sided rectangle that is proportional to the full rectangle. This can be determined by looking at the diagonal of this implied rectangle; it’s either part of the full rectangle or it’s part of the first proportional vertical reciprocal.







.........

Continue until you are as close to the polar point as possible: 



This is the upper right, angular spiral for a rectangle of proportion 5 to 6. AR 1.20 (10x12, 20x24, 40x36).

Animated GIF (5 and then 2 second delays): 



Additional angular spirals with increasing aspect ratios:

 








Notes:

  1. The spirals are tighter for the rectangles closer to a square, such as a 4x5 . The smaller the ratio of length to width, the more turns in the angular spiral. For this reason, the Greeks preferred ratios closer to 1.191. We demonstrated here with 1.2.

  2. The golden rectangle (10x16.18...) is the only one that can be divided into a square plus a smaller, vertical rectangle with the same proportion as the original rectangle.

  3. The next step would be to construct quarter circle curves (next lesson) but for now construct free-hand curves to be used when planning your painting to draw the viewer's eye around the canvas perhaps, leading to the polar point or the main focus point.

  4. An alternative way to look at these rectangles is to visualize the “gnomons” of these proportional rectangles. In the case of the golden rectangle, the gnomons are squares. For all the other rectangles the gnomons are the left-over rectangles that together with the smaller proportional rectangle, make up the next larger proportional rectangle. A definition of a “Gnomon” is that part which when added to a rectangle enlarges the rectangle to a larger one that has the same proportion.

    Anyway, to make a long story short, while doing these angular spirals we are implicitly constructing “whirling gnomons”. We will describe these further in the next lesson.  The gnomons get wider as the smaller vertical rectangles get thinner. 
 A goal will be to construct a logarithmic spiral.










 

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