Spirals
Part II
©Lorena Loo
Part II
©Lorena Loo
A rectangle of so-called golden proportions, length to width in the ratio of phi to 1, purportedly appears as the most pleasing of proportioned rectangles to human aesthetics. I say "proportedly" because there is some debate whether this is actually the case, even amongst some mathematicians.2
Nonetheless, it is the particular properties of this rectangle rather than its alleged aesthetically pleasing proportions that is of greatest interest.
If you divide a golden rectangle into a square and rectangle, the new rectangle will also have golden proportions. Subdivide the new rectangle into a square and rectangle and once again get a another but smaller golden rectangle. This continual subdivision into whirling squares can be carried on indefinitely, at least in theory.
Nonetheless, it is the particular properties of this rectangle rather than its alleged aesthetically pleasing proportions that is of greatest interest.
If you divide a golden rectangle into a square and rectangle, the new rectangle will also have golden proportions. Subdivide the new rectangle into a square and rectangle and once again get a another but smaller golden rectangle. This continual subdivision into whirling squares can be carried on indefinitely, at least in theory.
The golden triangle is a type of isoceles triangle (two opposite sides are of equal length) such that its sides are phi times the length of its base. Successive whirling golden triangles will also give rise to a logarithmic spiral. Successive points of the sides of the whirling golden triangles will lie on such a spiral. As with the golden rectangle, we can closely approximate this logarithmic spiral by drawing successive circular arcs. In this case, the arcs will be 3/10 of a full circle as opposed to one-quarter. These circular arcs are depicted in the animation below where the red dots represent the centers of successive arcs.
© Copyright Lorena Loo
Spirals are the most beautiful of forms in nature and in mathematics. If you believe the yogis, they are also that of which matter comes into being and out of which it returns to the pure energy field.
1Mark R. Showalter; Saturn's Strangest Ring Becomes Curiouser and Curiouser; Science, Vol 310, 25 November 2005; pp 1287-1288
2Two papers written by mathematicians to debunk common misconceptions about the golden ratio are:
George Markowsky; Misconceptions about the Golden Ratio; College Mathematics Journal, 23.1 (1992): 2-19
Clement Falbo; The Golden Ratio-A Contrary Viewpoint; College Mathematics Journal, March 2005, pp 123,134
1Mark R. Showalter; Saturn's Strangest Ring Becomes Curiouser and Curiouser; Science, Vol 310, 25 November 2005; pp 1287-1288
2Two papers written by mathematicians to debunk common misconceptions about the golden ratio are:
George Markowsky; Misconceptions about the Golden Ratio; College Mathematics Journal, 23.1 (1992): 2-19
Clement Falbo; The Golden Ratio-A Contrary Viewpoint; College Mathematics Journal, March 2005, pp 123,134
The parameters of the golden rectangle's whirling squares can be used to form the most beautiful of all mathematical curves, the logarithmic spiral. Also called spira mirabilis, this spiral has been a common occurrence in the natural world for millions of years. The diagonal corners of the successive "whirling" squares of a golden rectangle happen to be points lying on a logarithmic spiral. The intersection of the diagonals of two successive golden rectangles determines the pole of this spiral, the point of the spiral's ultimate destination as it infinitely spirals inward.
The two orange lines are diagonals of two successive golden rectangles. Their intersection is the pole of the golden mean spiral.
If you rotate through a given angle (one complete revolution, say), the distance from the pole to the points on the logarithmic spiral "grows" multiplicatively by a fixed amount. Consider that amount the growth factor. This is what characterizes a logarithmic spiral from all other types of spirals. In the particular case of a golden rectangle, the growth factor is related to phi (or powers of phi). So the golden rectangle's log spiral is a particular case of a logarithmic spiral.
It is possible to closely approximate such a logarithmic spiral by drawing successive quarter circles using a compass as illustrated in the animation below. The red dots indicate the centers of the successive quarter circles. The ratio of the area of each square to the area defined by its corresponding quarter circle is . If you have read some of the other articles on this web
site, you will have encountered that same ratio elsewhere.
A true logarithmic spiral would not be entirely contained within the boundaries of a golden rectangle and its whirling squares. Still the quarter circles method mentioned above is a very good approximation to this particular case of logarithmic spiral.
It is possible to closely approximate such a logarithmic spiral by drawing successive quarter circles using a compass as illustrated in the animation below. The red dots indicate the centers of the successive quarter circles. The ratio of the area of each square to the area defined by its corresponding quarter circle is . If you have read some of the other articles on this web
site, you will have encountered that same ratio elsewhere.
A true logarithmic spiral would not be entirely contained within the boundaries of a golden rectangle and its whirling squares. Still the quarter circles method mentioned above is a very good approximation to this particular case of logarithmic 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.
Images:
Battle Ram courtesy of Susan Mongold Briggs, Tongue River Farm; http://www.icelandicsheep.com/products.html
Low pressure system off Greenland: Credit: Jacques Descloitres, MODIS Rapid Response Team, NASA/GSFC; http://veimages.gsfc.nasa.gov//6204/Iceland.A2003247.1410.1km.jpg
Venus double vortex & Saturn single vortex: Credit: NASA/JPL/Space Science Institute/University of Arizona; Link to Saturn image: http://www.nasa.gov/images/content/162420main_pia08333-browse.jpg
Mossy spiral; Credit: NASA; http://www.nasa.gov/vision/earth/livingthings/16jul_firemoss.html
Snail Shell; Courtesy of Natures Desktop Wallpapers; http://www.natures-desktop.com/natures-desktop-animals3.php
Bronze Coiled Serpent from China: The Inner Reaches of Outer Space: Metaphor As Myth and As Religion; Joseph Campbell; HarperPerennial; 1986
Battle Ram courtesy of Susan Mongold Briggs, Tongue River Farm; http://www.icelandicsheep.com/products.html
Low pressure system off Greenland: Credit: Jacques Descloitres, MODIS Rapid Response Team, NASA/GSFC; http://veimages.gsfc.nasa.gov//6204/Iceland.A2003247.1410.1km.jpg
Venus double vortex & Saturn single vortex: Credit: NASA/JPL/Space Science Institute/University of Arizona; Link to Saturn image: http://www.nasa.gov/images/content/162420main_pia08333-browse.jpg
Mossy spiral; Credit: NASA; http://www.nasa.gov/vision/earth/livingthings/16jul_firemoss.html
Snail Shell; Courtesy of Natures Desktop Wallpapers; http://www.natures-desktop.com/natures-desktop-animals3.php
Bronze Coiled Serpent from China: The Inner Reaches of Outer Space: Metaphor As Myth and As Religion; Joseph Campbell; HarperPerennial; 1986
