Artist, geometer, and educator Mark A. Reynolds investigated Leonardo da Vinci’s frequent use of the octagon. This article uses excerpts from Mark Reynolds’ writing on the subject as appears on his informative and attractive website. For the full article, the reader is encouraged to visit— http://markareynolds.com/?p=89
“…in his (Leonardo’s) studies and sketches of the centralized church, …we can find influences specifically from Brunelleschi, as well as from other fifteenth-century architects working with this type of religious structure…[we also see] his almost obsessive and frequently repetitious drawing of octagonal shapes and forms in his notebooks throughout his career… The object is to provide us with more insight as to why the octagon held so much fascination for Leonardo as one of the ultimate geometric expressions of grandeur and practicality in spatial organization, design, and development.
“It is obvious to anyone spending time looking through Leonardo’s sketchbooks that he drew octagons and octagonal systems almost obsessively throughout his long artistic career…Even when he did not draw the completed octagon, we still find Leonardo’s preoccupation with octagonal geometry. In rapid sketches and in considered drawings, from floor plans to mechanical devices, we find octagonal shapes taking precedence over other geometric systems. At other times, Leonardo’s interest in the square and its diagonal, requisites for √2 geometry and the octagon, can also be found scattered throughout his studies. One well-known example is in the Codex Atlanticus (Cod. Atl), ff. 190 v-b. It shows the head of a warrior for The Battle of Anghiari, with notes on geometry that include several square and diagonal constructions. There is a cube with a diagonal on one of the square faces. The cube also has an internal diagonal that runs inside the form, from front corner to rear corner of the cube, which demonstrates the √2 and √3 lengths within the form.
“Da Vinci used the octagon in his architectural drawings almost to the exclusion of any other, save a rare foray into a hexagon or an occasional dodecagon. There were barely any architectural investigations done with the pentagonal system, other than a couple of fortress and defense constructions. There are times when Leonardo explores hexagonal geometry, occasionally in a temple plan, but more often when working with tessellations and patterns, including knot designs and embroidery. We do also find drawings relating to the circle and the square. Yet the octagon is related to these two shapes, and so, we are left with the predominance of the octagon and Leonardo’s relentless curiosity about it. But why choose the octagon over the other regular polygons when he knew of them all, how to draw them, their armatures and proportioning systems? Perhaps we can find a few possible answers from looking at the time in which he lived, and the prevailing state of the world of art, architecture, and geometry into which he was introduced and that he developed.
“In art schools, architecture and design students are required to have two right angled triangles, traditionally known as “set squares.” They are: 1.) the 45°/45°/90° triangle, and, 2.) the 30°/60°/90° triangle. For centuries, these tools have been traditionally used for ad Quadratum (“from the Square,” 45/45/90) and ad Triangulum (“from the Triangle,” 30/60/90) geometry. The two systems of proportioning were learned early in one’s training in geometric construction, and remained part of the geometer’s toolbox throughout one’s career. Although a somewhat broad generalization, we may say that we usually find the development of ad Triangulum in Eastern cultures, Byzantium, and in the early part of the European Middle Ages, while ad Quadratum seems to have found more of a home in Ancient Greece and Rome. The square, ‘square root two’ geometry, and the theta (√2 + 1) rectangle continued to be developed and utilized in European countries well into the Renaissance. (The subcontinent and China had always had an interest in the application of ad Quadratum geometry as well.) The square, both as a system in and of itself, and as an integral part of all rectangles, was commonplace, and frequently, the circle and the square could be found in various combinations with each other and employed in various rectangular ratios. But it was Leonardo’s ability to combine these other shapes with the octagon, and how these earlier uses of octagonal systems and symmetries may have influenced his own work that is of particular note. It is with this backdrop of ad quadratum geometry in Europe then that we begin our discussion of the octagon.