Lunes, rosettes and Leonardo, 09th of November 2009.

Leonardo again. A lot of posts on Leonardo recenlty is a consequence of a work on a larger project, and pieces published here are only a small part of a larger picture. Comming soon I hope. Anyhow ... Lunes. Lune is a piece of surface whose borders are two circular arcs. Only concave shapes are typically considered to be lunes, i.e. shapes in which the centres of curvature of the both arcs are on the same side of the shape. The shape got its name from the Moon, of course.

In the context of Leonardo da Vinci, the lunes are interesting because Leonardo often used them in his designs of ornaments, and in the failed (of course) mathematical attempts of quadrature of the circle (one such interesting attempt is related to Hippocrates of Chios). I represented one of his numerous designs (from Codex Atlanticus) related to combination of polygons and circles (famous vitruvian man belongs to this category) in the "modern" and colored edition in the image below, where one can clearly see the lunes and Hippocrates' proof that the quadrature of dark blue areas is possible.

Especially interesting are the lunes whose both arcs have the same curvatures, because one can nicely fill up the circular piece of plane with them. And since I intend to construct the rosette, it it important to me. I first want to inscribe a certain number of circles in a lune, as shown in the image below.

Clearly, one can inscribe infinite number of such circles in a lune. Determination of positions and radii of the circles in not too difficult, and one can always use Mathematica in case of problems. One obtains the quadratic equation for the radius of the next circle that depends on the radii of previously inscribed circles. Once we inscribe circles in a lune, we can color them as we wish, and then rotate the lune around its tip certain number of times so that one completely covers the circle in such a way (formally, one gets completely covered surrounding of the center but the shape of the obtained structure is only in the limit of an infinitely thin lune really a circle). It is also obvious that lunes of arbitrary thicknesses are not allowed. And that's how one obtains a particular design of a rosette or some kind of vitrail (stained glass).

Of course, one can play with details and make rosette within rosette within rosette ..., changing in the process the algorithms for circle coloring and similar.

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Last updated on 09th of November 2009.