1_2: BETTER DONE THAN PERFECT - SUNFLOWER’S SPIRALS T he spirals in sunflowers, daisies and other flowers are a wonderful display of mathematics from the natural world. I’ve always wondered: those spirals should converge in the center, with even smaller seeds, like in an Escher drawing. Is that really so? Do the spirals converge with smaller and smaller seeds? No. The infinitesimal doesn’t exist (at least in biology). Once the forming ovaries approach the center, they become too near to form nice spirals, and pack together “as it comes” breaking the elegant symmetry of the periphery. This can be understood by thinking of the development of the flower from the multiplicating meristem, where cells find their place by a fight for place; happens that if the cells are many, they fit together into nice patterns, while in the centre, they are few remaining with little place, and little care for harmony. What to learn from here? Efficiency is preferable to perfection; pure mathematics doesn’t exist in reality, where it has to live with many compromises. About compromises, the flower in question is not an actual sunflower, but probably an aster, from a nearby garden (the same as from the 1_1 image); spirals are the same anyway. Subject: Flower, likely Asteracea, yellow garden cultivar. Section right above flower base. Lens: Macro 90mm f/2.8 Theme 1: HEXAGONE +/- EPSILON While humans like squares, in biology the most common geometry for tesselating surfaces is by hexagons; this likely originates spontaneously by the packing of spherical cells over a plane. And when the surface has a curvature, complete covering is achieved by adding or removing sides, leading to a few pentagons or heptagons. I’m not sure what inspirations natural tessellations can provide; after all, covering by squares is much more efficient from a design and manufacturing perspective. And if we really need to cover a sphere with hexagons and pentagons, well, football balls have been devised already, long time ago. Anyway! Hexagons minimize the contour, as probably do pentagons and heptagons when the surface is curved. Are truly optimal if we need to place objects somewhere, like windmills. We may use the number of pentagons and heptagons to measure the curvature of a surface; sounds cumbersome but may provide a breakthrough somewhere. Truth here is, biological tessellations simply make for beautiful patterns to photograph.