Dividing the Circle by Six: Hexagons, Triangles and Squares
Proportions of circle to square, triangle to square, and the compass that describes a portion
The sheet argues that six is the most perfect division of the circle, because a single opening of the compass steps the circumference exactly six times and enters the diameter twice. Around this thesis Leonardo sets circles with inscribed triangles, pentagons and hexagons, squares built on the sides of triangles and on a diameter, and two small diagrams of a compass describing a portion of a curve. Recurring propositions state that the proportion between circles equals that between the squares on their diameters, and between similar parts equals that between their similar wholes. A flower-of-life rosette at the foot demonstrates the six-fold subdivision carried to its limit.
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The number six as the perfect division of the circle
The greatest perfection of dividing the circle into equal parts is the number six, for one and the same opening of the compass measures the circumference exactly six times and enters twice into the diameter, which cannot happen with any other opening. Three diameters through the centre cut the circle into six sectors, and removing their six portions leaves six equilateral triangles; but first it makes the hexagon.
Proportion of circle to circle as square to square
Such proportion holds from circle to circle as from square to square made by the multiplication of the diameter of that circle by itself. Similarly, the proportion among equilateral triangles equals that among the squares built on their sides.
Compass describing a portion of a curve
Two small diagrams show a compass (sesto) opened to describe a portion of a circle, labelled with the letters a, d, c, b and, in the second, a further point f. The accompanying rule directs that such arcs be always made proportioned to their chords, the opening of the compass being set to the side of the triangle.
Like parts to like wholes; hexagon complements
Such proportion holds from like parts to like parts as from their like wholes to their like wholes. Here follow the six portions of the hexagons, which are the complements to the six sectors of the circle that encloses its six sectors within itself.