Hexagons, Triangles, and Portions of the Circle
Dividing hexagons into equilateral triangles and circles into sixfold sectors
The sheet, written upside-down in two columns, develops a geometry of the number six: every hexagon is shown to be infinitely divisible into equilateral triangles, of which six is always an aliquot part. Small diagrams accompany the argument — a hexagon split into six triangles, squares subdivided into smaller squares, equilateral triangles raised on the sides of squares, and two-angled figures ("bisangoli") that are half-triangles or portions of a circle. A circle with an inscribed six-pointed star and a hexagon within a circle illustrate how the equilateral triangle enters a circle six times to leave six circular sectors. A column of numerical calculations stands in the upper corner.
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Every hexagon divisible into equilateral triangles
Every hexagonal surface — a surface of six sides — is held to be infinitely divisible into equilateral triangles, and six is always an aliquot part of their number. The claim is illustrated by a small hexagon partitioned into six triangles meeting at the center.
Adding equal quantities to the six sides
Adding equal quantities to the six sides of a hexagon makes it grow equally, so each triangle resolves into sixfold numbers. Six is therefore called an aliquot, or multiplicative, part and the constant divisor of the equal equilateral triangles into which hexagons resolve.
Two-angled figures and portions of the circle
The "bisangolo" or two-angled figure is defined in three cases: an equilateral one (a–b), an unequal-sided one, and a third made from half the first, which is also called a portion of a circle. The most useful portion is the one built on a circle's semidiameter, since the circumference contains its curved side exactly six times.
The equilateral triangle six times in the circle
The equilateral triangle goes six times into the circle, touching twelve angles on the circumference while their other six angles meet the center. What remains between the enclosed hexagon and the circumference is six circular portions joined to a triangle's side, a figure Leonardo names a sector of the circle.
Column of calculations
A block of paired figures records multiplications and running sums — among them 7 × 14 = 84 and totals such as 1000, 250, and 444. They appear to work out the sixfold multiples used in the triangle-and-hexagon argument.