Shape and Volume: Leonardo’s elegant geometry

Leonardo's geometric studies are divided into two main areas. On the one hand, studies of plane geometry which consist of the transformation of rectilinear figures into curvilinear (or vice versa) of equivalent area.

In Leonardo’s hands these studies acquire great pictorial beauty.

And on the other are Leonardo’s studies of solid geometry, and the transformation of a given solid figure into another of a different shape of equal volume.

Here Leonardo’s drawings appear like virtual geometric sculptures.

Here is a typical study of the geometric transformation of plane figures of different shapes (in this case all curvilinear) but of equivalent area. 

The empty lunula, or crescent-shaped area in the upper part of the drawing, is equivalent to the three dashed leaf-shaped figures. The overall configuration of this geometric study is very similar to that of the heart valves, which Leonardo studied in anatomy.

In this image what at first glance looks like a simple decorative drawing is another study of plane geometry. Leonardo draws figures of different form equivalent to each other by area.

In each basic circular element, the dashed areas are equivalent to the central empty area. Together they make a very elegant pattern, animated by the alternation of empty and dark areas, similar to the chiaroscuro of Leonardo’s paintings.

Here is a study of the transformation of a cube into a pyramid with square base.

Starting with a cube, Leonardo tries to create a pyramid of equivalent volume, with a height greater than that of the three sides of the cube. He asks himself how much the square base of the pyramid will have to decrease in comparison to the base of the cube.

This drawing shows the transformation of a parallelepiped, a cube-like solid, into a pyramid with square base.

Leonardo studies the relationship between the height of the pyramid and its base with respect to the base of the parallelepiped.

This page looks at the transformation of a square-based parallelepiped into a cube of equal volume (top drawing) and of a cube into a parallelepiped.

Leonardo plans to go through the transformation of the starting figure into a cylinder, a passage not represented in the drawing.

These drawings show studies of the transformation of a parallelepiped into a square-based pyramid. Leonardo also plans to study how the square base of a pyramid varies with its height, while always keeping the overall volume constant.

Notice how in one of the text notes, Leonardo indicates the word “square” with a small square drawing plus the final letters "ta".

The main geometrical problems Leonardo explores in the Codex Arundel are: the 1:2 ratio between a semicircle and a complete inscribed circle; various curvilinear, rectilinear or mixed forms; the transformation and relationship between cube and pyramid and between cylinder and cone of equal height. 

On the right a spiral introduces, into this perfect world of geometry, a form more representative of the dynamism of the physical and natural world.

The sheet opens with a study of the equivalence of plane geometry. Leonardo intends to demonstrate the equivalence between the circumference of the small circle and the arc of a semicircle in which the first is inscribed.

Here are more studies on the transformation of solid bodies of various shapes and, in particular, of the cube into a cylinder and various other solid figures.

A note and drawing at the bottom right of this page concern the largest cylinder that can be obtained from a cube. Immediately above this drawing Leonardo considers the relationship between a cylinder and a coincident base cone.

These studies sit between geometry and statics (a branch of mechanics that concerns how loads act on physical objects). Here Leonardo analyses the accidental centre of gravity of a solid wedge-shaped geometric figure. While the natural centre of gravity of a body is determined by its weight and shape, its “accidental” centre comes when the body is suspended from a given point

These drawings show a vertical line or a balance system to which the geometric body is variously suspended, from which Leonardo studies the behaviour of the accidental centre of gravity of the solid object.