Definition 13

A boundary is that which is an extremity of anything.

Definition 14

A figure is that which is contained by any boundary or boundaries.


These are rather nebulous definitions since they are based on the undefined terms “extremity” and “contained by.” Euclid deals with two kinds of figures in the Elements: plane figures and solid figures. Plane figures are defined in the upcoming definitions: circles and semicircles in I.Def.15 and I.Def.18, rectilinear figures in I.Def.19 and particular kinds of rectilinear figures such as triangles and quadrilaterals following that. Specific solid figures such as spheres, cones, pyramids, and various polyhedra are defined in Book XI. Plane figures are not solid figures since they are not contained by any boundaries in space. Thus, implicit to the concept of figure is the ambient plane or space of the figure.

Extremities, boundaries, and topology

Euclid deals with three kinds of extremities, or boundaries. There are the ends of lines (I.Def.3), the edges of surfaces (I.Def.3), and the surfaces of solids (XI.Def.2). A finite line has two points as its boundaries. A circle is defined in I.Def.15 as is a plane figure and has its circumference as its boundary. A sphere is defined in XI.Def.14 as a solid figure and has a spherical surface as its boundary.

The modern subject of topology studies space in a different way than geometry does. The geometric concepts of straightness, distance, and angle are excluded from topology, but the concept of boundary is central to topology. In topology, a sphere remains a sphere even when it’s squeezed or stretched.

Not everything has a boundary. For instance, the circumference of a circle has no boundary. Also a spherical surface has no boundary. In topology, a finite region with no boundary is called a cycle. Circles and spherical surfaces are cycles. In general, if something is a boundary, it has no boundary itself. So boundaries are cycles. But not all cycles are boundaries.

Topology uses cycles and boundaries to distinguish various spaces. For instance, on a spherical surface, every circle is the boundary of a region on that surface. But on a toroidal surface (rotate a circle around a line in the plane of the circle that doesn’t meet the circle), there are circles (for instance, that circle mentioned parenthetically) that don’t bound any region on the surface. Thus, spherical surfaces are topologically different from toroidal surfaces.

Figures and their boundaries

The definition of figure needs to be fleshed out. In order to be a figure, a region must be bounded, that is, held in by a boundary. For instance, an infinite plane is unbounded, so it is not intended to be a figure. Neither is the region between two parallel lines even though that region has the two parallel lines as its extremities.

Other figures may be considered if other ambient spaces are allowed, although Euclid only uses plane and solid figures. For a one-dimensional example, a line segment with its endpoints as its boundary could be considered to be a figure in an infinite line. Also, a hemisphere could be considered to be a figure on the surface of a sphere with the equator as its boundary.

Nonconnected, nonsimply-connected, and nonconvex figures

Nowhere in the Elements does a nonconnected figure occur. It’s apparent that figures are supposed to be connected.

For an example of a nonconnected figure, consider the following. Given a circle and a line that doesn’t intersect that plane, when that circle is rotated around the line in space, a solid results called a torus. The intersection of that torus with the original plane is the figure that consists of the original circle and another on the other side of the line. Considered as a single figure it is disconnected. It would be called two figures in the Elements.

Nonsimply-connected figures are those with holes in them. An example of such a figure is an annulus, also called a ring, which is the figure between two concentric circles. See proposition XII.16 for an illustration. There’s no indication that Euclid considered nonsimply-connected figures.

Nowhere in the Elements does a nonconvex figure explicitly occur. A pentagram is a five-pointed star. See the Guide to proposition IV.11 for an illustration. To study nonconvex polygons, Euclid would have had to admit angles greater than 180°, which he didn’t.