Book XII, Proposition 17

Transcribed from [1]:

Given two spheres about the same centreto inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. [1: 441]

Heath [1: 441] translated Euclid’s explanation to the following (read the full description here):

Inscribe a regular 2n-gon in an “equatorial circle,” then inscribe 2n-gons in n “meridian circles” using as vertices the two poles and two of the preceding points (opposite on the “equator-circle”). To complete the polyhedron, connect the vertices of these n polygons with lines parallel to the equator.

The first part is my interpretation of the problem based on [2], [3], and [4] and explains how to model the polyhedron inscribed in the greater sphere; the second part describes how to model the lesser sphere.

Modeling the polyhedron circumscribed by the greater sphere

The images below illustrate the situation for n = 2 and n = 3 on the left and the right, respectively. The first images show a sphere and the diameter perpendicular to plane xy. Between its antipodes, the semicircle is perpendicular to the y-axis. Each of the 2n-gons (an octagon and a dodecagon, respectively, named here as the first polygon) is inscribed in the intersection of plane xy with the sphere.

The spheres were omitted in the second pair of images, and a second polygon (resulting from the rotation of the first polygon around axis y) is shown. The semicircle was divided into 2n.

Planes parallel to xy (or lines parallel to x) containing each point allow us to determine the center of new 2n-gons whose radius will be the distance between these points and the previous.

Shown in blue, we now have 2n-2 polygons parallel to the first (the “lost” polygons have the antipodes of the diameter as center and radius 0).

Taking z as the axis of 2n-fold symmetry, we rotate the second polygon around it. Thus, we obtain 2n polygons (shown in green) perpendicular to xy that share their vertices with the previous polygons.

In each case, the convex hull is a polyhedron with n rings. A ring is a sequence of faces between the polygons parallel to the great circle of the sphere with the first polygon. The following are interactive 3D models of each.

Campano da Novara studied the polyhedron resulting from n = 3 in the 13th century, later named Campanus Sphere. It was illustrated by Leonardo de Vinci in Divine Proportione, first published in 1509, in solid and vacuum modes. Pacioli named it the septuaginta duarum basium, meaning it has seventy-two bases, or faces, as we call them.

The video shows the resulting polyhedron considering n between 1 and 6:

Modeling the lesser sphere

To model the lesser sphere, we consider the plane defined by its center and the median of one of the polyhedron’s faces in one of the larger rings (shown below in blue). The radius of the sphere tangent to that face is determined in this plane, considering the right triangle whose hypotenuse has the sphere’s center and the lower midpoint for vertices. The longest leg of this triangle is perpendicular to the median.

The image below shows a concentric sphere with the polyhedron but not tangent to the blue face because its radius is not the length of the longest leg.

The image below shows the same sphere with a radius equal to the longest leg of the triangle. The great circle belonging to the triangle’s plane is tangent to the median, and thus, the polyhedron does not touch the lesser sphere at its surface.

Interestingly, despite the quadrilateral being cyclic, its tangent point with the sphere is not the anticentre, the vertex centroid, the edge centroid, or the area centroid.

I will later include a photo of a 3D-printed model of the sphere and the polyhedron.

References:

  1. Euclid, Heath, T. L., & Densmore, D. (2017). Euclid’s elements: All Thirteen books complete in one volume: The thomas L. heath translation. Green Lion Press (pp. 441-446)
  2. Cromwell, P. (1997). Polyhedra. Cambridge, U.K.: Cambridge University Press (pp. 106-107)
  3. Mohanty, Y. (2020, November 12). Campanus’ Sphere. Geometiles®. Retrieved August 09, 2022, from https://geometiles.com/campanus-sphere/ and https://geometiles.com/dev/wp-content/uploads/2020/11/About-Campanus-Sphere_version4.pdf
  4. Ricol, R. C. (n.d.). Campanus’ sphere and other polyhedra inscribed in a sphere. matematicasVisuales. Retrieved August 09, 2022, from http://www.matematicasvisuales.com/english/html/geometry/space/sphereCampanus.html