This
document was originally written for my own use to supplement the book
“Woodturning Wizardry” by David Springett.
In that book David gives measurements for a 62mm diameter sphere and for
no other. He also has given only a very
few of the formulas that are useful when working with spheres of arbitrary
diameter. All required formulas and
tables of measurements for spherical diameters likely to be used are given
here. Of course, to work with spheres of
other than 62mm, one will have to make or have made the special tools that are
required. The exceptions, described in
David’s book, are the Pomanders and the Singapore Ball which only require one
special tool.
"Uniformly
spaced" points on a sphere necessarily form a regular polyhedron. One can uniformly space 2, 4, 6, 8, 12 and 20
points on a sphere. See Figure 1.
A
regular polyhedron is a solid, three-dimensional figure each face of
which is a regular polygon having equal sides and equal angles. Every face has the same number of vertices,
and the same number of faces meet at every vertex. An inscribed (inside)
sphere touches the center of every
face, and a circumscribed sphere (outside) touches every vertex.
There
are exactly five of these figures, also called the Platonic Solids: the
tetrahedron, octahedron, cube, icosahedron and dodecahedron having 4, 6, 8, 12
and 20 vertices respectively. A diameter
marks two evenly spaced points. The
faces of these five solid figures are either equilateral triangles (tetrahedron,
octahedron and icosahedron), square (cube), or pentagon (dodecahedron).
To
lay out uniformly spaced points on the surface of a sphere, it is necessary to
determine the dimensions of the regular polyhedron which the given sphere
circumscribes. The radius to which a
compass must be set to mark out the points becomes an exercise in determining
the lengths of two related straight lines:
1)
an edge or side
of the regular polygon that makes up the faces of the inscribed regular
polyhedron and
2)
that of
determining the distance from the projection of the center of a face of the
solid onto the sphere to a vertex of the solid.
In other words, the distance from a vertex of the polygon to the center
point of a face of the polygon that has been projected onto the surface of the
sphere along a radial line. Expressing
this distance another way, this distance is that from a vertex to the single
point at which the inscribed sphere touches a face which included that vertex.
For
example, refer to Figure 1. The
centroids of the pentagonal faces of the dodecahedron, projected upon the
surface of the sphere, become the 12 primary points. The 5 vertices of each pentagonal face become
the constellation points that surround each primary point. The 20 constellation points are sometimes
referred to as the secondary points.
Formulas
are given here with little justification and no derivation. If you are interested in the details, then
see the companion article “Figures Of Interest To Woodturners”.