Sphere packing, circle packing, and kagome weaving

A packing of spheres centered on a surface generates an attractive triangulation. Image quoted from J. Wallner and H. Pottmann, "Geometric Computing for Freeform Architecture."

Any triangulation of a surface directs a kagome weaving. Some of the prettiest triangulations result from sphere packings or circle packings (which are nearly the same thing.)

A circle packing is a configuration of circles having specified patterns of tangencies. Adding the condition that circle interiors may not overlap, such a circle packing can be called a packing of discs. A disc packing on the plane is associated with a plane graph, called its carrier, that has a vertex at each circle center and an edge connecting the vertices of any two circles that are tangent to each other. A packing of discs is called compact if its carrier is a triangulation.

The incircles in the carrier graph of a plane compact circle packing constitute a circle packing having a carrier graph dual to the first. The two sets of circles are orthogonal. (Only the red circle belongs to the original packing.) Image quoted from M. Hoebinger, " Packing of circles and spheres on surfaces."

Mathias Hoebinger has shown that a compact disc packing in the plane has as its carrier a triangulation in which circles inscribed in the triangles (incircles) are mutually tangent. In effect, a second (non-compact) circle packing lurks inside the carrier of the first. The carrier graphs of the two packings are graph duals of each other. The points of tangency of the two packings coincide, and the two sets of circles are orthogonal at those points. See diagram above.

The graphenes of theoretical carbon chemistry  are closely related to circle packings, and both topics are related to deltahedral surfaces, which are the surfaces we can kagome-weave with straight weavers. For example the haeckelites, which are graphenes having 5-, 6-, and 7-carbon rings, can have the same arrangements as disc packings with two sizes of disks. These 3-regular patterns can be converted into triangulations by dualization. Triangulations direct kagome weaves.

Haeckelite nanotubes. Image quoted from Terrones et al. "New Metallic Allotropes of Planar and Tubular Carbon."

A compact packing of discs of two sizes (5- and 7-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5- and 7-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5- and 8-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4- and 8-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5-, 6- and  9-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (5-, 12-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 10-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 12-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."

A compact packing of discs of two sizes (4-, 18-carbon rings.) Image quoted from T. Kennedy, "Compact packings of the plane with two sizes of discs."