An interesting measure of the size of a circle is the possible number of neighboring circles. One can place circles in a rectangular grid, with identical circles touching above, below, to the left, and to the right of the central circle. However, a tighter arrangement can be made, with hexagonal symmetry. Here, each circle is surrounded by six equivalent ones. This arrangement has the special property that all the neighbors are touching two other circles off the original one. In other words, six circles will exactly surround one, in two dimensions. Items forced on to a flat surface, if short on space, will get six neighbors, and tend to take on the shapes of hexagons themselves, as in a honeycomb.

**A circle encircled by six circles shows the kissing number in two dimensions.**Another example of the kissing number in a plane is carbon atoms in graphite, as in pencil "lead". Here, layers are made of carbon atoms forming three bonds in the plane to similar atoms. The atoms form flat hexagonal rings, which are surrounded by identical hexagons.

**Carbon atoms in two dimensions form hexagonal sheets, called graphite.**While carbon in the graphite form is essentially a two-dimensional structure, the diamond form of carbon is a covalent network solid, extending equally in three dimensions.

**Carbon atoms in diamond extend through three dimensions.**In the last few years, a third fundamental form of the element carbon has been discovered, in which individual atomic bonds are made in two dimensions, but the overall shape obtained is three dimensional. Specifically, a group of 60 carbon atoms will wrap itself into a sphere (changing some hexagons into pentagons). This forms a compound called buckminsterfullerene, or a buckyball. This is named after the inventor of the similarly shaped geodesic dome, Buckminster Fuller.

A

**buckyball**, or buckminsterfullerene, the spherical C60 form of pure carbon atoms, Registry number [99685-96-8], in the Chemical Abstracts Service

The problem of stacking or packing balls efficiently has been an interesting puzzle for four centuries. While people have stacked things for millennia, it was in 1611 that Kepler posed the Sphere-Packing Problem [19]. What kind of stacking of spheres can be proven to be the densest possible?

A first layer of spheres can be arranged in the rectangular or hexagonal patterns of circles in a plane, described above. Such layers can be stacked exactly atop one another, yielding respectively the arrangements called the body-centered (or 3-dimensional) cubic lattice, and the face-centered cubic lattice or cubic close-packed form. However, if flat layers are stacked repeatedly in a staggered way, a third, most dense pattern emerges, called the hexagonal lattice, or hexagonal close-packed

**Oranges or atoms in three dimensions will stack in these three arrangements.**All three of these packings are found in nature, as are mixtures of these with less regular forms. Interestingly, elemental metals use all forms, in patterns that do not always correlate with electronic symmetry. For example, potassium, chromium and tungsten prefer the body-centered cubic form, as does iron at room temperature. However, iron at other temperatures takes on the cubic close-packed form, as do copper, silver, and gold. In contrast, the elements in the same period directly below iron (i.e., ruthenium and osmium) prefer the hexagonal close-packed arrangement, as do most of the rare-earth elements, as well as zinc and titanium.

The number of circles that can surround a circle in any given dimension is called the "kissing number" by mathematicians. On a one-dimensional line, this is two (left and right). In a two-dimensional plane, as we discussed, the kissing number is six. The kissing number is 12 in three dimensions, as in the above hexagonal close-packed arrangement (with six in the plane, three more above, and three others below). In theory, a four-dimensional sphere should have a kissing number of 24, in eight dimensions it is 240, and in 24-dimensional space the kissing number is 196,560 circles touching the center circle at once. The kissing number of 6 for a circle means that it is possible to construct a regular hexagon (and consequently a six-pointed star) by using the radius to cut the circle into six equal arcs. Unlike the simplicity of the construction of a six-pointed star, it is a challenge to use only a compass and straight edge to construct a pentagram, a five-pointed star extending from a regular pentagon. To do this task, we need to know how to find the "Golden Mean" of a line segment. The Golden Mean is a cut in a line segment such that the ratio of the larger section to the smaller one is the same as the ratio of the entire line segment to the larger one. It is possible to find the Golden Mean of a given line segment using a compass and straight edge. The Golden Mean has been used extensively in art and architecture from the ancient times of the Egyptians and Greeks up to today. The larger section of the Golden Mean of the radius of any circle divides the circle into ten equal arcs, which enables us to construct the pentagram. The complexity of these procedures, or the harmony and order of the completed form, created the association of mysticism with the combination of a pentagram inscribed in a circle, the chosen symbol of the order of the Pythagoreans, sometimes associated in western tradition with witchcraft

**The pentagram of the Pythagoreans**