Sphere packing in the context of Cannonball

Ice is water that is frozen into a solid state, typically forming at or below temperatures of 0 °C, 32 °F, or 273.15 K. It occurs naturally on Earth, on other planets, in Oort cloud objects, and as interstellar ice. As a naturally occurring crystalline inorganic solid with an ordered structure, ice is considered to be a mineral. Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaque bluish-white color.

Virtually all of the ice on Earth is of a hexagonal crystalline structure denoted as ice I_h (spoken as "ice one h"). Depending on temperature and pressure, at least nineteen phases (packing geometries) can exist. The most common phase transition to ice I_h occurs when liquid water is cooled below 0 °C (273.15 K, 32 °F) at standard atmospheric pressure. When water is cooled rapidly (quenching), up to three types of amorphous ice can form. Interstellar ice is overwhelmingly low-density amorphous ice (LDA), which likely makes LDA ice the most abundant type in the universe. When cooled slowly, correlated proton tunneling occurs below −253.15 °C (20 K, −423.67 °F) giving rise to macroscopic quantum phenomena.

Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them, resulting in an infinite sphere packing.

The fractal dimension is approximately 2.473946 (±1 in the last digit).

In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ${\displaystyle {\mathcal {H}}}$, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.

Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}$. Every modular form is attached to a Galois representation.

In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.

The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.

In crystallography, interstitial sites, holes or voids are the empty space that exists between the packing of atoms (spheres) in the crystal structure.

The holes are easy to see if you try to pack circles together; no matter how close you get them or how you arrange them, you will have empty space in between. The same is true in a unit cell; no matter how the atoms are arranged, there will be interstitial sites present between the atoms. These sites or holes can be filled with other atoms (interstitial defect). The picture with packed circles is only a 2D representation. In a crystal lattice, the atoms (spheres) would be packed in a 3D arrangement. This results in different shaped interstitial sites depending on the arrangement of the atoms in the lattice.