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Macro photography of a natural snowflake. A snowflake is a single ice crystal that is large enough to fall through the Earth's atmosphere as snow. [1] [2] [3] Snow appears white in color despite being made of clear ice.
The hexagonal snowflake, a crystalline formation of ice, has intrigued people throughout history. This is a chronology of interest and research into snowflakes. Artists, philosophers, and scientists have wondered at their shape, recorded them by hand or in photographs, and attempted to recreate hexagonal snowflakes.
Snowflakes nucleate around particles in the atmosphere by attracting supercooled water droplets, which freeze in hexagonal-shaped crystals. Snowflakes take on a variety of shapes, basic among these are platelets, needles, columns and rime. As snow accumulates into a snowpack, it may blow into drifts.
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island [1] [2]) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" [3] by the Swedish mathematician Helge von Koch.
The number of vertices, edges, and faces of GP(m,n) can be computed from m and n, with T = m 2 + mn + n 2 = (m + n) 2 − mn, depending on one of three symmetry systems: [1] The number of non-hexagonal faces can be determined using the Euler characteristic, as demonstrated here.
A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism. A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such a dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes.
The hexagonal tiling honeycomb, {6,3,3}, has hexagonal tiling, {6,3}, facets with vertices on a horosphere. One such facet is shown in as seen in this Poincaré disk model. In H 3 hyperbolic space, paracompact regular honeycombs have Euclidean tiling facets and vertex figures that act like finite polyhedra.
For an antiprism with congruent regular n-gon bases, twisted by an angle of 180 / n degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes.