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A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").
The structures of minerals provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a crystal , defined by crystal structure , crystal system , and point group ; for example, there are exactly 14 Bravais lattices for ...
Computational design of low-dimensional materials: In 2011, he predicted 20+ metallic boron monolayer structures and created systematic naming series for these monolayers, including α, β, χ, and δ series; [18] two in these series, χ 3-borophene and β 12-borophene were later confirmed by experiments.
This layer-by-layer growth is two-dimensional, indicating that complete films form prior to growth of subsequent layers. [2] [3] Stranski–Krastanov growth is an intermediary process characterized by both 2D layer and 3D island growth. Transition from the layer-by-layer to island-based growth occurs at a critical layer thickness which is ...
A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius is a monotonic continuous function of angle : r = r ( φ ) . {\displaystyle r=r(\varphi )\;.} The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant ).
An example of the cubic crystals typical of the rock-salt structure [broken anchor]. Time-lapse of growth of a citric acid crystal. The video covers an area of 2.0 by 1.5 mm and was captured over 7.2 min. The interface between a crystal and its vapor can be molecularly sharp at temperatures well below the melting point.
The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains.
The model, named after Murray Eden, was first described in 1961 [2] as a way of studying biological growth, and was simulated on a computer for clusters up to about 32,000 cells. By the mid-1980s, clusters with a billion cells had been grown, and a slight anisotropy had been observed.