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The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.
Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions. [2] An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic.
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [2]
All group 4 subjects (except computer science and environmental systems and societies; see below) follow roughly the same format. Each subject has its Subject Specific Core (SSC), i.e., material taught at both the standard and higher levels. Students sitting the Higher Level examination study the Additional Higher Level (AHL) material.
Advanced Placement (AP) Biology (also known as AP Bio) is an Advanced Placement biology course and exam offered by the College Board in the United States. For the 2012–2013 school year, the College Board unveiled a new curriculum with a greater focus on "scientific practices".
The history of aperiodic crystals can be traced back to the early 20th century, when the science of X-ray crystallography was in its infancy. At that time, it was generally accepted that the ground state of matter was always an ideal crystal with three-dimensional space group symmetry, or lattice periodicity.
Since then, understanding of zinc in human biology has advanced to the point that it is considered as important as iron. Modern advancements in analytical technology have made it clear the importance of biometals in signalling pathways and the initial thoughts on the chemical basis of life. [2]