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Examples include the seven days of creation and so seven days that make up a week, and the seven lamps on the Temple Menorah. One variation on the use of seven is the use of the number six in numerology, used as a final hallmark in a series leading to a seven (e.g. mankind is created on the sixth day in Genesis, out of the seven days of creation).
Ten 3-fold axes and six 5-fold axes (icosahedral symmetries I and I h) According to the crystallographic restriction theorem, only a limited number of point groups are compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 of the 7 others. Together, these make up the 32 so-called crystallographic point groups.
We now confine our attention to the plane in which the symmetry acts (Scherrer 1946), illustrated with lattice vectors in the figure. Lattices restrict polygons Compatible: 6-fold (3-fold), 4-fold (2-fold) Incompatible: 8-fold, 5-fold. Now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon.
The day-year principle was partially employed by Jews [7] as seen in Daniel 9:24–27, Ezekiel 4:4-7 [8] and in the early church. [9] It was first used in Christian exposition in 380 AD by Ticonius, who interpreted the three and a half days of Revelation 11:9 as three and a half years, writing 'three days and a half; that is, three years and six months' ('dies tres et dimidium; id est annos ...
Therefore, the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc. 3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry ...
The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals: implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles ...
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A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al 71 Ni 24 Fe 5. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching ...