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Johann F. C. Hessel
Johann Friedrich Christian Hessel (27 April 1796 – 3 June 1872) was a German physician (MD, University of Würzburg, 1817) and professor of mineralogy (PhD, University of Heidelberg, 1821) at the University of Marburg. Contributions to Mineralogy and Crystallography The origins of geometric crystallography (the field concerned with the structures of crystalline solids), for which Hessel's work was noteworthy, can be traced back to eighteenth and nineteenth century mineralogy. Hessel also made contributions to classical mineralogy (the field concerned with the chemical compositions and physical properties of minerals), as well. Derivation of the Crystal Classes In 1830, Hessel proved that, as a consequence of Haüy’law of rational intercepts morphological forms can combine to give exactly 32 kinds of crystal symmetry in Euclidean space, since only two-, three-, four-, and six-fold rotation axes can occur. A crystal form here denotes a set of symmetrically equivalent planes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallographic Point Group
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected. In the classification of crystals, each point group defines a so-called (geometric) crystal class. There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calcium Fluoride
Calcium fluoride is the inorganic compound of the elements calcium and fluorine with the formula CaF2. It is a white insoluble solid. It occurs as the mineral fluorite (also called fluorspar), which is often deeply coloured owing to impurities. Chemical structure The compound crystallizes in a cubic motif called the fluorite structure. Ca2+ centres are eight-coordinate, being centered in a cube of eight F− centres. Each F− centre is coordinated to four Ca2+ centres in the shape of a tetrahedron. Although perfectly packed crystalline samples are colorless, the mineral is often deeply colored due to the presence of F-centers. The same crystal structure is found in numerous ionic compounds with formula AB2, such as CeO2, cubic ZrO2, UO2, ThO2, and PuO2. In the corresponding anti-structure, called the antifluorite structure, anions and cations are swapped, such as Be2C. Gas phase The gas phase is noteworthy for failing the predictions of VSEPR theory; the molecule is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lead(II) Sulfide
Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cut, lead is a shiny gray with a hint of blue. It tarnishes to a dull gray color when exposed to air. Lead has the highest atomic number of any stable element and three of its isotopes are endpoints of major nuclear decay chains of heavier elements. Lead is toxic, even in small amounts, especially to children. Lead is a relatively unreactive post-transition metal. Its weak metallic character is illustrated by its amphoteric nature; lead and lead oxides react with acids and bases, and it tends to form covalent bonds. Compounds of lead are usually found in the +2 oxidation state rather than the +4 state common with lighter members of the carbon group. Exceptions are mostly limited to organolead compounds. Like the lighter members of the group, le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Formula For Polyhedra
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Antoine Jean L'Huilier
Simon Antoine Jean L'Huilier (or L'Huillier) (24 April 1750 in Geneva – 28 March 1840 in Geneva) was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the generalization of Euler's formula for planar graphs. He won the mathematics section prize of the Berlin Academy of Sciences for 1784 in response to a question on the foundations of the calculus. The work was published in his 1787 book ''Exposition elementaire des principes des calculs superieurs''. (A Latin version was published in 1795.) Although L'Huilier won the prize, Joseph Lagrange, who had suggested the question and was the lead judge of the submissions, was disappointed in the work, considering it "the best of a bad lot." Lagrange would go on to publish his own work on foundations. L'Huilier and Cauchy L'Huilier introduced the abbreviation "lim" for limit that reappeared in 1821 in Cours d'Analyse by Augustin Louis Cauchy, who wou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Habit
In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. The habit of a crystal is dependent on its crystallographic form and growth conditions, which generally creates irregularities due to limited space in the crystallizing medium (commonly in rocks).Klein, Cornelis, 2007, ''Minerals and Rocks: Exercises in Crystal and Mineral Chemistry, Crystallography, X-ray Powder Diffraction, Mineral and Rock Identification, and Ore Mineralogy,'' Wiley, third edition, Wenk, Hans-Rudolph and Andrei Bulakh, 2004, ''Minerals: Their Constitution and Origin,'' Cambridge, first edition, Recognizing the habit can aid in mineral identification and description, as the crystal habit is an external representation of the internal ordered atomic arrangement. Most natural crystals, however, do not display ideal habits and are commonly malformed. Hence, it is also important to describe the quality of the shape of a mineral specimen: * Euhedral: a cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (physics)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image of a subset A under the function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system. Often, vertical translations ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Niggli
Paul Niggli (26 June 1888 – 13 January 1953) was a Swiss crystallographer, mineralogist, and petrologist who was a leader in the field of X-ray crystallography. Education and career Niggli was born in Zofingen and studied at the Swiss Federal Institute of Technology (ETH) in Zurich and the University of Zurich, where he obtained a doctorate. His 1919 book, ''Geometrische Kristallographie des Diskontinuums'', played a seminal role in the refinement of space group theory. In this book, Niggli demonstrated that although X-ray reflection conditions do not always uniquely determine the space group to which a crystal belongs, they do reveal a small number of possible space groups to which it could belong. Niggli used morphological methods to account for internal structure and, in his 1928 ''Kristallographische und Strukturtheoretische Grundbegriffe,'' he took up what is essentially the reverse process, the task of establishing the connection between space lattices and external cry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Barlow (geologist)
William Barlow FRS (8 August 1845 – 28 February 1934) was an English amateur geologist specialising in crystallography. He was born in Islington, in London, England. His father became wealthy as a speculative builder as well as a building surveyor, allowing William to have a private education. After his father died in 1875, William and his brother inherited this fortune, allowing him to pursue his interest in crystallography without a need to labour for a living. William examined the forms of crystalline structures, and deduced that there were only 230 forms of symmetrical crystal arrangements, known as space groups. His results were published in 1894, after they had been independently announced by Evgraf Fedorov and Arthur Schönflies, although his approach did display some novelty. His structural models of simple compounds such as NaCl and CsCl were later confirmed using X-ray crystallography. He served as the president of the English Mineralogical Society from 1915 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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