Misplaced Pages

Dodecahedron

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

In geometry , a dodecahedron (from Ancient Greek δωδεκάεδρον ( dōdekáedron ) ; from δώδεκα ( dṓdeka )  'twelve' and ἕδρα ( hédra )  'base, seat, face') or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid . There are also three regular star dodecahedra , which are constructed as stellations of the convex form. All of these have icosahedral symmetry , order 120.

#139860

77-520: Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron , a common crystal form in pyrite , has pyritohedral symmetry , while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry . The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with

154-457: A protein chain three dihedral angles are defined: The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined). The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the C atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively. The vast majority of

231-427: A torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond . Every set of three non-colinear atoms of a molecule defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify

308-455: A triangular gyrobianticupola. It has D 3d symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match. The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It

385-435: A Fe face-centered cubic sublattice into which the S 2 ions are embedded. (Note though that the iron atoms in the faces are not equivalent by translation alone to the iron atoms at the corners.) The pyrite structure is also seen in other MX 2 compounds of transition metals M and chalcogens X = O , S , Se and Te . Certain dipnictides with X standing for P , As and Sb etc. are also known to adopt

462-429: A cube have the coordinates (±1, ±1, ±1). The coordinates of the 12 additional vertices are ( 0, ±(1 + h ), ±(1 − h ) ) , ( ±(1 + h ), ±(1 − h ), 0 ) and ( ±(1 − h ), 0, ±(1 + h ) ) . h is the height of the wedge -shaped "roof" above the faces of that cube with edge length 2. An important case is h = ⁠ 1 / 2 ⁠ (a quarter of the cube edge length) for perfect natural pyrite (also

539-421: A cubic convex hull at one limit of collinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal. It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The endo-dodecahedron is concave and equilateral; it can tessellate space with

616-410: A dihedral angle such that replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0 {\displaystyle -\mathbf {b} _{0}} changes the sign of the angle, which can be between − π and π . In some scientific areas such as polymer physics , one may consider a chain of points and links between consecutive points. If

693-448: A flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called "wings") are upwardly inclined to the lateral axis; when downwardly inclined they are said to be at a negative dihedral angle. When the two intersecting planes are described in terms of Cartesian coordinates by the two equations the dihedral angle, φ {\displaystyle \varphi } between them

770-602: A foul odor and corrosion of copper wiring. In the United States, in Canada, and more recently in Ireland, where it was used as underfloor infill, pyrite contamination has caused major structural damage. Concrete exposed to sulfate ions, or sulfuric acid, degrades by sulfate attack : the formation of expansive mineral phases, such as ettringite (small needle crystals exerting a huge crystallization pressure inside

847-534: A sample of pyrite was placed against a circular file to strike the sparks needed to fire the gun. Pyrite is used with flintstone and a form of tinder made of stringybark by the Kaurna people of South Australia , as a traditional method of starting fires. Pyrite has been used since classical times to manufacture copperas ( ferrous sulfate ). Iron pyrite was heaped up and allowed to weather (an example of an early form of heap leaching ). The acidic runoff from

SECTION 10

#1732791017140

924-584: A simple liquid-phase exfoliation route. This is the first study to demonstrate the production of non-layered 2D-platelets from 3D bulk FeS 2 . Furthermore, they have used these 2D-platelets with 20% single walled carbon-nanotube as an anode material in lithium-ion batteries, reaching a capacity of 1000 mAh/g close to the theoretical capacity of FeS 2 . In 2021, a natural pyrite stone has been crushed and pre-treated followed by liquid-phase exfoliation into two-dimensional nanosheets, which has shown capacities of 1200 mAh/g as an anode in lithium-ion batteries. From

1001-465: Is cubic and was among the first crystal structures solved by X-ray diffraction . It belongs to the crystallographic space group Pa 3 and is denoted by the Strukturbericht notation C2. Under thermodynamic standard conditions the lattice constant a {\displaystyle a} of stoichiometric iron pyrite FeS 2 amounts to 541.87 pm . The unit cell is composed of

1078-404: Is a common accessory mineral in igneous rocks, where it also occasionally occurs as larger masses arising from an immiscible sulfide phase in the original magma. It is found in metamorphic rocks as a product of contact metamorphism . It also forms as a high-temperature hydrothermal mineral , though it occasionally forms at lower temperatures. Pyrite occurs both as a primary mineral, present in

1155-626: Is a dodecahedron with chiral tetrahedral symmetry (T). Like the regular dodecahedron , it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form. Abstractions sharing

1232-465: Is about 1 atm . A newer commercial use for pyrite is as the cathode material in Energizer brand non-rechargeable lithium metal batteries . Pyrite is a semiconductor material with a band gap of 0.95 eV . Pure pyrite is naturally n-type, in both crystal and thin-film forms, potentially due to sulfur vacancies in the pyrite crystal structure acting as n-dopants. During the early years of

1309-458: Is accelerated by the action of Acidithiobacillus bacteria which oxidize pyrite to first produce ferrous ions ( Fe ), sulfate ions ( SO 4 ), and release protons (   H , or H 3 O ). In a second step, the ferrous ions ( Fe ) are oxidized by O 2 into ferric ions ( Fe ) which hydrolyze also releasing   H ions and producing FeO(OH). These oxidation reactions occur more rapidly when pyrite

1386-423: Is accounted for by point symmetry groups C 3 i and C 3 , respectively. The missing center of inversion at S lattice sites has important consequences for the crystallographic and physical properties of iron pyrite. These consequences derive from the crystal electric field active at the sulfur lattice site, which causes a polarization of S ions in the pyrite lattice. The polarisation can be calculated on

1463-464: Is affected by the values φ and ψ . For instance, there are direct steric interactions between the C γ of the side chain in the gauche rotamer and the backbone nitrogen of the next residue when ψ is near -60°. This is evident from statistical distributions in backbone-dependent rotamer libraries . Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called

1540-460: Is also a parallelohedral spacefiller . Another important rhombic dodecahedron, the Bilinski dodecahedron , has twelve faces congruent to those of the rhombic triacontahedron , i.e. the diagonals are in the ratio of the golden ratio . It is also a zonohedron and was described by Bilinski in 1960. This figure is another spacefiller, and can also occur in non-periodic spacefillings along with

1617-432: Is also known as the syn - or cis -conformation; antiperiplanar as anti or trans ; and synclinal as gauche or skew . For example, with n - butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn -conformation shown above, with a dihedral angle of 60° is less stable than the anti -conformation with a dihedral angle of 180°. For macromolecular usage

SECTION 20

#1732791017140

1694-554: Is also the fastest growing in terms of the unroasted iron pyrites imports, with a CAGR of +27.8% from 2007 to 2016. In July 2020 scientists reported that they have observed a voltage-induced transformation of normally diamagnetic pyrite into a ferromagnetic material, which may lead to applications in devices such as solar cells or magnetic data storage. Researchers at Trinity College Dublin , Ireland have demonstrated that FeS 2 can be exfoliated into few-layers just like other two-dimensional layered materials such as graphene by

1771-428: Is an iron sulfide with the chemical formula Fe S 2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral . Pyrite's metallic luster and pale brass-yellow hue give it a superficial resemblance to gold , hence the well-known nickname of fool's gold . The color has also led to the nicknames brass , brazzle , and brazil , primarily used to refer to pyrite found in coal . The name pyrite

1848-407: Is based on one that is itself created by enlarging 24 of the 48 faces of the disdyakis dodecahedron .) The crystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges. The following points are vertices of a tetartoid pentagon under tetrahedral symmetry : under

1925-526: Is brighter yellow with a greenish hue when wet and is softer (3.5–4 on Mohs' scale). Arsenopyrite (FeAsS) is silver white and does not become more yellow when wet. Iron pyrite is unstable when exposed to the oxidizing conditions prevailing at the Earth's surface: iron pyrite in contact with atmospheric oxygen and water, or damp, ultimately decomposes into iron oxyhydroxides ( ferrihydrite , FeO(OH)) and sulfuric acid ( H 2 SO 4 ). This process

2002-443: Is brittle, gold is malleable. Natural gold tends to be anhedral (irregularly shaped without well defined faces), whereas pyrite comes as either cubes or multifaceted crystals with well developed and sharp faces easy to recognise. Well crystallised pyrite crystals are euhedral ( i.e. , with nice faces). Pyrite can often be distinguished by the striations which, in many cases, can be seen on its surface. Chalcopyrite ( CuFeS 2 )

2079-456: Is defined by and satisfies 0 ≤ φ < π . {\displaystyle 0\leq \varphi <\pi .} In this case, switching the two half-planes gives the same result, and so does replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0 . {\displaystyle -\mathbf {b} _{0}.} In chemistry (see below), we define

2156-548: Is derived from the Greek πυρίτης λίθος ( pyritēs lithos ), 'stone or mineral which strikes fire', in turn from πῦρ ( pŷr ), 'fire'. In ancient Roman times, this name was applied to several types of stone that would create sparks when struck against steel ; Pliny the Elder described one of them as being brassy, almost certainly a reference to what is now called pyrite. By Georgius Agricola 's time, c.  1550 ,

2233-414: Is dual to the quasiregular cuboctahedron (an Archimedean solid ) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces. The rhombic dodecahedron has several stellations , the first of which

2310-449: Is finely dispersed (framboidal crystals initially formed by sulfate reducing bacteria (SRB) in argillaceous sediments or dust from mining operations). Pyrite oxidation by atmospheric O 2 in the presence of moisture ( H 2 O ) initially produces ferrous ions ( Fe ) and sulfuric acid which dissociates into sulfate ions and protons , leading to acid mine drainage (AMD). An example of acid rock drainage caused by pyrite

2387-425: Is given by: and satisfies 0 ≤ φ ≤ π / 2. {\displaystyle 0\leq \varphi \leq \pi /2.} It can easily be observed that the angle is independent of d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} . Alternatively, if n A and n B are normal vector to

Dodecahedron - Misplaced Pages Continue

2464-458: Is not a formally recognised mineral, and is named after the Peruvian scientist Jose J. Bravo (1874–1928). Pyrite is distinguishable from native gold by its hardness, brittleness and crystal form. Pyrite fractures are very uneven , sometimes conchoidal because it does not cleave along a preferential plane. Native gold nuggets , or glitters, do not break but deform in a ductile way. Pyrite

2541-432: Is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}. The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex. The convex regular dodecahedron also has three stellations , all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra . They are the small stellated dodecahedron {5/2, 5},

2618-430: Is sprayed onto the exposed coal surfaces to reduce the hazard of dust explosions . This has the secondary benefit of neutralizing the acid released by pyrite oxidation and therefore slowing the oxidation cycle described above, thus reducing the likelihood of spontaneous combustion. In the long term, however, oxidation continues, and the hydrated sulfates formed may exert crystallization pressure that can expand cracks in

2695-419: Is the 2015 Gold King Mine waste water spill . Pyrite oxidation is sufficiently exothermic that underground coal mines in high-sulfur coal seams have occasionally had serious problems with spontaneous combustion . The solution is the use of buffer blasting and the use of various sealing or cladding agents to hermetically seal the mined-out areas to exclude oxygen. In modern coal mines, limestone dust

2772-417: Is the angle between two intersecting planes or half-planes . In chemistry , it is the clockwise angle between half-planes through two sets of three atoms , having two atoms in common. In solid geometry , it is defined as the union of a line and two half-planes that have this line as a common edge . In higher dimensions , a dihedral angle represents the angle between two hyperplanes . The planes of

2849-571: Is used to make marcasite jewelry . Marcasite jewelry, using small faceted pieces of pyrite, often set in silver , has been made since ancient times and was popular in the Victorian era . At the time when the term became common in jewelry making, "marcasite" referred to all iron sulfides including pyrite, and not to the orthorhombic FeS 2 mineral marcasite which is lighter in color, brittle and chemically unstable, and thus not suitable for jewelry making. Marcasite jewelry does not actually contain

2926-441: The face angle , is measured as the internal angle with respect to the polyhedron. An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling . An angle greater than 180° exists on concave portions of a polyhedron. Every dihedral angle in an edge-transitive polyhedron has

3003-423: The great dodecahedron {5, 5/2}, and the great stellated dodecahedron {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the great icosahedron {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of

3080-653: The molecular conformation . Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn (s), those corresponding to angles between ±90° and 180° anti (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal (c) and those between 0° and ±30° or ±150° and 180° are called periplanar (p). The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation

3157-459: The tetartoid with tetrahedral symmetry : A pyritohedron is a dodecahedron with pyritohedral (T h ) symmetry. Like the regular dodecahedron , it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure). However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of

Dodecahedron - Misplaced Pages Continue

3234-430: The vector quadruple product formula, and the fact that a scalar triple product is zero if it contains twice the same vector: Given the definition of the cross product , this means that φ {\displaystyle \varphi } is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say

3311-432: The 20th century, pyrite was used as a mineral detector in radio receivers, and is still used by crystal radio hobbyists. Until the vacuum tube matured, the crystal detector was the most sensitive and dependable detector available—with considerable variation between mineral types and even individual samples within a particular type of mineral. Pyrite detectors occupied a midway point between galena detectors and

3388-644: The Iberian Peninsula. In the beliefs of the Thai people (especially those in the south), pyrite is known as Khao tok Phra Ruang , Khao khon bat Phra Ruang (ข้าวตอกพระร่วง, ข้าวก้นบาตรพระร่วง) or Phet na tang , Hin na tang (เพชรหน้าทั่ง, หินหน้าทั่ง). It is believed to be a sacred item that has the power to prevent evil, black magic or demons. Dihedral angle Right Interior Exterior Adjacent Vertical Complementary Supplementary Dihedral A dihedral angle

3465-452: The arrangement of the metal and diatomic anions differ from that of pyrite. Despite its name, chalcopyrite ( CuFeS 2 ) does not contain dianion pairs, but single S sulfide anions. Pyrite usually forms cuboid crystals, sometimes forming in close association to form raspberry-shaped masses called framboids . However, under certain circumstances, it can form anastomosing filaments or T-shaped crystals. Pyrite can also form shapes almost

3542-426: The basis of higher-order Madelung constants and has to be included in the calculation of the lattice energy by using a generalised Born–Haber cycle . This reflects the fact that the covalent bond in the sulfur pair is inadequately accounted for by a strictly ionic treatment. Arsenopyrite has a related structure with heteroatomic As–S pairs rather than S-S pairs. Marcasite also possesses homoatomic anion pairs, but

3619-438: The concrete pores) and gypsum creates inner tensile forces in the concrete matrix which destroy the hardened cement paste, form cracks and fissures in concrete, and can lead to the ultimate ruin of the structure. Normalized tests for construction aggregate certify such materials as free of pyrite or marcasite. Pyrite is the most common of sulfide minerals and is widespread in igneous, metamorphic, and sedimentary rocks. It

3696-598: The convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular great stellated dodecahedron where all edges and angles are equal again, and the faces have been distorted into regular pentagrams . On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces. A tetartoid (also tetragonal pentagonal dodecahedron , pentagon-tritetrahedron , and tetrahedric pentagon dodecahedron )

3773-424: The dihedral angle between two consecutive such half-planes. If u 1 , u 2 and u 3 are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (− π , π ] . This dihedral angle is defined by or, using the function atan2 , This dihedral angle does not depend on the orientation of the chain (order in which

3850-422: The dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b 0 , b 1 and b 2 such that P + b 0 , P + b 1 and P + b 2 belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes

3927-611: The face of a perfect crystal (which is rarely found in nature). Height = 5 2 ⋅ Long side {\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}} Width = 4 3 ⋅ Long side {\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}} Short sides = 7 12 ⋅ Long side {\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}} The eight vertices of

SECTION 50

#1732791017140

4004-480: The following conditions: The regular dodecahedron is a tetartoid with more than the required symmetry. The triakis tetrahedron is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.) A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular anticupola connected base-to-base, called

4081-419: The gold is "invisible gold" incorporated into the pyrite (see Carlin-type gold deposit ). It has been suggested that the presence of both gold and arsenic is a case of coupled substitution but as of 1997 the chemical state of the gold remained controversial. Pyrite gained a brief popularity in the 16th and 17th centuries as a source of ignition in early firearms , most notably the wheellock , where

4158-585: The heap was then boiled with iron to produce iron sulfate. In the 15th century, new methods of such leaching began to replace the burning of sulfur as a source of sulfuric acid . By the 19th century, it had become the dominant method. Pyrite remains in commercial use for the production of sulfur dioxide , for use in such applications as the paper industry , and in the manufacture of sulfuric acid. Thermal decomposition of pyrite into FeS ( iron(II) sulfide ) and elemental sulfur starts at 540 °C (1,004 °F); at around 700 °C (1,292 °F), p S 2

4235-564: The lengths of edges or the angles between edges or faces.) Topologically distinct dodecahedra (excluding pentagonal and rhombic forms) Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector , based upon a suggestion from Albert Einstein . Regular dodecahedrons are sometimes used as dice, when they are known as d12s, especially in games such as Dungeons and Dragons . Pyrite The mineral pyrite ( / ˈ p aɪ r aɪ t / PY -ryte ), or iron pyrite , also known as fool's gold ,

4312-420: The mineral marcasite. The specimens of pyrite, when it appears as good quality crystals, are used in decoration. They are also very popular in mineral collecting. Among the sites that provide the best specimens are Soria and La Rioja provinces (Spain). In value terms, China ($ 47 million) constitutes the largest market for imported unroasted iron pyrites worldwide, making up 65% of global imports. China

4389-450: The more mechanically complicated perikon mineral pairs. Pyrite detectors can be as sensitive as a modern 1N34A germanium diode detector. Pyrite has been proposed as an abundant, non-toxic, inexpensive material in low-cost photovoltaic solar panels. Synthetic iron sulfide was used with copper sulfide to create the photovoltaic material. More recent efforts are working toward thin-film solar cells made entirely of pyrite. Pyrite

4466-715: The original sediments, and as a secondary mineral, deposited during diagenesis . Pyrite and marcasite commonly occur as replacement pseudomorphs after fossils in black shale and other sedimentary rocks formed under reducing environmental conditions. Pyrite is common as an accessory mineral in shale, where it is formed by precipitation from anoxic seawater, and coal beds often contain significant pyrite. Notable deposits are found as lenticular masses in Virginia, U.S., and in smaller quantities in many other locations. Large deposits are mined at Rio Tinto in Spain and elsewhere in

4543-444: The oxidation state of molybdenum is Mo . The mineral arsenopyrite has the formula Fe As S. Whereas pyrite has [S 2 ] units, arsenopyrite has [AsS] units, formally derived from deprotonation of arsenothiol (H 2 AsSH). Analysis of classical oxidation states would recommend the description of arsenopyrite as Fe [AsS] . Iron-pyrite FeS 2 represents the prototype compound of the crystallographic pyrite structure. The structure

4620-412: The peptide bonds in proteins are trans , though the peptide bond to the nitrogen of proline has an increased prevalence of cis compared to other amino-acid pairs. The side chain dihedral angles are designated with χ n (chi- n ). They tend to cluster near 180°, 60°, and −60°, which are called the trans , gauche , and gauche conformations. The stability of certain sidechain dihedral angles

4697-528: The perspective of classical inorganic chemistry , which assigns formal oxidation states to each atom, pyrite and marcasite are probably best described as Fe [S 2 ] . This formalism recognizes that the sulfur atoms in pyrite occur in pairs with clear S–S bonds. These persulfide [ S–S ] units can be viewed as derived from hydrogen disulfide , H 2 S 2 . Thus pyrite would be more descriptively called iron persulfide, not iron disulfide. In contrast, molybdenite , Mo S 2 , features isolated sulfide S centers and

SECTION 60

#1732791017140

4774-442: The planes, one has where n A  ·  n B is the dot product of the vectors and | n A | | n B | is the product of their lengths. The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite. However the absolute values can be and should be avoided when considering

4851-431: The point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle. A simpler formula for the same dihedral angle is the following (the proof is given below) or equivalently, This can be deduced from previous formulas by using

4928-443: The points are sequentially numbered and located at positions r 1 , r 2 , r 3 , etc. then bond vectors are defined by u 1 = r 2 − r 1 , u 2 = r 3 − r 2 , and u i = r i+1 − r i , more generally. This is the case for kinematic chains or amino acids in a protein structure . In these cases, one is often interested in the half-planes defined by three consecutive points, and

5005-685: The pyrite structure. The Fe atoms are bonded to six S atoms, giving a distorted octahedron. The material is a semiconductor . The Fe ions is usually considered to be low spin divalent state (as shown by Mössbauer spectroscopy as well as XPS). The material as a whole behaves as a Van Vleck paramagnet , despite its low-spin divalency. The sulfur centers occur in pairs, described as S 2 . Reduction of pyrite with potassium gives potassium dithioferrate , KFeS 2 . This material features ferric ions and isolated sulfide (S ) centers. The S atoms are tetrahedral, being bonded to three Fe centers and one other S atom. The site symmetry at Fe and S positions

5082-478: The pyritohedron in the Weaire–Phelan structure ). Another one is h = ⁠ 1 / φ ⁠ = 0.618... for the regular dodecahedron . See section Geometric freedom for other cases. Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the compound of two dodecahedra . The pyritohedron has a geometric degree of freedom with limiting cases of

5159-414: The rhombic dodecahedra, are space-filling . There are numerous other dodecahedra . While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner. The convex regular dodecahedron

5236-403: The rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra. There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing

5313-643: The rock and lead eventually to roof fall . Building stone containing pyrite tends to stain brown as pyrite oxidizes. This problem appears to be significantly worse if any marcasite is present. The presence of pyrite in the aggregate used to make concrete can lead to severe deterioration as pyrite oxidizes. In early 2009, problems with Chinese drywall imported into the United States after Hurricane Katrina were attributed to pyrite oxidation, followed by microbial sulfate reduction which released hydrogen sulfide gas ( H 2 S ). These problems included

5390-431: The same abstract regular polyhedron ; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron. In crystallography , two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry , and

5467-564: The same as a regular dodecahedron , known as pyritohedra, and this suggests an explanation for the artificial geometrical models found in Europe as early as the 5th century BC. Cattierite ( Co S 2 ), vaesite ( Ni S 2 ) and hauerite ( Mn S 2 ), as well as sperrylite ( Pt As 2 ) are similar in their structure and belong also to the pyrite group. Bravoite is a nickel-cobalt bearing variety of pyrite, with > 50% substitution of Ni for Fe within pyrite. Bravoite

5544-576: The same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes. Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite , and it may be an inspiration for the discovery of the regular Platonic solid form. The true regular dodecahedron can occur as a shape for quasicrystals (such as holmium–magnesium–zinc quasicrystal ) with icosahedral symmetry , which includes true fivefold rotation axes. The name crystal pyrite comes from one of

5621-415: The solid's topology and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In Conway polyhedron notation this is a gyro tetrahedron.) A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron . (The tetartoid shown here

5698-423: The symbols T, C, G , G , A and A are recommended (ap, sp, +sc, −sc, +ac and −ac respectively). A Ramachandran plot (also known as a Ramachandran diagram or a [ φ , ψ ] plot), originally developed in 1963 by G. N. Ramachandran , C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure . In

5775-487: The term had become a generic term for all of the sulfide minerals . Pyrite is usually found associated with other sulfides or oxides in quartz veins , sedimentary rock , and metamorphic rock , as well as in coal beds and as a replacement mineral in fossils , but has also been identified in the sclerites of scaly-foot gastropods . Despite being nicknamed "fool's gold", pyrite is sometimes found in association with small quantities of gold. A substantial proportion of

5852-419: The two common crystal habits shown by pyrite (the other one being the cube ). In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for

5929-424: The usual cases) are φ = π {\displaystyle \varphi =\pi } , φ = + π / 3 {\displaystyle \varphi =+\pi /3} and φ = − π / 3 {\displaystyle \varphi =-\pi /3} , which are called the trans , gauche , and gauche conformations. In stereochemistry ,

#139860