John Horton Conway
John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.
Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.
There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.
In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.
He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.
Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope
Graphite, archaically referred to as plumbago, is a crystalline form of the element carbon with its atoms arranged in a hexagonal structure. It occurs in this form and is the most stable form of carbon under standard conditions. Under high pressures and temperatures it converts to diamond. Graphite is used in lubricants, its high conductivity makes it useful in electronic products such as electrodes and solar panels. The principal types of natural graphite, each occurring in different types of ore deposits, are Crystalline small flakes of graphite occurs as isolated, plate-like particles with hexagonal edges if unbroken; when broken the edges can be angular. Ordered pyrolytic graphite refers to graphite with an angular spread between the graphite sheets of less than 1°; the name "graphite fiber" is sometimes used to refer to carbon fibers or carbon fiber-reinforced polymer. Graphite occurs in metamorphic rocks as a result of the reduction of sedimentary carbon compounds during metamorphism, it occurs in igneous rocks and in meteorites.
Minerals associated with graphite include quartz, calcite and tourmaline. The principal export sources of mined graphite are in order of tonnage: China, Canada and Madagascar. In meteorites, graphite occurs with silicate minerals. Small graphitic crystals in meteoritic iron are called cliftonite; some microscopic grains have distinctive isotopic compositions, indicating that they were formed before the Solar system. They are one of about 12 known types of mineral that predate the Solar System and have been detected in molecular clouds; these minerals were formed in the ejecta when supernovae exploded or low- to intermediate-sized stars expelled their outer envelopes late in their lives. Graphite may be the third oldest mineral in the Universe. Solid carbon comes in different forms known as allotropes depending on the type of chemical bond; the two most common are graphite. In diamond the bonds are sp3 and the atoms form tetrahedra with each bound to four nearest neighbors. In graphite they are sp2 orbital hybrids and the atoms form in planes with each bound to three nearest neighbors 120 degrees apart.
The individual layers are called graphene. In each layer, the carbon atoms are arranged in a honeycomb lattice with separation of 0.142 nm, the distance between planes is 0.335 nm. Atoms in the plane are bonded covalently, with only three of the four potential bonding sites satisfied; the fourth electron is free to migrate in the plane. However, it does not conduct in a direction at right angles to the plane. Bonding between layers is via weak van der Waals bonds, which allows layers of graphite to be separated, or to slide past each other; the two known forms of graphite and beta, have similar physical properties, except that the graphene layers stack differently. The alpha graphite may be either buckled; the alpha form can be converted to the beta form through mechanical treatment and the beta form reverts to the alpha form when it is heated above 1300 °C. The equilibrium pressure and temperature conditions for a transition between graphite and diamond is well established theoretically and experimentally.
The pressure changes linearly between 1.7 GPa at 0 K and 12 GPa at 5000 K. However, the phases have a wide region about this line where they can coexist. At normal temperature and pressure, 20 °C and 1 standard atmosphere, the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. However, at temperatures above about 4500 K, diamond converts to graphite. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at 2000 K, a pressure of 35 GPa is needed; the acoustic and thermal properties of graphite are anisotropic, since phonons propagate along the bound planes, but are slower to travel from one plane to another. Graphite's high thermal stability and electrical and thermal conductivity facilitate its widespread use as electrodes and refractories in high temperature material processing applications. However, in oxygen-containing atmospheres graphite oxidizes to form carbon dioxide at temperatures of 700 °C and above.
Graphite is hence useful in such applications as arc lamp electrodes. It can conduct electricity due to the vast electron delocalization within the carbon layers; these valence electrons are free to move. However, the electricity is conducted within the plane of the layers; the conductive properties of powdered graphite allow its use as pressure sensor in carbon microphones. Graphite and graphite powder are valued in industrial applications for their self-lubricating and dry lubricating properties. There is a common belief that graphite's lubricating properties are due to the loose interlamellar coupling between sheets in the structure. However, it has been shown that in a vacuum environment, graphite degrades as a lubricant, due to the hypoxic conditions; this observation led to the hypothesis that the lubrication is due to the presence of fluids between the layers, such as air and water, which are adsorbed from the
Graphene is an allotrope of carbon consisting of a single layer of carbon atoms arranged in a hexagonal lattice. Graphene can be considered as an indefinitely large aromatic molecule, the ultimate case of the family of flat polycyclic aromatic hydrocarbons. Graphite, the most common allotrope of carbon, is a stack of graphene layers held together with weak bonds. Fullerenes and carbon nanotubes, two other forms of carbon, have structures similar to that of graphene. Graphene has many uncommon properties, it is the strongest material tested, conducts heat and electricity efficiently, is nearly transparent, yet opaque for a 1-atom-thick layer. Graphene shows a large and nonlinear diamagnetism, greater than that of graphite, can be levitated by neodymium magnets, it is a semimetal with small overlap between the conduction bands. Scientists theorized about graphene for years, it had been produced unintentionally in small quantities for centuries through the use of pencils and other similar graphite applications.
It was observed in electron microscopes in 1962, but it was studied only while supported on metal surfaces. The material was rediscovered and characterized in 2004 by Andre Geim and Konstantin Novoselov at the University of Manchester. Research was informed by existing theoretical descriptions of its composition and properties; this work resulted in the two winning the Nobel Prize in Physics in 2010 "for groundbreaking experiments regarding the two-dimensional material graphene." The name "graphene" is a combination of "graphite" and the suffix -ene, named by Hanns-Peter Boehm and colleagues, who produced and observed single-layer carbon foils in 1962. Boehm et al. introduced the term graphene in 1986 to describe single sheets of graphite. Graphene can be considered an "infinite alternant" polycyclic aromatic hydrocarbon; the International Union of Pure and Applied Chemistry notes, "previously, descriptions such as graphite layers, carbon layers, or carbon sheets have been used for the term graphene...it is incorrect to use for a single layer a term which includes the term graphite, which would imply a three-dimensional structure.
The term graphene should be used only when the reactions, structural relations or other properties of individual layers are discussed."Geim defined "isolated or free-standing graphene" as "a single atomic plane of graphite, which – and this is essential – is sufficiently isolated from its environment to be considered free-standing." This definition is narrower than the IUPAC definition and refers to cloven and suspended graphene. Other forms such as graphene grown on various metals, can become free-standing if, for example, suspended or transferred to silicon dioxide or silicon carbide. Graphene is a crystalline allotrope of carbon with 2-dimensional properties, its carbon atoms are packed densely in a regular atomic-scale chicken wire pattern. Each atom has four bonds: one σ bond with each of its three neighbors and one π-bond, oriented out of plane; the atoms are about 1.42 Å apart. Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices; this perspective was used to calculate the band structure for a single graphite layer using a tight-binding approximation.
Graphene's stability is due to its packed carbon atoms and a sp2 orbital hybridization – a combination of orbitals s, px and py that constitute the σ-bond. The final pz electron makes up the π-bond; the π-bonds hybridize together to form the π ∗ - bands. These bands are responsible for most of graphene's notable electronic properties, via the half-filled band that permits free-moving electrons. Graphene sheets in solid form show evidence in diffraction for graphite's layering; this is true of some single-walled nanostructures. However, unlayered graphene with only rings has been found in the core of presolar graphite onions. TEM studies show faceting at defects in flat graphene sheets and suggest a role for two-dimensional crystallization from a melt. Graphene can self-repair holes in its sheets when exposed to molecules containing carbon, such as hydrocarbons. Bombarded with pure carbon atoms, the atoms align into hexagons filling the holes; the atomic structure of isolated, single-layer graphene is studied by TEM on sheets of graphene suspended between bars of a metallic grid.
Electron diffraction patterns showed the expected honeycomb lattice. Suspended graphene showed "rippling" of the flat sheet, with amplitude of about one nanometer; these ripples may be intrinsic to the material as a result of the instability of two-dimensional crystals, or may originate from the ubiquitous dirt seen in all TEM images of graphene. Atomic resolution real-space images of isolated, single-layer graphene on SiO2 substrates are available via scanning tunneling microscopy. Photoresist residue, which must be removed to obtain atomic-resolution images, may be the "adsorbates" observed in TEM images, may explain the observed rippling. Rippling on SiO2 is caused by conformation of graphene to the underlying SiO2 and is not intrinsic. Ab initio calculations show that a graphene sheet is thermodynamically unstable if its size is less than about 20 nm and becomes the most stable fullerene only for molecules larger than 24,000 atoms. Analogs are two-dimensional systems. Analogs can be systems in which the physics is easier to manipulate.
In those systems
Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. These cylindrical carbon molecules have unusual properties, which are valuable for nanotechnology, electronics and other fields of materials science and technology. Owing to the material's exceptional strength and stiffness, nanotubes have been constructed with a length-to-diameter ratio of up to 132,000,000:1 larger than that for any other material. In addition, owing to their extraordinary thermal conductivity and mechanical and electrical properties, carbon nanotubes find applications as additives to various structural materials. For instance, nanotubes form a tiny portion of the material in some baseball bats, golf clubs, car parts, or damascus steel. Nanotubes are members of the fullerene structural family, their name is derived from their long, hollow structure with the walls formed by one-atom-thick sheets of carbon, called graphene. These sheets are rolled at specific and discrete angles, the combination of the rolling angle and radius decides the nanotube properties, for example, whether the individual nanotube shell is a metal or semiconductor.
Nanotubes are categorized as multi-walled nanotubes. Individual nanotubes align themselves into "ropes" held together by van der Waals forces, more pi-stacking. Applied quantum chemistry orbital hybridization, best describes the chemical bonding in nanotubes; the chemical bonding of nanotubes involves sp2-hybrid carbon atoms. These bonds, which are similar to those of graphite and stronger than those found in alkanes and diamond, provide nanotubes their unique strength. There is no consensus on some terms describing carbon nanotubes in scientific literature: both "-wall" and "-walled" are being used in combination with "single", "double", "triple", or "multi", the letter C is omitted in the abbreviation, for example, multi-walled carbon nanotube. D = a π = 78.3 p m. SWNTs are an important variety of carbon nanotubes because most of their properties change with the values, this dependence is non-monotonic. In particular, their band gap can vary from zero to about 2 eV and their electrical conductivity can show metallic or semiconducting behavior.
Single-walled nanotubes are candidates for miniaturizing electronics. The most basic building block of these systems is an electric wire, SWNTs with diameters of an order of a nanometer can be excellent conductors. One useful application of SWNTs is in the development of the first intermolecular field-effect transistors; the first intermolecular logic gate using SWCNT FETs was made in 2001. A logic gate requires both a p-FET and an n-FET; because SWNTs are p-FETs when exposed to oxygen and n-FETs otherwise, it is possible to expose half of an SWNT to oxygen and protect the other half from it. The resulting SWNT acts as a not logic gate with both p- and n-type FETs in the same molecule. Prices for single-walled nanotubes declined from around $1500 per gram as of 2000 to retail prices of around $50 per gram of as-produced 40–60% by weight SWNTs as of March 2010; as of 2016, the retail price of as-produced 75% by weight SWNTs was $2 per gram, cheap enough for widespread use. SWNTs are forecast to make a large impact in electronics applications by 2020 according to The Global Market for Carbon Nanotubes report.i Multi-walled nanotubes consist of multiple rolled layers of graphene.
There are two models. In the Russian Doll model, sheets of graphite are arranged in concentric cylinders, e.g. a single-walled nanotube within a larger single-walled nanotube. In the Parchment model, a single sheet of graphite is rolled in around itself, resembling a scroll of parchment or a rolled newspaper; the interlayer distance in multi-walled nanotubes is close to the distance between graphene layers in graphite 3.4 Å. The Russian Doll structure is observed more commonly, its individual shells can be described as SWNTs, which can be semiconducting. Because of statistical probability and restrictions on the relative diameters of the individual tubes, one of the shells, thus the whole MWNT, is a zero-gap metal. Double-walled carbon nanotubes form a special class of nanotubes because their morphology and properties are similar to those of SWNTs but they are more resistant to chemicals; this is important when it is necessary to graft chemical functions to the surface of the nanotubes to add properties to the CNT.
Covalent functionalization of SWNTs will break some C=C double bonds, leaving "holes" in the structure on the nanotube and thus modifying both its mechanical and electrical properties. In the case of DWNTs, only the outer wall is modified. DWNT synthesis on the gram-scale by the CCVD technique was first proposed in 2003 from the selective reduction of oxide solutions in methane and hydrogen; the telescopic motion ability of inner shells and their unique mechanical properties will permit the use of multi-walled
Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is π 3 2 ≃ 0.74048. The same packing density can be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction; the Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only in case of 2, 3, 8 and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them; the cubic and hexagonal arrangements are close to one another in energy, it may be difficult to predict which form will be preferred from first principles.
There are two simple regular lattices. They are called face-centered hexagonal close-packed, based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; the fcc lattice is known to mathematicians as that generated by the A3 root system. The problem of close-packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after a question on piling cannonballs on ships was posed to him by Sir Walter Raleigh on their expedition to America. Cannonballs were piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base; the cannonball problem asks which flat square arrangements of cannonballs can be stacked into a square pyramid. Édouard Lucas formulated the problem as the Diophantine equation ∑ n = 1 N n 2 = M 2 or 1 6 N = M 2 and conjectured that the only solutions are N = 1, M = 1, N = 24, M = 70.
Here N is the number of layers in the pyramidal stacking arrangement and M is the number of cannonballs along an edge in the flat square arrangement. In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres and two smaller gaps surrounded by four spheres; the distances to the centers of these gaps from the centers of the surrounding spheres is √3⁄2 for the tetrahedral, √2 for the octahedral, when the sphere radius is 1. Relative to a reference layer with positioning A, two more positionings B and C are possible; every sequence of A, B, C without immediate repetition of the same one is possible and gives an dense packing for spheres of a given radius. The most regular ones are fcc = ABC ABC ABC... hcp = AB AB AB AB.... There is an uncountably infinite number of disordered arrangements of planes that are sometimes collectively referred to as "Barlow packings", after crystallographer William BarlowIn close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch of one sphere diameter.
The distance between sphere centers, projected on the z axis, is: pitch Z = 6 ⋅ d 3 ≈ 0.816 496 58 d, where d is the diameter of a sphere. The coordination number of hcp and fcc is 12 and their atomic packing factors are equal to the number mentioned above, 0.74. When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact; the distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere and r2 is the radius of the second. In close packing all of the spheres share a common radius, r; therefore two centers would have a distance 2r. To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to hcp; the box would be placed on the x-y-z coordinate space.
First form a row of spheres. The centers will all lie on a straight line, their x-coordinate will vary by 2r since the distance between each center of the spheres are touching is 2r. The y-coordinate and z-coordinate wil
Silicene is a two-dimensional allotrope of silicon, with a hexagonal honeycomb structure similar to that of graphene. Contrary to graphene, silicene has a periodically buckled topology. Although theorists had speculated about the existence and possible properties of free-standing silicene, researchers first observed silicon structures that were suggestive of silicene in 2010. Using a scanning tunneling microscope they studied self-assembled silicene nanoribbons and silicene sheets deposited onto a silver crystal, Ag and Ag, with atomic resolution; the images revealed hexagons in a honeycomb structure similar to that of graphene, however, were shown to originate from the silver surface mimicking the hexagons. Density functional theory calculations showed that silicon atoms tend to form such honeycomb structures on silver, adopt a slight curvature that makes the graphene-like configuration more likely. However, such a model has been invalidated for Si/Ag: the Ag surface displays a missing-row reconstruction upon Si adsorption and the honeycomb structures observed are tip artefacts.
This was followed by the discovery of dumbbell reconstruction in silicene which explains the formation mechanisms of layered silicene and silicene on Ag. In 2015, a silicene field-effect transistor made its debut that opens up new opportunities for two-dimensional silicon for various fundamental science studies and electronic applications. Silicon and carbon are similar atoms, they lie next to each other in the same group on the periodic table and have an s2 p2 electronic structure. The 2D structures of silicene and graphene are quite similar but have important differences. While both form hexagonal structures, graphene is flat, while silicene forms a buckled hexagonal shape, its buckled structure gives silicene a tuneable band gap by applying an external electric field. Silicene's hydrogenation reaction is more exothermic than graphene's. Another difference is that since silicon's covalent bonds do not have pi-stacking, silicene does not cluster into a graphite-like form; the formation of a buckled structure in silicene unlike planar structure of graphene has been attributed to strong Pseudo Jahn-Teller distortions arising due to vibronic coupling between spaced filled and empty electronic states.
Silicene and graphene have similar electronic structures. Both have a Dirac cone and linear electronic dispersion around the Dirac points. Both have a quantum spin Hall effect. Both are expected to have the characteristics of massless Dirac fermions that carry charge, but this is only predicted for silicene and has not been observed because it is expected to only occur with free-standing silicene which has not been synthesized, it is believed that the substrate on which the silicene is made has a substantial effect on its electronic properties. Unlike carbon atoms in graphene, silicon atoms tend to adopt sp3 hybridization over sp2 in silicene, which makes it chemically active on the surface and allows its electronic states to be tuned by chemical functionalization. Compared with graphene, silicene has several prominent advantages: a much stronger spin–orbit coupling, which may lead to a realization of quantum spin Hall effect in the experimentally accessible temperature, a better tunability of the band gap, necessary for an effective field effect transistor operating at room temperature, an easier valley polarization and more suitability for valleytronics study.
Early studies of silicene showed that different dopants within the silicene structure provide the ability to tune its band gap. The band gap in epitaxial silicene has been tuned by oxygen adatoms from zero-gap-type to semiconductor-type. With a tunable band gap, specific electronic components could be made-to-order for applications that require specific band gaps; the band gap can be brought down to 0.1 eV, smaller than the band gap found in traditional field effect transistors. Inducing n-type doping within silicene requires an alkali metal dopant. Varying the amount adjusts the band gap. Maximum doping increases the band gap 0.5eV. Due to heavy doping, the supply voltage must be c. 30V. Alkali metal-doped silicene can only produce n-type semiconductors. Neutral doping is required to produce devices such as light emitting diodes. LEDs use a p-i-n junction to produce light. A separate dopant must be introduced to generate p-type doped silicene. Iridium doped silicene allows p-type silicene to be created.
Through platinum doping, i-type silicene is possible. With the combination of n-type, p-type and i-type doped structures, silicene has opportunities for use in electronics. Power dissipation within traditional metal oxide semiconductor field effect transistors generates a bottleneck when dealing with nano-electronics. Tunnel field-effect transistors may become an alternative to traditional MOSFETs because they can have a smaller subthreshold slope and supply voltage, which reduce power dissipation. Computational studies showed. Silicene TFETs have an on-state current over 1mA/μm, a sub-threshold slope of 77 mV/decade and a supply voltage of 1.7 V. With this much increased on-state current and reduced supply voltage, power dissipation within these devices is far below that of traditional MOSFETs and its peer TFETs. 2D silicene is not planar featuring chair-like puckering distortions in
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees; the triangular tiling has Schläfli symbol of. Conway calls it a deltille, named from the triangular shape of the Greek letter delta; the triangular tiling can be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane; the other two are the hexagonal tiling. There are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314. There is one class of Archimedean colorings, 111112, not 1-uniform, containing alternate rows of triangles where every third is colored; the example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.
The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb; the A*2 lattice can be constructed by the union of all three A2 lattices, equivalent to the A2 lattice. + + = dual of = The vertices of the triangular tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing; the packing density is π⁄√12 or 90.69%. The voronoi cell of a triangular tiling is a hexagon, so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling. With identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color; the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid; these can be expanded to Platonic solids: five and three triangles on a vertex define an icosahedron and tetrahedron respectively.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols, continuing into the hyperbolic plane. It is topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, continuing into the hyperbolic plane. Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct. There are 4 regular complex apeirogons. Regular complex apeirogons have edges, where edges can contain 2 or more vertices. Regular apeirogons pr are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, vertex figures are r-gonal; the first is made of 2-edges, next two are triangular edges, the last has overlapping hexagonal edges. There are three Laves tilings made of single type of triangles: Triangular tiling honeycomb Simplectic honeycomb Tilings of regular polygons List of uniform tilings Isogrid Coxeter, H.
S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. P35 John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. "Triangular Grid". MathWorld. Weisstein, Eric W. "Regular tessellation". MathWorld. Weisstein, Eric W. "Uniform tessellation". MathWorld. Klitzing, Richard. "2D Euclidean tilings x3o6o - trat - O2"