Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System after Mercury. In English, Mars carries a name of the Roman god of war, is referred to as the "Red Planet" because the reddish iron oxide prevalent on its surface gives it a reddish appearance, distinctive among the astronomical bodies visible to the naked eye. Mars is a terrestrial planet with a thin atmosphere, having surface features reminiscent both of the impact craters of the Moon and the valleys and polar ice caps of Earth; the days and seasons are comparable to those of Earth, because the rotational period as well as the tilt of the rotational axis relative to the ecliptic plane are similar. Mars is the site of Olympus Mons, the largest volcano and second-highest known mountain in the Solar System, of Valles Marineris, one of the largest canyons in the Solar System; the smooth Borealis basin in the northern hemisphere covers 40% of the planet and may be a giant impact feature. Mars has two moons and Deimos, which are small and irregularly shaped.
These may be captured asteroids, similar to a Mars trojan. There are ongoing investigations assessing the past habitability potential of Mars, as well as the possibility of extant life. Future astrobiology missions are planned, including the Mars 2020 and ExoMars rovers. Liquid water cannot exist on the surface of Mars due to low atmospheric pressure, less than 1% of the Earth's, except at the lowest elevations for short periods; the two polar ice caps appear to be made of water. The volume of water ice in the south polar ice cap, if melted, would be sufficient to cover the entire planetary surface to a depth of 11 meters. In November 2016, NASA reported finding a large amount of underground ice in the Utopia Planitia region of Mars; the volume of water detected has been estimated to be equivalent to the volume of water in Lake Superior. Mars can be seen from Earth with the naked eye, as can its reddish coloring, its apparent magnitude reaches −2.94, surpassed only by Jupiter, the Moon, the Sun.
Optical ground-based telescopes are limited to resolving features about 300 kilometers across when Earth and Mars are closest because of Earth's atmosphere. Mars is half the diameter of Earth with a surface area only less than the total area of Earth's dry land. Mars is less dense than Earth, having about 15% of Earth's volume and 11% of Earth's mass, resulting in about 38% of Earth's surface gravity; the red-orange appearance of the Martian surface is caused by rust. It can look like butterscotch. Like Earth, Mars has differentiated into a dense metallic core overlaid by less dense materials. Current models of its interior imply a core with a radius of about 1,794 ± 65 kilometers, consisting of iron and nickel with about 16–17% sulfur; this iron sulfide core is thought to be twice as rich in lighter elements as Earth's. The core is surrounded by a silicate mantle that formed many of the tectonic and volcanic features on the planet, but it appears to be dormant. Besides silicon and oxygen, the most abundant elements in the Martian crust are iron, aluminum and potassium.
The average thickness of the planet's crust is about 50 km, with a maximum thickness of 125 km. Earth's crust averages 40 km. Mars is a terrestrial planet that consists of minerals containing silicon and oxygen and other elements that make up rock; the surface of Mars is composed of tholeiitic basalt, although parts are more silica-rich than typical basalt and may be similar to andesitic rocks on Earth or silica glass. Regions of low albedo suggest concentrations of plagioclase feldspar, with northern low albedo regions displaying higher than normal concentrations of sheet silicates and high-silicon glass. Parts of the southern highlands include detectable amounts of high-calcium pyroxenes. Localized concentrations of hematite and olivine have been found. Much of the surface is covered by finely grained iron oxide dust. Although Mars has no evidence of a structured global magnetic field, observations show that parts of the planet's crust have been magnetized, suggesting that alternating polarity reversals of its dipole field have occurred in the past.
This paleomagnetism of magnetically susceptible minerals is similar to the alternating bands found on Earth's ocean floors. One theory, published in 1999 and re-examined in October 2005, is that these bands suggest plate tectonic activity on Mars four billion years ago, before the planetary dynamo ceased to function and the planet's magnetic field faded, it is thought that, during the Solar System's formation, Mars was created as the result of a stochastic process of run-away accretion of material from the protoplanetary disk that orbited the Sun. Mars has many distinctive chemical features caused by its position in the Solar System. Elements with comparatively low boiling points, such as chlorine and sulphur, are much more common on Mars than Earth. After the formation of the planets, all were subjected to the so-called "Late Heavy Bombardment". About 60% of the surface of Mars shows a record of impacts from that era, whereas much of the remaining surface is underlain by immense impact basins caused by those events.
There is evidence of an enormous impact basin in the northern hemisphere of Mars, spanning 10,600 by 8,500 km, or four times the size of the Moon's South Pole – Aitk
Johannes Kepler was a German astronomer and astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, his books Astronomia nova, Harmonices Mundi, Epitome Astronomiae Copernicanae; these works provided one of the foundations for Newton's theory of universal gravitation. Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg, he became an assistant to the astronomer Tycho Brahe in Prague, the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He taught mathematics in Linz, was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting telescope, was mentioned in the telescopic discoveries of his contemporary Galileo Galilei, he was a corresponding member of the Accademia dei Lincei in Rome. Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy and physics.
Kepler incorporated religious arguments and reasoning into his work, motivated by the religious conviction and belief that God had created the world according to an intelligible plan, accessible through the natural light of reason. Kepler described his new astronomy as "celestial physics", as "an excursion into Aristotle's Metaphysics", as "a supplement to Aristotle's On the Heavens", transforming the ancient tradition of physical cosmology by treating astronomy as part of a universal mathematical physics. Kepler was born on December 27, the feast day of St John the Evangelist, 1571, in the Free Imperial City of Weil der Stadt, his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline, his father, Heinrich Kepler, earned a precarious living as a mercenary, he left the family when Johannes was five years old. He was believed to have died in the Eighty Years' War in the Netherlands.
His mother, Katharina Guldenmann, an innkeeper's daughter, was a herbalist. Born prematurely, Johannes claimed to have been sickly as a child, he impressed travelers at his grandfather's inn with his phenomenal mathematical faculty. He was introduced to astronomy at an early age, developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, writing that he "was taken by mother to a high place to look at it." In 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being "called outdoors" to see it and that the moon "appeared quite red". However, childhood smallpox left him with weak vision and crippled hands, limiting his ability in the observational aspects of astronomy. In 1589, after moving through grammar school, Latin school, seminary at Maulbronn, Kepler attended Tübinger Stift at the University of Tübingen. There, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590.
He proved himself to be a superb mathematician and earned a reputation as a skilful astrologer, casting horoscopes for fellow students. Under the instruction of Michael Maestlin, Tübingen's professor of mathematics from 1583 to 1631, he learned both the Ptolemaic system and the Copernican system of planetary motion, he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the principal source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and astronomy at the Protestant school in Graz, he accepted the position in April 1594, at the age of 23. Kepler's first major astronomical work, Mysterium Cosmographicum, was the first published defense of the Copernican system. Kepler claimed to have had an epiphany on July 19, 1595, while teaching in Graz, demonstrating the periodic conjunction of Saturn and Jupiter in the zodiac: he realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe.
After failing to find a unique arrangement of polygons that fit known astronomical observations, Kepler began experimenting with 3-dimensional polyhedra. He found that each of the five Platonic solids could be inscribed and circumscribed by spherical orbs. By ordering the solids selectively—octahedron, dodecahedron, cube—Kepler found that the spheres could be placed at intervals corresponding to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler rejected this formula, because it was not precise enough. As
An epiphany is an experience of a sudden and striking realization. The term is used to describe scientific breakthrough, religious or philosophical discoveries, but it can apply in any situation in which an enlightening realization allows a problem or situation to be understood from a new and deeper perspective. Epiphanies are studied by psychologists and other scholars those attempting to study the process of innovation. Epiphanies are rare occurrences and follow a process of significant thought about a problem, they are triggered by a new and key piece of information, but a depth of prior knowledge is required to allow the leap of understanding. Famous epiphanies include Archimedes's discovery of a method to determine the density of an object and Isaac Newton's realization that a falling apple and the orbiting moon are both pulled by the same force; the word epiphany referred to insight through the divine. Today, this concept is more used without such connotations, but a popular implication remains that the epiphany is supernatural, as the discovery seems to come from the outside.
The word's secular usage may owe much of its popularity to Irish novelist James Joyce. The Joycean epiphany has been defined as "a sudden spiritual manifestation, whether from some object, event, or memorable phase of the mind — the manifestation being out of proportion to the significance or logical relevance of whatever produces it." The author used epiphany as a literary device within each entry of his short story collection Dubliners. Joyce had first expounded on epiphany's meaning in the fragment Stephen Hero, although this was only published posthumously in 1944. For the philosopher Emmanuel Lévinas, epiphany or a manifestation of the divine is seen in another's face. In traditional and pre-modern cultures, initiation rites and mystery religions have served as vehicles of epiphany, as well as the arts; the Greek dramatists and poets would, in the ideal, induct the audience into states of catharsis or kenosis, respectively. In modern times an epiphany lies behind the title of William Burroughs' Naked Lunch, a drug-influenced state, as Burroughs explained, "a frozen moment when everyone sees what is at the end of the fork."
Both the Dadaist Marcel Duchamp and the Pop Artist Andy Warhol would invert expectations by presenting commonplace objects or graphics as works of fine art by presenting them in a way no one had thought to do before. Epiphanies can come in many different forms, are generated by a complex combination of experience, knowledge and context. A contemporary example of an epiphany in education might involve the process by which a student arrives at some form of new insight or clarifying thought. Despite this popular image, epiphany is the result of significant work on the part of the discoverer, is only the satisfying result of a long process; the surprising and fulfilling feeling of epiphany is so surprising because one cannot predict when one's labor will bear fruit, our subconscious can play a significant part in delivering the solution. A common myth predicts. Not all innovations occur through epiphanies. Most innovations occur without epiphany, epiphanies contribute little towards finding the next one.
Crucially, epiphany cannot be predicted, or controlled. Although epiphanies are only a rare occurrence, crowning a process of significant labor, there is a common myth that epiphanies of sudden comprehension are responsible for leaps in technology and the sciences. Famous epiphanies include Archimedes' realization of how to estimate the volume of a given mass, which inspired him to shout "Eureka!". The biographies of many mathematicians and scientists include an epiphanic episode early in the career, the ramifications of which were worked out in detail over the following years. For example Albert Einstein was struck as a young child by being given a compass, realizing that some unseen force in space was making it move. Another better, example from Einstein's life occurred in 1905 after he had spent an evening unsuccessfully trying to reconcile Newtonian physics and Maxwell's equations. While taking a streetcar home, he looked behind him at the receding clocktower in Bern and realized that if the car sped up close to the speed of light, he would see the clock slow down.
Einstein had a second epiphany two years in 1907 which he called "the happiest thought of my life" when he imagined an elevator falling, realized that a passenger would not be able to tell the difference between the weightlessness of falling, the weightlessness of space - a thought which allowed him to generalize his theory of relativity to include gravity as a curvature in spacetime. A similar flash of holistic understanding in a prepared mind was said to give Charles Darwin his "hunch", Darwin stated that he always remembered the spot in the road where his carriage was when the epiphany struck. Another famous epiphany myth is associated wit
Nicolaus Copernicus was a Renaissance-era mathematician and astronomer who formulated a model of the universe that placed the Sun rather than the Earth at the center of the universe, in all likelihood independently of Aristarchus of Samos, who had formulated such a model some eighteen centuries earlier. The publication of Copernicus' model in his book De revolutionibus orbium coelestium, just before his death in 1543, was a major event in the history of science, triggering the Copernican Revolution and making a pioneering contribution to the Scientific Revolution. Copernicus was born and died in Royal Prussia, a region, part of the Kingdom of Poland since 1466. A polyglot and polymath, he obtained a doctorate in canon law and was a mathematician, physician, classics scholar, governor and economist. In 1517 he derived a quantity theory of money—a key concept in economics—and in 1519 he formulated an economic principle that came to be called Gresham's law. Nicolaus Copernicus was born on 19 February 1473 in the city of Thorn, in the province of Royal Prussia, in the Crown of the Kingdom of Poland.
His father was a merchant from Kraków and his mother was the daughter of a wealthy Toruń merchant. Nicolaus was the youngest of four children, his brother Andreas became an Augustinian canon at Frombork. His sister Barbara, named after her mother, became a Benedictine nun and, in her final years, prioress of a convent in Chełmno, his sister Katharina married the businessman and Toruń city councilor Barthel Gertner and left five children, whom Copernicus looked after to the end of his life. Copernicus never married and is not known to have had children, but from at least 1531 until 1539 his relations with Anna Schilling, a live-in housekeeper, were seen as scandalous by two bishops of Warmia who urged him over the years to break off relations with his "mistress". Copernicus' father's family can be traced to a village in Silesia near Nysa; the village's name has been variously spelled Kopernik, Copernic, Kopernic and today Koperniki. In the 14th century, members of the family began moving to various other Silesian cities, to the Polish capital, Kraków, to Toruń.
The father, Mikołaj the Elder the son of Jan, came from the Kraków line. Nicolaus was named after his father, who appears in records for the first time as a well-to-do merchant who dealt in copper, selling it in Danzig, he moved from Kraków to Toruń around 1458. Toruń, situated on the Vistula River, was at that time embroiled in the Thirteen Years' War, in which the Kingdom of Poland and the Prussian Confederation, an alliance of Prussian cities and clergy, fought the Teutonic Order over control of the region. In this war, Hanseatic cities like Danzig and Toruń, Nicolaus Copernicus's hometown, chose to support the Polish King, Casimir IV Jagiellon, who promised to respect the cities' traditional vast independence, which the Teutonic Order had challenged. Nicolaus' father was engaged in the politics of the day and supported Poland and the cities against the Teutonic Order. In 1454 he mediated negotiations between Poland's Cardinal Zbigniew Oleśnicki and the Prussian cities for repayment of war loans.
In the Second Peace of Thorn, the Teutonic Order formally relinquished all claims to its western province, which as Royal Prussia remained a region of the Crown of the Kingdom of Poland until the First and Second Partitions of Poland. Copernicus's father married Barbara Watzenrode, the astronomer's mother, between 1461 and 1464, he died about 1483. Nicolaus' mother, Barbara Watzenrode, was the daughter of a wealthy Toruń patrician and city councillor, Lucas Watzenrode the Elder, Katarzyna, mentioned in other sources as Katarzyna Rüdiger gente Modlibóg; the Modlibógs were a prominent Polish family, well known in Poland's history since 1271. The Watzenrode family, like the Kopernik family, had come from Silesia from near Świdnica, after 1360 had settled in Toruń, they soon became one of most influential patrician families. Through the Watzenrodes' extensive family relationships by marriage, Copernicus was related to wealthy families of Toruń, Gdańsk and Elbląg, to prominent Polish noble families of Prussia: the Czapskis, Działyńskis, Konopackis and Kościeleckis.
Lucas and Katherine had three children: Lucas Watzenrode the Younger, who would become Bishop of Warmia and Copernicus's patron. Lucas Watzenrode the Elder, a wealthy merchant and in 1439–62 president of the judicial bench, was a decided opponent of the Teutonic Knights. In 1453 he was the delegate from Toruń at the Grudziądz conference that planned the uprising against them. During the ensuing Thirteen Years' War, he supported the Prussian cities' war effort with substantial monetary subsidies, with political activity in Toruń and Danzig, by fighting in battles at Łasin and Malbork, he died in 1462. Lucas Watzenrode the Younger, the astronomer's maternal uncle and patron, was educated at the University of Kraków and at the universities of Cologne and Bologna, he was a bitter opponent of the Teutonic Order, its Grand Master once referred to him as "the devil incarn
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius about nine times that of Earth, it has only one-eighth the average density of Earth, but with its larger volume Saturn is over 95 times more massive. Saturn is named after the Roman god of agriculture. Saturn's interior is composed of a core of iron–nickel and rock; this core is surrounded by a deep layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium, a gaseous outer layer. Saturn has a pale yellow hue due to ammonia crystals in its upper atmosphere. Electrical current within the metallic hydrogen layer is thought to give rise to Saturn's planetary magnetic field, weaker than Earth's, but has a magnetic moment 580 times that of Earth due to Saturn's larger size. Saturn's magnetic field strength is around one-twentieth of Jupiter's; the outer atmosphere is bland and lacking in contrast, although long-lived features can appear.
Wind speeds on Saturn can reach 1,800 km/h, higher than on Jupiter, but not as high as those on Neptune. In January 2019, astronomers reported that a day on the planet Saturn has been determined to be 10h 33m 38s + 1m 52s− 1m 19s , based on studies of the planet's C Ring; the planet's most famous feature is its prominent ring system, composed of ice particles, with a smaller amount of rocky debris and dust. At least 62 moons are known to orbit Saturn, of which 53 are named; this does not include the hundreds of moonlets in the rings. Titan, Saturn's largest moon, the second-largest in the Solar System, is larger than the planet Mercury, although less massive, is the only moon in the Solar System to have a substantial atmosphere. Saturn is a gas giant because it is predominantly composed of helium, it lacks a definite surface. Saturn's rotation causes it to have the shape of an oblate spheroid, its equatorial and polar radii differ by 10%: 60,268 km versus 54,364 km. Jupiter and Neptune, the other giant planets in the Solar System, are oblate but to a lesser extent.
The combination of the bulge and rotation rate means that the effective surface gravity along the equator, 8.96 m/s2, is 74% that at the poles and is lower than the surface gravity of Earth. However, the equatorial escape velocity of nearly 36 km/s is much higher than that for Earth. Saturn is the only planet of the Solar System, less dense than water—about 30% less. Although Saturn's core is denser than water, the average specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times Earth's mass, Saturn is 95 times Earth's mass. Together and Saturn hold 92% of the total planetary mass in the Solar System. Despite consisting of hydrogen and helium, most of Saturn's mass is not in the gas phase, because hydrogen becomes a non-ideal liquid when the density is above 0.01 g/cm3, reached at a radius containing 99.9% of Saturn's mass. The temperature and density inside Saturn all rise toward the core, which causes hydrogen to be a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a small rocky core surrounded by hydrogen and helium with trace amounts of various volatiles.
This core is more dense. Examination of Saturn's gravitational moment, in combination with physical models of the interior, has allowed constraints to be placed on the mass of Saturn's core. In 2004, scientists estimated that the core must be 9–22 times the mass of Earth, which corresponds to a diameter of about 25,000 km; this is surrounded by a thicker liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that transitions to a gas with increasing altitude. The outermost layer consists of gas. Saturn has a hot interior, reaching 11,700 °C at its core, it radiates 2.5 times more energy into space than it receives from the Sun. Jupiter's thermal energy is generated by the Kelvin–Helmholtz mechanism of slow gravitational compression, but such a process alone may not be sufficient to explain heat production for Saturn, because it is less massive. An alternative or additional mechanism may be generation of heat through the "raining out" of droplets of helium deep in Saturn's interior.
As the droplets descend through the lower-density hydrogen, the process releases heat by friction and leaves Saturn's outer layers depleted of helium. These descending droplets may have accumulated into a helium shell surrounding the core. Rainfalls of diamonds have been suggested to occur within Saturn, as well as in Jupiter and ice giants Uranus and Neptune; the outer atmosphere of Saturn contains 3.25 % helium by volume. The proportion of helium is deficient compared to the abundance of this element in the Sun; the quantity of elements heavier than helium is not known but the proportions are assumed to match the primordial abundances from the formation of the Solar System. The total mass of these heavier elements is estimated to be 19–31 times the mass of the Earth, with a significant fraction located in Saturn's core region. Trace amounts of ammonia, ethane, propane and methane have been detected in Saturn's atmosphere; the upper clouds are composed of ammonia crystals, while the lower level clouds appear to consist of either ammonium hydrosulfide or water.
Ultraviolet radiation from the Sun causes methane ph
In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, a Platonic solid. There are three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120; the pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry; the elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are a large number of other dodecahedra; the convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol. The dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex; the convex regular dodecahedron 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, the great dodecahedron, the great stellated dodecahedron; the small stellated dodecahedron and great dodecahedron are dual to each other. All of these regular star dodecahedra have regular pentagrammic faces; the convex regular dodecahedron and great stellated dodecahedron are different realisations of the same 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, the tetartoid with tetrahedral symmetry: A pyritohedron is a dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not constrained to be regular, the underlying atomic arrangement has no true fivefold symmetry axes.
Its 30 edges are divided into two sets -- containing 6 edges of 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, 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 with icosahedral symmetry, which includes true fivefold rotation axes. Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube; the coordinates of the eight vertices of the original cube are: The coordinates of the 12 vertices of the cross-edges are: where h is the height of the wedge-shaped "roof" above the faces of the cube. When h = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed; when h = 0, the cross-edges are absorbed in the facets of the cube, the pyritohedron reduces to a cube.
When h = −1 + √5/2, the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = −1 − √5/2, the conjugate of this value, the result is a regular great stellated dodecahedron. A reflected pyritohedron is made by swapping; the two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case; the pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear edges, 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. A tetartoid is a dodecahedron with chiral tetrahedral symmetry. 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, half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form, its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, the bisection lines are slanted retaining 3-fold rotation at the 8 corners. The following points are vertices of a tetartoid pentagon under tetrahedral symmetry:, it can be seen as a tetrahedron, with edges divided into 3 segments, along with a center point of each triangular face. In Conway polyhedron notation it can be seen as a gyro tetrahedron. A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedra constructed from two triangular anticupola connected base-to-base, called a triangular gyrobianticupo
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids, it has 6 faces, 12 edges, 8 vertices. The cube is a square parallelepiped, an equilateral cuboid and a right rhombohedron, it is a regular square prism in three orientations, a trigonal trapezohedron in four orientations. The cube is dual to the octahedron, it has octahedral symmetry. The cube is the only convex polyhedron; the cube has four special orthogonal projections, centered, on a vertex, edges and normal to its vertex figure. The first and third correspond to the B2 Coxeter planes; the cube can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are while the interior consists of all points with −1 < xi < 1 for all i.
In analytic geometry, a cube's surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of edge length a: As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids with a given surface area. A cube has the largest volume among cuboids with the same total linear size. For a cube whose circumscribing sphere has radius R, for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have: ∑ i = 1 8 d i 4 8 + 16 R 4 9 = 2. Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube, they were unable to solve this problem, in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces; the highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color; the lowest symmetry D2h is a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors; the cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is unique among the Platonic solids in having faces with an number of sides and it is the only member of that group, a zonohedron; the cube can be cut into six identical square pyramids.
If these square pyramids are attached to the faces of a second cube, a rhombic dodecahedron is obtained. The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions; the quotient of the cube by the antipodal map yields the hemicube. If the original cube has edge length 1, its dual polyhedron has edge length 2 / 2; the cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form the stella octangula; the int