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
Sir Roger Penrose is an English mathematical physicist and philosopher of science. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford and Emeritus Fellow of Wadham College, Oxford. Penrose has made contributions to the mathematical physics of general cosmology, he has received several prizes and awards, including the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems. Born in Colchester, Roger Penrose is a son of psychiatrist and geneticist Lionel Penrose and Margaret Leathes, the grandson of the physiologist John Beresford Leathes and his Russian wife, Sonia Marie Natanson, who had left St. Petersburg in the late 1880s, his uncle was artist Roland Penrose. Penrose is the brother of physicist Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951.
Having started research under the professor of geometry and astronomy, Sir W. V. D. Hodge, Penrose finished his PhD at Cambridge in 1958, with a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd, he devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form", exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects inspired it. Escher's Waterfall, Ascending and Descending were in turn inspired by Penrose; as reviewer Manjit Kumar puts it: As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that loops up and down. An article followed and a copy was sent to Escher.
Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces. Having become a reader at Birkbeck College, London it was in 1964 that, in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime"; until work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be soluble explicitly, there was doubt about whether such cases were typical. One approach to this issue was by the use of perturbation theory, as developed under the leadership of John Archibald Wheeler at Princeton; the other, more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most its conformal structure, since it is the latter — as determined by the lay of the lightcones — that determines the trajectories of lightlike geodesics, hence their causal relationships.
The importance of Penrose's epoch-making paper "Gravitational collapse and space-time singularities" was not only its result. It showed a way to obtain general conclusions in other contexts, notably that of the cosmological Big Bang, which he dealt with in collaboration with Dennis Sciama's most famous student, Stephen Hawking, it was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture, to the effect that any ensuing singularities would be confined within a well-behaved event horizon surrounding a hidden space-time region for which Wheeler coined the term black hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as the Penrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such as quasars. Following up his "weak cosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis".
Together with the BKL conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. From 1979 dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of the second law of thermodynamics. Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation; this effect has come to be called Penrose -- Terrell rotation. In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature. Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on
Waterfall (M. C. Escher)
Waterfall is a lithograph by the Dutch artist M. C. Escher, first printed in October 1961, it shows a perpetual motion machine where water from the base of a waterfall appears to run downhill along the water path before reaching the top of the waterfall. While most two-dimensional artists use relative proportions to create an illusion of depth, Escher here and elsewhere uses conflicting proportions to create a visual paradox; the watercourse supplying the waterfall has the structure of two Penrose triangles. A Penrose triangle is an impossible object designed by Oscar Reutersvärd in 1934, found independently by Roger Penrose in 1958; the image depicts a watermill with an elevated waterwheel as the main feature. The aqueduct flows behind it; the walls of the aqueduct step downward. The aqueduct turns three times, first to the left to the right, to the left again; the viewer looks down at the scene diagonally, which means that from the viewer's perspective the aqueduct appears to be slanted upward.
The viewer is looking across the scene diagonally from the lower right, which means that from the viewer's perspective the two left-hand turns are directly in line with each other, while the waterwheel, the forward turn and the end of the aqueduct are all in line. The second left-hand turn is supported by pillars from the first, while the other two corners are supported by a tower of pillars that begins at the waterwheel; the water falls off the edge of the aqueduct and over the waterwheel in an impossible infinite cycle. The use of the Penrose stairs is paralleled by Escher's Ascending and Descending, where instead of the flow of water, two lines of monks endlessly march uphill or downhill around the four flights of stairs; the two support towers continue above the aqueduct and are topped by two compound polyhedra, revealing Escher's interest in mathematics as an artist. The one on the left is a compound of three cubes; the one on the right is known as Escher's solid. Below the mill is a garden of giant plants.
This is a magnified view of a cluster of moss and lichen that Escher drew in ink as a study in 1942. The background seems to be a climbing expanse of terraced farmland. Escher's Solid—from Wolfram MathWorld Escher's Solid Includes a great deal of metric data The Polyhedra of M. C. Escher from George W. Hart
M. C. Escher
Maurits Cornelis Escher was a Dutch graphic artist who made mathematically-inspired woodcuts and mezzotints. Despite wide popular interest, Escher was for long somewhat neglected in the art world in his native Netherlands, he was 70. In the twenty-first century, he became more appreciated, with exhibitions across the world, his work features mathematical objects and operations including impossible objects, explorations of infinity, symmetry, perspective and stellated polyhedra, hyperbolic geometry, tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, conducted his own research into tessellation. Early in his career, he drew inspiration from nature, making studies of insects and plants such as lichens, all of which he used as details in his artworks, he traveled in Italy and Spain, sketching buildings, townscapes and the tilings of the Alhambra and the Mezquita of Cordoba, became more interested in their mathematical structure.
Escher's art became well known among scientists and mathematicians, in popular culture after it was featured by Martin Gardner in his April 1966 Mathematical Games column in Scientific American. Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums, he was one of the major inspirations of Douglas Hofstadter's Pulitzer Prize-winning 1979 book Gödel, Bach. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today, he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven. Although he excelled at drawing, his grades were poor, he took piano lessons until he was thirteen years old. In 1918, he went to the Technical College of Delft.
From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and the art of making woodcuts. He studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra and Ravello. In the same year, he traveled through Spain, visiting Madrid and Granada, he was impressed by the Italian countryside and, in Granada, by the Moorish architecture of the fourteenth-century Alhambra. The intricate decorative designs of the Alhambra, based on geometrical symmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of tessellation and became a powerful influence on his work. Escher returned to Italy and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924.
The couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta had two more sons – Arthur and Jan, he travelled visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the Amalfi coast in 1931 and 1934, Gargano and Sicily in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns, it was here that he became fascinated, to the point of obsession, with tessellation, explaining: It remains an absorbing activity, a real mania to which I have become addicted, from which I sometimes find it hard to tear myself away. The sketches he made in the Alhambra formed a major source for his work from that time on, he studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys.
His art correspondingly changed from being observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same his early work shows his interest in the nature of space, the unusual and multiple points of view. In 1935, the political climate in Italy became unacceptable to Escher, he had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d'Œx, where they remained for two years; the Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" in 1935, again in 1949 he designed Netherlands stamps. These were for the 75th anniversary of the Universal Postal Union. Escher, who had
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, or logic puzzles. Puzzles are created to be a form of entertainment but they can arise from serious mathematical or logistical problems. In such cases, their solution may be a significant contribution to mathematical research; the 1989 edition of the Oxford English Dictionary dates the word puzzle to the end of the 16th century. Its first documented use was in a book titled The Voyage of Robert Dudley...to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, by Abram Kendall, master; the word came to be used as a noun. The word puzzle comes from pusle, meaning "bewilder, confound", a frequentive of the obsolete verb pose in the sense of "perplex".
The use of the word to mean "a toy contrived to test one's ingenuity" is recent. Puzzles can be divided into categories. For example, a maze is a type of tour puzzle; some other categories are construction puzzles, stick puzzles, tiling puzzles, disentanglement puzzles, lock puzzles, folding puzzles, combination puzzles, mechanical puzzles. A chess problem is a puzzle. Examples are the eight queens puzzle. Jigsaw puzzles. Lateral thinking puzzles called "situation puzzles" Mathematical puzzles include the missing square puzzle and many impossible puzzles — puzzles which have no solution, such as the Seven Bridges of Königsberg, the three cups problem, three utilities problem Mechanical puzzles such as the Rubik's Cube and Soma cube Metapuzzles are puzzles which unite elements of other puzzles. Paper-and-pencil puzzles such as Uncle Art's Funland, connect the dots, nonograms Also the logic puzzles published by Nikoli: Sudoku, Kakuro, Hashiwokakero, Hitori, Light Up, Number Link, Ripple Effect and Kuromasu.
Peg solitaire. Rubik's Cube and other combination puzzles can be stimulating toys for children or recreational activities for adults. Sangaku Sliding puzzles such as the 15 Puzzle. Puzz-3D is a three-dimensional variant of this type. Sokoban Spot the difference Tangram Word puzzles, including anagrams, crossword puzzles and word search puzzles. Tabletop and digital word puzzles include Bananagrams, Bonza, Letterpress, Puzzlage, Ruzzle, Upwords, WordSpot, Words with Friends. Wheel of Fortune is a game show centered on a word puzzle. Solutions of puzzles require the recognition of patterns and the adherence to a particular kind of ordering. People with a high level of inductive reasoning aptitude may be better at solving such puzzles than others, but puzzles based upon inquiry and discovery may be solved more by those with good deduction skills. Deductive reasoning improves with practice. Mathematical puzzles involves BODMAS. BODMAS is an acronym and it stands for Bracket, Of, Multiplication and Subtraction.
In certain regions, PEDMAS is the synonym of BODMAS. It explains the order of operations to solve an expression; some mathematical puzzle requires Top to Bottom convention to avoid the ambiguity in the order of operations. It is an elegantly simple idea that relies, as sudoku does, on the requirement that numbers appear only once starting from top to bottom as coming along. Puzzle makers are people; some notable creators of puzzles are: Ernő Rubik Sam Loyd Henry Dudeney Boris Kordemsky David J. Bodycombe Will Shortz Lloyd King Martin Gardner Raymond Smullyan Jigsaw puzzles are the most popular form of puzzle. Jigsaw puzzles were invented around 1760, when John Spilsbury, a British engraver and cartographer, mounted a map on a sheet of wood, which he sawed around the outline of each individual country on the map, he used the resulting pieces as an aid for the teaching of geography. After becoming popular among the public, this kind of teaching aid remained the primary use of jigsaw puzzles until about 1820.
The largest puzzle is made by German game company Ravensburger. The smallest puzzle made was created at LaserZentrum Hannover, it is the size of a sand grain. By the early 20th century and newspapers had found that they could increase their readership by publishing puzzle contests, beginning with crosswords and in modern days sudoku. There are organizations and events that cater to puzzle enthusiasts, such as: Nob Yoshigahara Puzzle Design Competition World Puzzle Championship National Puzzlers' League Puzzlehunts such as the Maze of Games List of impossible puzzles List of Nikoli puzzle types Riddle Puzzles at DMOZ
Lionel Sharples Penrose, FRS was a British psychiatrist, medical geneticist, paediatrician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penrose was the Galton professor of eugenics at University College London, emeritus professor, he was cited by professor Bryan Sykes in Adam's Curse: A Future Without Men Penrose was educated at the Downs School and the Quaker Leighton Park School, St John's College, Cambridge On leaving school in 1916, he served, as a conscientious objector, with the Friends' Ambulance Unit/British Red Cross in France until the end of the First World War. He went on to study at Cambridge. At Cambridge he gained a first class degree in moral sciences before leaving for Vienna for a year, to study at the psychological department at the University of Vienna. In 1928 qualified with the conjoint in 1928 at St Thomas' Hospital before qualifying for a Doctor of Medicine in 1930. Penrose undertook research into schizophrenia, designing tests of intelligence that were non-verbal in nature, that are still in current use, was one of the earliest researcher on the phenylketonuria condition in the 1930s.
Penrose's "Colchester Survey", produced as the report in 1938, in collaboration with the MRC called th MRC special report: No.229, Clinical and genetic study of 1,280 cases of mental defect, was the earliest serious attempt to study the genetics of mental retardation. He found that the relatives of patients with severe mental retardation were unaffected but some of them were affected with similar severity to the original patient, whereas the relatives of patients with mild mental retardation tended to have mild or borderline disability. Penrose went on to identify and study many of the genetic and chromosomal causes of mental retardation; this body of work culminated in The Biology of Mental Defect. Penrose was a central figure in British medical genetics following World War II. From 1945 to 1965 he occupied the Galton Chair at the Galton Laboratory at University College London, he received a number of awards and honours including the 1960 Albert Lasker Award for Basic Medical Research. The Lasker citation read: "Professor Penrose and his associates have been responsible over the years for studies which touch all aspects of human genetics, include genetic analyses of most of the known hereditary diseases, contributions to mathematical genetics, biochemical genetics, the study of gene linkage in man, theoretical work on the mutagenic effect of ionizing radiations.
Most their attention has been turned to abnormalities of human chromosomes associated with congenital defects mongolism." Penrose's Law states that the population size of prisons and psychiatric hospitals are inversely related, although this is viewed as something of an oversimplification. Penrose, a member of the Society of Friends, was a lead figure in the Medical Association for the Prevention of War in the 1950s. Penrose developed the Penrose method, a method for apportioning seats in a global assembly based on the square root of each nation's population; such a voting system is based on the voting power of any voter decreasing with the size of the voting body as one over its square root. See Penrose square root law. Penrose was interested in different facets of biology, for example fingerprint and cytogenetics, which were a result of his research into the problems of mental defect Down syndrome, he did intensive research on the latter, communicating the results of his investigations in 1963 and winning the Joseph P. Kennedy Jr. Foundation Award for his contributions to the understanding of the causes of mental retardation.
Penrose married Margaret Leathes in 1928 and they had four children: Oliver Penrose, born 1929, physicist. After Penrose's death, Margaret married the mathematician Max Newman, she died in 1989. Penrose' father was James Doyle Penrose, his mother was Elisabeth Josephine Penrose and his brother was Sir Roland Penrose, both British artists
An impossible trident known as an impossible fork, a blivet, devil's tuning fork, etc. is a drawing of an impossible object, a kind of an optical illusion. It appears to have three cylindrical prongs at one end which mysteriously transform into two rectangular prongs at the other end. In 1964 D. H. Schuster reported that he noticed an ambiguous figure of a new kind in the advertising section of an aviation journal, he dubbed it a "three-stick clevis". He described the novelty as follows: "Unlike other ambiguous drawings, an actual shift in visual fixation is involved in its perception and resolution." The word "poiuyt" appeared on the March 1965 cover of Mad magazine bearing the four-eyed Alfred E. Neuman balancing the impossible fork on his finger with caption "Introducing'The Mad Poiuyt' ". An anonymously-contributed version described as a "hole location gauge" was printed in the June 1964 issue of Analog Science Fiction and Fact, with the comment that "this outrageous piece of draftsmanship evidently escaped from the Finagle & Diddle Engineering Works".
The term "blivet" for the impossible fork was popularized by Worm Runner's Digest magazine. In 1967 Harold Baldwin published there an article, "Building better blivets", in which he described the rules for the construction of drawings based on the impossible fork. In December 1968 American optical designer and artist Roger Hayward wrote a humorous submission "Blivets: Research and Development" for The Worm Runner's Digest in which he presented various drawings based on the blivet, he "explained" the term as follows: "The blivet was first discovered in 1892 in Pfulingen, Germany, by a cross-eyed dwarf named Erasmus Wolfgang Blivet." He published there a sequel, Blivets — the Makings