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Compound Of Three Octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ''Stars''. Construction A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1: . The remaining octahedron edges cross each other in pairs, within the interior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Three Octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ''Stars''. Construction A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1: . The remaining octahedron edges cross each other in pairs, within the interior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stellated Rhombic Dodecahedron
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces. Escher's solid can tessellate space to form the stellated rhombic dodecahedral honeycomb. Stellation, solid, and compound The first stellation of the rhombic dodecahedron has 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron, meaning that each of its faces lies in the same plane as one of the rhombus faces of the rhombic dodecahedron, with each face containing the rhombus in the same plane, and that it has the same symmetries as the rhombic dodecahedron. It is the first stellation, meaning that no other self-intersecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George W
George Walker Bush (born July 6, 1946) is an American politician who served as the 43rd president of the United States from 2001 to 2009. A member of the Republican Party, Bush family, and son of the 41st president George H. W. Bush, he previously served as the 46th governor of Texas from 1995 to 2000. While in his twenties, Bush flew warplanes in the Texas Air National Guard. After graduating from Harvard Business School in 1975, he worked in the oil industry. In 1978, Bush unsuccessfully ran for the House of Representatives. He later co-owned the Texas Rangers of Major League Baseball before he was elected governor of Texas in 1994. As governor, Bush successfully sponsored legislation for tort reform, increased education funding, set higher standards for schools, and reformed the criminal justice system. He also helped make Texas the leading producer of wind powered electricity in the nation. In the 2000 presidential election, Bush defeated Democratic incum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chameleon
Chameleons or chamaeleons (family Chamaeleonidae) are a distinctive and highly specialized clade of Old World lizards with 202 species described as of June 2015. The members of this family are best known for their distinct range of colors, being capable of shifting to different hues and degrees of brightness. The large number of species in the family exhibit considerable variability in their capacity to change color. For some, it is more of a shift of brightness (shades of brown); for others, a plethora of color-combinations (reds, yellows, greens, blues) can be seen. Chameleons are distinguished by their zygodactylous feet, their prehensile tail, their laterally compressed bodies, their head casques, their projectile tongues, their swaying gait, and crests or horns on their brow and snout. Chameleons' eyes are independently mobile, and because of this there are two separate, individual images that the brain is analyzing of the chameleon’s environment. When hunting prey, they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Brückner
Johannes Max Brückner (5 August 1860 – 1 November 1934) was a German geometer, known for his collection of polyhedral models. Education and career Brückner was born in Hartau, in the Kingdom of Saxony, a town that is now part of Zittau, Germany. He completed a Ph.D. at Leipzig University in 1886, supervised by Felix Klein and Wilhelm Scheibner, with a dissertation concerning conformal maps. After teaching at a grammar school in Zwickau, he moved to the gymnasium in Bautzen. Brückner is known for making many geometric models, particularly of stellated and uniform polyhedra, which he documented in his book ''Vielecke und Vielflache: Theorie und Geschichte'' (''Polygons and polyhedra: Theory and History'', Leipzig: B. G. Teubner, 1900). The shapes first studied in this book include the final stellation of the icosahedron and the compound of three octahedra, made famous by M. C. Escher's print ''Stars''. Joseph Malkevitch lists the publication of this book, which documen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piero Della Francesca
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca is chiefly appreciated for his art. His painting is characterized by its serene humanism, its use of geometric forms and perspective. His most famous work is the cycle of frescoes ''The History of the True Cross'' in the church of San Francesco in the Tuscan town of Arezzo. Biography Early years Piero was born Piero di Benedetto in the town of Borgo Santo Sepolcro, modern-day Tuscany, to Benedetto de' Franceschi, a tradesman, and Romana di Perino da Monterchi, members of the Florentine and Tuscan Franceschi noble family. His father died before his birth, and he was called Piero della Francesca after his mother, who was referred to as "la Francesca" due to her marriage into the Franceschi family (similar to how Lisa Gherardini became kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Quinque Corporibus Regularibus
''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, in the Latin language; its title means '' he little bookon the five regular solids''. It is one of three books known to have been written by della Francesca. Along with the Platonic solids, ''De quinque corporibus regularibus'' includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including Luca Pacioli's 1509 ''Divina proportione'' (which incorporated without credit an Italian translation of della Francesca's work). Lost for many years, ''De quinque corporibus regularibus'' was rediscovered i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piero Della Francesca - Libellus De Quinque Corporibus Regularibus - P41b (cropped)
Piero is an Italian given name. Notable people with the name include: *Piero Angela (1928–2022), Italian television host *Piero Barucci (born 1933), Italian academic and politician *Piero del Pollaiuolo (c. 1443–1496), Italian painter *Piero della Francesca (c1415–1492), Italian artist of the Early Renaissance * Piero De Benedictis (born 1945), Italian-born Argentine and Colombian folk singer *Piero Ciampi (1934–1980), Italian singer *Piero di Cosimo (1462-1522), also known as Piero di Lorenzo, Italian Renaissance painter *Piero di Cosimo de' Medici (1416–1469), ''de facto'' ruler of Florence from 1464 to 1469 *Piero Ferrari (born 1945), Italian businessman *Piero Focaccia (born 1944), Italian pop singer *Piero Fornasetti (1913–1988), Italian painter *Piero Gardoni (1934–1994), Italian professional footballer *Piero Golia (born 1974), Italian conceptual artist *Piero Gros (born 1954), Italian alpine skier *Piero the Unfortunate (1472–1503), Gran maestro of Florence *P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stella Octangula
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's ''De Divina Proportione,'' 1509. It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2. It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
