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Stars (M. C. Escher)
''Stars'' is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space. The compound of three octahedra used for the central cage in ''Stars'' had been studied before in mathematics, and Escher likely learned of it from the book ''Vielecke und Vielflache'' by Max Brückner. Escher used similar compound polyhedral forms in several other works, including ''Crystal'' (1947), ''Study for Stars'' (1948), ''Double Planetoid'' (1949), and ''Waterfall'' (1961). The design for ''Stars'' was likely influenced by Escher's own interest in both geometry and astronomy, by a long history of using geometric forms to model the heavens, and by a drawing style used by Leonardo da Vinci. Commentators have interpreted the cage's compound shape as a reference to double and triple stars in astronomy, or to twinned crystals in crystallography. The image contrasts the celestial order of its polyhedral shapes with the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wood Engraving
Wood engraving is a printmaking technique, in which an artist works an image or ''matrix'' of images into a block of wood. Functionally a variety of woodcut, it uses relief printing, where the artist applies ink to the face of the block and prints using relatively low pressure. By contrast, ordinary engraving, like etching, uses a metal plate for the matrix, and is printed by the intaglio method, where the ink fills the ''valleys'', the removed areas. As a result, wood engravings deteriorate less quickly than copper-plate engravings, and have a distinctive white-on-black character. Thomas Bewick developed the wood engraving technique in Great Britain at the end of the 18th century. His work differed from earlier woodcuts in two key ways. First, rather than using woodcarving tools such as knives, Bewick used an engraver's burin (graver). With this, he could create thin delicate lines, often creating large dark areas in the composition. Second, wood engraving traditionally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constellation
A constellation is an area on the celestial sphere in which a group of visible stars forms Asterism (astronomy), a perceived pattern or outline, typically representing an animal, mythological subject, or inanimate object. The origins of the earliest constellations likely go back to prehistory. People used them to relate stories of their beliefs, experiences, creation myth, creation, or mythology. Different cultures and countries adopted their own constellations, some of which lasted into the early 20th century before today's constellations were internationally recognized. The recognition of constellations has changed significantly over time. Many changed in size or shape. Some became popular, only to drop into obscurity. Some were limited to a single culture or nation. The 48 traditional Western constellations are Greek. They are given in Aratus' work ''Phenomena'' and Ptolemy's ''Almagest'', though their origin probably predates these works by several centuries. Constellation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol . Dimensions If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is :r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is :r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a while the midradius, which touches the middle of each edge, is :r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a These quantities may also be expressed as :r_u = a\, \frac \phi :r_i = a\, \frac :r_m = a\, \frac where ''ϕ'' is the golden rat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timaeus (dialogue)
''Timaeus'' (; grc-gre, Τίμαιος, Timaios, ) is one of Plato's dialogues, mostly in the form of long monologues given by Critias and Timaeus, written 360 BC. The work puts forward reasoning on the possible nature of the physical world and human beings and is followed by the dialogue ''Critias''. Participants in the dialogue include Socrates, Timaeus, Hermocrates, and Critias. Some scholars believe that it is not the Critias of the Thirty Tyrants who appears in this dialogue, but his grandfather, who is also named Critias. It has been suggested from some traditions (Diogenes Laertius (VIII 85) from Hermippus of Smyrna (3rd century BC) and Timon of Phlius ( 320 – 235 BC)) that ''Timaeus'' was influenced by a book about Pythagoras, written by Philolaus, although this assertion is generally considered false. Introduction The dialogue takes place the day after Socrates described his ideal state. In Plato's works, such a discussion occurs in the ''Republic''. Socrates fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution of higher learning on the European continent. Along with his teacher, Socrates, and his student, Aristotle, Plato is a central figure in the history of Ancient Greek philosophy and the Western and Middle Eastern philosophies descended from it. He has also shaped religion and spirituality. The so-called neoplatonism of his interpreter Plotinus greatly influenced both Christianity (through Church Fathers such as Augustine) and Islamic philosophy (through e.g. Al-Farabi). In modern times, Friedrich Nietzsche diagnosed Western culture as growing in the shadow of Plato (famously calling Christianity "Platonism for the masses"), while Alfred North Whitehead famously said: "the safest general characterization of the European philosophical tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Star
A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in which case they are called ''visual binaries''. Many visual binaries have long orbital periods of several centuries or millennia and therefore have orbits which are uncertain or poorly known. They may also be detected by indirect techniques, such as spectroscopy (''spectroscopic binaries'') or astrometry (''astrometric binaries''). If a binary star happens to orbit in a plane along our line of sight, its components will eclipse and transit each other; these pairs are called ''eclipsing binaries'', or, together with other binaries that change brightness as they orbit, ''photometric binaries''. If components in binary star systems are close enough they can gravitationally distort their mutual outer stellar atmospheres. In some cases, thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refracting Telescope
A refracting telescope (also called a refractor) is a type of optical telescope that uses a lens (optics), lens as its objective (optics), objective to form an image (also referred to a dioptrics, dioptric telescope). The refracting telescope design was originally used in spyglasses and astronomy, astronomical telescopes but is also used for long-focus lens, long-focus camera lenses. Although large refracting telescopes were very popular in the second half of the 19th century, for most research purposes, the refracting telescope has been superseded by the reflecting telescope, which allows larger apertures. A refractor's magnification is calculated by dividing the focal length of the objective lens by that of the eyepiece. Refracting telescopes typically have a lens at the front, then a optical train, long tube, then an eyepiece or instrumentation at the rear, where the telescope view comes to focus. Originally, telescopes had an objective of one element, but a century later, tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonardo Polyhedra
Leonardo is a masculine given name, the Italian, Spanish, and Portuguese equivalent of the English, German, and Dutch name, Leonard. People Notable people with the name include: * Leonardo da Vinci (1452–1519), Italian Renaissance scientist, inventor, engineer, sculptor, and painter Artists * Leonardo Schulz Cardoso, Brazilian singer * Emival Eterno da Costa (born 1963), Brazilian singer known as Leonardo * Leonardo de Mango (1843–1930), Italian-born Turkish painter * Leonardo DiCaprio (born 1974), American actor * Leonardo Pieraccioni (born 1965), Italian actor and director Athletes * Leonardo Araújo (born 1969), usually known as Leonardo, Brazilian World Cup-winning footballer, and former sporting director of Paris Saint Germain * Leonardo Fioravanti (born 1997), Italian surfer * Leonardo Lourenço Bastos (born 1975), Brazilian footballer * Leonardo Bittencourt, German footballer * Leonardo Bonucci (born 1987), Italian footballer * Leonardo Candi (born 1997), I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Spiral ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009. # a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. # a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a . The first definition describes a |
Rhombic Dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly times the short face-diagonal length; thus, the acute angles on each face measure arccos(), or approximately 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the ''Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
