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Rhombic Hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically identical to the convex deltoidal hexecontahedron which has kite faces. Dissection The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length ''a'' to be V = (10 + 2\sqrt 5)a^3 and the area to be A = (24\sqrt 5)a^2. : Construction A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of ( ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically identical to the convex deltoidal hexecontahedron which has kite faces. Dissection The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length ''a'' to be V = (10 + 2\sqrt 5)a^3 and the area to be A = (24\sqrt 5)a^2. : Construction A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of ( ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Construction Of Rhombic Hexecontahedron From Rhombic Triacontahedron
Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and comes from Latin ''constructio'' (from ''com-'' "together" and ''struere'' "to pile up") and Old French ''construction''. To construct is the verb: the act of building, and the noun is construction: how something is built, the nature of its structure. In its most widely used context, construction covers the processes involved in delivering buildings, infrastructure, industrial facilities and associated activities through to the end of their life. It typically starts with planning, financing, and design, and continues until the asset is built and ready for use; construction also covers repairs and maintenance work, any works to expand, extend and improve the asset, and its eventual demolition, dismantling or decommissioning. The construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem< ...
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Instituto Nacional De Matemática Pura E Aplicada
The Instituto Nacional de Matemática Pura e Aplicada (IMPA; en, National Institute for Pure and Applied Mathematics) is widely considered to be the foremost research and educational institution of Brazil in the area of mathematics. It is located in the city of Rio de Janeiro, and was formerly known simply as ''Instituto de Matemática Pura e Aplicada'', hence its official abbreviation. It is a research and education institution qualified as a Social Organization (SO) under the auspices of the Ministry of Science, Technology, Innovations and Communications (MCTIC) and the Ministry of Education (MEC) of Brazil. Currently located in the Jardim Botânico neighborhood (South Zone) of Rio de Janeiro, Brazil, IMPA was founded on October 15, 1952. It was the first research unit of the National Research Council (CNPq), a federal funding agency created a year earlier. Its logo is a stylized Möbius strip, reproducing a large sculpture of a Möbius strip on display within the IMPA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook interface Wolfram Mathematica (called ''Mathematica'' by some of its users) is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WolframAlpha
WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mathematica, a technical computing platform. WolframAlpha gathers data from academic and commercial websites such as the CIA's ''The World Factbook'', the United States Geological Survey, a Cornell University Library publication called ''All About Birds'', ''Chambers Biographical Dictionary'', Dow Jones, the ''Catalogue of Life'', CrunchBase, Best Buy, and the FAA to answer queries. A Spanish version was launched in 2022. Technology Overview Users submit queries and computation requests via a text field. WolframAlpha then computes answers and relevant visualizations from a knowledge base of curated, structured data that come from other sites and books. It is able to respond to particularly phrased natural language fact-based questions. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Rhombic Triacontahedron
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes). It can be constructed from the convex solid by expanding the faces by factor of \varphi^3 \approx 4.236, where \varphi\! is the golden ratio. This solid is to the compound of great icosahedron and great stellated dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs. What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a rhombic hexecontahedron. The rhombs have two angles of \arccos(\frac\sqrt)\approx 63.434\,948\,822\,92^, and two of \arccos(-\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Hexecontahedron Stellation Diagram
Rhombic may refer to: *Rhombus, a quadrilateral whose four sides all have the same length (often called a diamond) *Rhombic antenna, a broadband directional antenna most commonly used on shortwave frequencies * polyhedra formed from rhombuses, such as the rhombic dodecahedron or the rhombic triacontahedron or the rhombic dodecahedral honeycomb or the rhombic icosahedron or the rhombic hexecontahedron or the rhombic enneacontahedron or the trapezo-rhombic dodecahedron * other things that exhibit the shape of a rhombus, such as rhombic tiling, Rhombic Chess, rhombic drive, Rhombic Skaapsteker, rhombic egg eater ''Dasypeltis scabra'', known as the common egg eater, egg-eating snake or rhombic egg eater, is a species of colubrid snake endemic to Africa. Geographic range ''Dasypeltis scabra'' is found in sub-Saharan Africa. It can also be found in Saudi A ..., rhombic night adder, forest rhombic night adder {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
