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Triangular Truncated Trapezohedron
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces. Geometry This polyhedron can be constructed by truncating two opposite vertices of a cube, of a trigonal trapezohedron (a convex polyhedron with six congruent rhombus sides, formed by stretching or shrinking a cube along one of its long diagonals), or of a rhombohedron or parallelepiped (less symmetric polyhedra that still have the same combinatorial structure as a cube). In the case of a cube, or of a trigonal trapezohedron where the two truncated vertices are the ones on the stretching axes, the resulting shape has three-fold rotational symmetry. Dürer's solid This polyhedron is sometimes called Dürer's solid, from its appearance in Albrecht Dürer's 1514 engraving '' Melencolia I''. The graph formed by its edges and vertices is called the Dürer graph. The shape of the solid depicted by Dürer is a subject of some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Trapezohedron
In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism. The vertices exist as 4 in four parallel planes, with alternating orientation in the middle creating the pentagons. The regular dodecahedron is the most common polyhedron in this class, being a Platonic solid, with 12 congruent pentagonal faces. A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron. Forms * Triangular truncated trapezohedron ( Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid * Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid *''Truncated pentagonal trapezohedron'' or regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chamfered Tetrahedron
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation it is represented by the letter . A polyhedron with edges will have a chamfered form containing new vertices, new edges, and new hexagonal faces. Chamfered Platonic solids In the chapters below the chamfers of the five Platonic solids are described in detail. Each is shown in a version with edges of equal length and in a canonical version where all edges touch the same midsphere. (They only look noticeably different for solids containing triangles.) The shown duals are dual to the canonical versions. Chamfered tetrahedron The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately trunc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picture Plane
In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" (or '' oculus'') and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest. Features In the technique of graphical perspective the picture plane has several features: :Given are an eye point O (from '' oculus''), a horizontal plane of reference called the ''ground plane'' γ and a picture plane π... The line of intersection of π and γ is called the ''ground line'' and denoted ''GR''. ... the orthogonal projection of O upon π is called the ''principal vanishing point P''...The line through ''P'' parallel to the ground line is called the ''horizon'' HZ The horizon frequently features vanishing points of lines appearing parallel in the foreground. Position The orientation of the picture plane is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling The Cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other tools. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, , the cube root of two. This is because a cube of side length 1 has a volume of , and a cube of twice that volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumsphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle''. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron is called the circumradius of , and the center point of this sphere is called the circumcenter of . Existence and optimality When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.. In ''De ... [...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] |
Regular Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification. The word ''crystal'' derives from the Ancient Greek word (), meaning both "ice" and "rock crystal", from (), "icy cold, frost". Examples of large crystals include snowflakes, diamonds, and table salt. Most inorganic solids are not crystals but polycrystals, i.e. many microscopic crystals fused together into a single solid. Polycrystals include most metals, rocks, ceramics, and ice. A third category of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calcite
Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on Scratch hardness, scratch hardness comparison. Large calcite crystals are used in optical equipment, and limestone composed mostly of calcite has numerous uses. Other polymorphs of calcium carbonate are the minerals aragonite and vaterite. Aragonite will change to calcite over timescales of days or less at temperatures exceeding 300 °C, and vaterite is even less stable. Etymology Calcite is derived from the German ''Calcit'', a term from the 19th century that came from the Latin word for Lime (material), lime, ''calx'' (genitive calcis) with the suffix "-ite" used to name minerals. It is thus etymologically related to chalk. When applied by archaeology, archaeologists and stone trade pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolf Von Engelhardt
Wolf Jürgen Baron von Engelhardt (9 February 1910, Tartu – 4 December 2008, Tübingen) was a German geologist and mineralogist. Baron von Engelhardt was a descendant of a Baltic Germans, Baltic German noble family Engelhardt family, Engelhardt. Biography In the years 1929-1935, he began the study of natural sciences, in particular geology, mineralogy and chemistry at the Universities of Halle, Berlin, and Göttingen. In Halle (Saale), Halle, he became a member of the Corps Guestphalia Halle. He received his doctorate on September 18, 1935, from the mineralogist Victor Moritz Goldschmidt with the topic geochemistry of barium. From 1935 to 1938 he worked as a research assistant at the Mineralogical Institute of the University of Rostock. On July 12, 1939, he obtained the habilitation at the University of Göttingen where he studied decay and construction of minerals in northern German Fuller's earth. From 1939 to 1944 he was a scientific assistant at the Mineralogical I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombi
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from grc, ῥόμβος, rhombos, meaning something that spins, which derives from the verb , romanized: , meaning "to turn round and round." The word was used both by Eucli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
