Misplaced Pages

#694305

18-415: The document divided the city of Berlin into nine operational defense zones (A through H, as slices of the outer city of Berlin and Z, its center, corresponding to the government district). It further divided the region into four concentric rings: an outer exclusion zone, extending well past Berlin's city limits; an outer defense zone extending roughly to the city limits; an inner defense zone extending out to

36-424: 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 P is called the circumradius of P , and

54-472: A midsphere , a sphere tangent to all edges of a polyhedron, and an inscribed sphere , a sphere tangent to all faces of a polyhedron. In the regular polyhedra , the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric . When the circumscribed sphere is the set of infinite limiting points of hyperbolic space , a polyhedron that it circumscribes is known as an ideal polyhedron . There are five convex regular polyhedra , known as

72-436: A set of concentric circles by a Möbius transformation . The ripples formed by dropping a small object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the targets used in target archery or similar sports provide another familiar example of concentric circles. Coaxial cable is a type of electrical cable in which the combined neutral and earth core completely surrounds

90-415: A sphere are concentric with each other and with the sphere. By Euler's theorem in geometry on the distance between the circumcenter and incenter of a triangle, two concentric circles (with that distance being zero) are the circumcircle and incircle of a triangle if and only if the radius of one is twice the radius of the other, in which case the triangle is equilateral . The circumcircle and

108-721: A type of mechanic sights commonly found on target rifles. They usually feature a large disk with a small-diameter hole near the shooter's eye, and a front globe sight (a circle contained inside another circle, called tunnel ). When these sights are correctly aligned, the point of impact will be in the middle of the front sight circle. Spheres: Apostol (2013) Regular polygons: Hardy, Godfrey Harold (1908), A Course of Pure Mathematics , The University Press, p. 107 Regular polyhedra: Gillard, Robert D. (1987), Comprehensive Coordination Chemistry: Theory & background , Pergamon Press, pp.  137, 139 , ISBN   9780080262321 . Circumsphere In geometry ,

126-415: Is a spherical shell . For a given point c in the plane, the set of all circles having c as their center forms a pencil of circles . Each two circles in the pencil are concentric, and have different radii. Every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into

144-445: Is a stub . You can help Misplaced Pages by expanding it . Concentric In geometry , two or more objects are said to be concentric when they share the same center . Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles , spheres , regular polygons , regular polyhedra , parallelograms, cones, conic sections, and quadrics. Geometric objects are coaxial if they share

162-509: Is the three-dimensional analogue of the circumscribed circle . All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere. The circumscribed sphere (when it exists) is an example of a bounding sphere , a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it in linear time . Other spheres defined for some but not all polyhedra include

180-716: The Berlin Ringbahn ; and the Citadel ( German : Zitadelle ), again, zone Z. In addition to the establishment of defense zones, this document also described the overall mechanism by which Berlin would be converted to a front line city. This included: The document also outlined the destruction of thousands of documents that were deemed "essential" to the Nazi war machine, this including documents pertaining to military and civilian logistics and installations, medical research, and other technological research. Adolf Hitler ordered

198-505: The Euclidean plane , two circles that are concentric necessarily have different radii from each other. However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevertheless be different circles. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth (approximated as a sphere). More generally, every two great circles on

SECTION 10

#1732791672695

216-410: The center point of this sphere is called the circumcenter of P . 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,

234-538: The execution of Fall Clausewitz on 20 April 1945. This set into motion preparations according to the Basic Order plan, and would have been followed later by the code word Kolberg , meaning full preparations should have been completed and the battle would have started. This article about a battle in German history is a stub . You can help Misplaced Pages by expanding it . This article related to Nazi Germany

252-402: The existence of a circumscribed sphere is sufficient, but it is not true: some bipyramids , for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a simple polyhedron has a circumscribed circle for each of its faces, it also has a circumscribed sphere. The circumscribed sphere

270-416: The incircle of a regular n -gon , and the regular n -gon itself, are concentric. For the circumradius-to-inradius ratio for various n , see Bicentric polygon#Regular polygons . The same can be said of a regular polyhedron 's insphere , midsphere and circumsphere . The region of the plane between two concentric circles is an annulus , and analogously the region of space between two concentric spheres

288-405: The live core(s) in system of concentric cylindrical shells. Johannes Kepler 's Mysterium Cosmographicum envisioned a cosmological system formed by concentric regular polyhedra and spheres. Concentric circles have been used on firearms surfaces as means of holding lubrication or reducing friction on components, similar to jewelling . Concentric circles are also found in diopter sights ,

306-404: The same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders , conic sections, and surfaces of revolution. Concentric objects are often part of the broad category of whorled patterns , which also includes spirals (a curve which emanates from a point, moving farther away as it revolves around the point). In

324-431: 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 solidorum elementis (circa 1630), René Descartes observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for

#694305