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Topic: Circumscribed sphere

Related Topics

Inscribed sphere

Sphere

Platonic solid

Polyhedron

Dihedral angle

Tetrahedron

Octahedron

Dodecahedron

Polyhedral dice

	Sphere
		More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere.
		For this reason, the sphere appears in nature: for instance bubbles and small water drops are spheres, because the surface tension tries to minimize surface area.
		The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere.
	www.brainyencyclopedia.com /encyclopedia/s/sp/sphere.html (617 words)

	Circumscribed sphere - Wikipedia, the free encyclopedia
		A circumscribed sphere is a sphere that encloses another solid object, such as a polyhedron.
		Any tetrahedron may have such a sphere drawn through its vertices.
		It is the three dimensional version of a Circumscribed circle.
	en.wikipedia.org /wiki/Circumscribed_sphere (62 words)

	Platonic solid - Wikipedia, the free encyclopedia
		The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere.
		The dodecahedron is also most like the sphere in the sense that it has the smallest central angle (ratio of the strut length to the radius of the circumscribed sphere), and the greatest surface area.
		In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron.
	en.wikipedia.org /wiki/Platonic_solid (1147 words)

	Tetrahedron
		A sphere of radius X is inscribed in a regular tetrahedron of arbitrary side length, and a sphere of radius Y is circumscribed about the same tetrahedron.
		Thus the ratio of the circumscribed sphere (the distance from the centroid to the apex) is three times the ratio of the inscribed sphere (the distance from the centroid to the base).
		The ratio of the circumscribed sphere to the inscribed sphere in a tetrahedron is 3:1, because the centroid (which is also the incenter and circumcenter) is one quarter of the way up the height, measured from the base to the apex.
	mcraeclan.com /MathHelp/GeometrySolidTetrahedron.htm (1065 words)

	Surface Projection (Site not responding. Last check: 2007-09-11)
		By a process of gnomonic (central) projection, the edges of the polyhedron generate a set of arcs of whose centre points are the centre of the polyhedron and thus the centre of the circumscribing sphere.
		Note that the sphere is tangential to the faces of the tetrahedron.
		The above image shows half of the circumscribed sphere of a cube in which the eight vertices of the cube touch the inside surface of the sphere.
	www.ul.ie /~cahird/polyhedronmode/surface.htm (602 words)

	archimedes3 (Site not responding. Last check: 2007-09-11)
		Archimedes determined the ratio of the volume of a sphere to the volume of the circumscribed cylinder.
		Archimedes shows by elementary geometry that if the two smaller circles are slid along the axis to a point at twice height of the cylinder, and the large circle is left where it is, their areas (thought of as masses) will exactly balance about a fulcrum (balance-point) at the center of the figure.
		If we imagine doing this for all the slices together, we will have balanced, on the right, the entire sphere and the entire cone, their masses concentrated at the end-point of the axis, and on the left the cylinder in its original position, since none of its slices had to be moved.
	80-www.ams.org.library.uor.edu /featurecolumn/archive/archimedes3.html (332 words)

	Somsiah's Four Tennis Ball Theorems (Site not responding. Last check: 2007-09-11)
		The surface area of a circumscribed right cylinder, including the bases, is equal to the surface area of two inscribed hemispheres, including the bases.
		The surface area of a circumscribed right cylinder, including the bases, is equal to the surface area of two inscribed hemispheres plus two included spheres of equal radius including the bases.
		The surface area of a circumscribed right cylinder, including the bases, is equal to the surface area of two inscribed hemispheres, plus any number of inscribed central spheres, including the bases.
	www.mcn.org /2/sesame/hour.html (534 words)

	Architecture of Life (Site not responding. Last check: 2007-09-11)
		By using triangulation to measure the arcs between two points on the earth based on their respective relationships to the sun's zenith point at high noon, Eratosthenes, his contemporary and friend was the first to prove the curve of the earth's surface.
		Kepler's writings about the planetary orbits described their paths in relationship to each other as the "harmony of the spheres." He was sure of the archetypal significance of the five regular polyhedra and applied them to the macro world of the heavens by introducing a celestial "nested" geometry.
		Kepler's studies of sphere packing were also integral to his understanding and articulation of periodicity, the regularly repeating motions of the planetary orbits on the macro scale and polyhedral shapes which recur in a three-dimensional lattice on the medio scale.
	euch3i.chem.emory.edu /proposal/www.cruzio.com/~devarco/life.htm (3313 words)

	Polyhedra - Fleurent G.M. - F and S Solids and their Compounds : Introduction (Site not responding. Last check: 2007-09-11)
		All vertices for both regular and semiregular solids lie on the circumscribed sphere of the solid.
		The dual of an Archimedean solid can also be derived from the planes tangent to the circumscribed sphere at points which coincide with the vertices of the Archimedean solid.
		The Schläfli symbol for Platonic solids is a useful description of the solid: m is the number of sides of the regular polygon the solid has for faces, n is the number of polygons surrounding a vertex.
	users.skynet.be /polyhedra.fleurent/Polyhedra/Text/03_Intro.htm (323 words)

	The Four Tennis Ball Theorems (Site not responding. Last check: 2007-09-11)
		Given: Construct a sphere of radius r, and a circumscribed right cylinder of diameter 2r and a height also 2r.
		Proof: The combined surface area of the two hemispheres is equal to the surface area of the sphere by definition.
		When a sphere elongates until the distance between the two new centers is equal to twice its radius [2R], then it may undergo a div1 operation with no further increase in surface area
	www.mcn.org /2/sesame/theorems.html (258 words)

	JOMAGEOLAB.htm
		Remember that the circumscribed circle of a triangle is the circle that goes through the 3 vertices of the triangle.
		Write a careful proof that the perpendicular bisectors of the sides of a triangle intersect in a point that is the center of the circumscribed circle in Euclidean geometry.
		Write a careful proof that the angle bisectors of a triangle intersect in a point that is the center of the inscribed circle in Euclidean geometry.
	www.joma.org /images/upload_library/4/vol2/upperlevel/geolab.html (633 words)

	Sculpture Based on Propellorized Polyhedra
		Each edge is cast to the center of the sphere and the bulk of the volume consists of long pyramidal openings that almost meet at the center.
		In a polyhedron midscribed to a sphere, the intersection of each face with the sphere is a circle inscribed in the face.
		That is where it is tangent to the sphere (otherwise, by symmetry, the edge would have two tangent points) hence two of the kite's angles are equal.
	www.georgehart.com /propello/propello.html (3258 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-09-11)
		I have searched on the web for different things but recently I have been looking for ways of calculating all of the lengths of the segments, the angles, etc. I was hoping you could shed some light on the subject.
		Given a regular icosahedron with edge length 1, consider the central projection of its edges and vertices on its circumscribed sphere.
		The radius of the circumscribed sphere is calculated to be R = Sqrt[(5+Sqrt[5])/8].
	mathforum.org /library/drmath/view/51704.html (574 words)

	The Particle: Standing wave atom model
		The inscribed sphere, will in turn be the circumscribed sphere of a smaller nested platonic structure, and so on, until a point is reached where the actual sides of the platonic equates to the smallest possible vibrating length in space, relating to planck length.
		Unlike the conventional model, where the space between electron shells is described as a void and empty space, in our model it is the space in between the inscribed and circumscribed spheres, which contain the inward and outward going spherical waves forming up the 3D standing wave shape.
		The TETRAHEDRAL SPHERE is the central projection of the tetrahedron onto the surface of the unit sphere.
	www.blazelabs.com /f-p-swave.asp (2951 words)

	Homework 6
		Proposition 3: Given two unequal magnitudes and a circle, it is possible to inscribe a polygon in the circle and to describe another about it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than that of the greater magnitude to the less.
		Proposition 23: The surface of the sphere is greater than the surface descibed by the revolution of the polygon inscribed in the great circle about the diameter of the great circle.
		Proposition 28: The surface of the figure circumscribed to the given sphere is greater than that of the sphere itself.
	nsm1.nsm.iup.edu /gsstoudt/history/ma350/hw6.html (650 words)

	The Particle: Platonic solids
		Leonhard Euler (1707-1783) who was a Swiss mathematician, noticed that no matter how one cuts a sphere into polygons, sometimes called a triangulation, there is a quantity which remains constant; in other words, there is a number related to the sphere independent of the triangulation.
		Each of the platonic solids is in fact a triangulation of the sphere into polygons.The Euler characteristic is given by F-E+V, where F is the number of polygonal faces, E is the number of edges, and V is the number of vertices in the triangulation.
		From the above calculations, it is shown that the ratios between both radius and volume of any circumcribed sphere to its inscribed sphere is a constant, not only for the case tetra-tetra, but also to the other two dual platonics, even if the platonic shape of the duals is not the same.
	www.blazelabs.com /f-p-solids.asp (2459 words)

	POLYHEDRA
		The circle and the sphere were considered to be the most harmonic figures; in ancient Greece but also even until the Renaissance, eg by Johannes Kepler.
		He also proved the same ratio of the surfaces of the sphere and the cylinder.Archimedes requested his friends that they would place over his tomb a cylinder containing a sphere (Plutarch AD 45-120).
		There are 13 spheres in an icosahedron and 12 spheres in a cuboctahedron.
	home.swipnet.se /polygon/hemsida.html (1445 words)

	The Particle (Site not responding. Last check: 2007-09-11)
		Other important elements are the octahedron (formed by six closest-packed spheres) and the vector equilibrium, which is the result of twelve spheres nested around a thirteenth, central sphere, in omnidirectional closest-packing, 60 degree co-ordinated configuration.
		This does not apply to the outer volume of space in between the inscribed and circumscribed spheres, where such volume has to be spun by the fast moving edges of the outer platonic.
		Although, the two spheres seem to occupy the same spherical volume, they are not, and the 'mirror' at the centre is neither part of the real world nor part of the virtual world, it has 3 dimensions of its own.
	bel.150m.com /particle.htm (14324 words)

	convex polyhedra 2 (Site not responding. Last check: 2007-09-11)
		The faces of these semi-regular polyhedra are regular polygons (of two or three types) ; their vertices are superimposable (but not regular) and lay on a sphere (circumscribed sphere); their edges have same length.
		A curiosity (Pugh - 1976): the Archimedean solids can all be circumscribed by a regular tetrahedron so that four of their faces lie on the faces of that tetrahedron.
		The hight of the pentagonal antiprisme is equal to the radius of the circle circumscribed to the pentagon.
	www.ac-noumea.nc /maths/amc/polyhedr/convex2_.htm (444 words)

	tetrahedra (Site not responding. Last check: 2007-09-11)
		Their volumes are then respectively the third and the sixth of the cube one's; with the same equilateral base, the second of this two regular pyramids is half as high as the first.
		are the radii of four spheres with centres the vertices and mutually tangent.
		The tangent points of these spheres are also the tangent points of the edges with the mid-sphere.
	www.ac-noumea.nc /maths/amc/polyhedr/tetra_.htm (1700 words)

	Answer to Problem of the Week for 2/23/2004
		A cube is inscribed in a sphere (or, in other words, the sphere is circumscribed about the cube).
		This means that the sphere passes through the eight vertices of the cube.
		If the surface area of the cube is 60 square inches, determine the surface area of the sphere.
	www.pen.k12.va.us /Div/Winchester/jhhs/math/probweek/p2004/a022304.html (57 words)

	Curves in 3D
		The circumscribed sphere above is an illustration of
		This sphere is tangent to the plane containing E at one of its foci, and tangent to the right cone along a circle.
		Then, the vertices of right cones which are circumscribed about a paraboloid lie on such focal parabolas.
	www.lems.brown.edu /vision/people/leymarie/Notes/CurvSurf/Curves.html (475 words)

	Date 07-03Ä94 (1921) FUFO To ALL Fr DON ALLEN Read NO Subj MESSAGE OF CYDONIA The +quot;Me
		The terminus of this wedge, together with the NW corner of the pyramid, are the only two points on the pyramid that, when connected, denote a line of latitude (see Fig.
		Again, putting this in simple terms: The geometry of a circumscribed tetrahedron is not only suggested by the alignments in Cydonia, but also by the siting latitude, size, shape, and orientation of the D&M Pyramid itself.
		The "Message of Cydonia" Verification of a highly-specific and redundant communication of "circumscribed tetrahedral geometry" -- including its obviously deliberate extension to the siting of the Cydonia Complex on the planet -- would be deemed a phenomenonal discovery.
	www.skepticfiles.org /mys2/hoaglnd1.htm (4364 words)

	New exercises and problems in Mathematics - April 2004 (Site not responding. Last check: 2007-09-11)
		Given the lengths of two sides of a triangle, and given that the corresponding medians are perpendicular, calculate the length of the third side.
		The surface area of the circumscribed sphere of a cube K
		denote the volume of the circumscribed sphere of the cube K
	www.komal.hu /verseny/2004-04/mat.e.shtml (609 words)

	Image Patterner - Sphere slicer (Site not responding. Last check: 2007-09-11)
		The output sphere is circumscribed outside the ideal-sphere, rather than inscribed within.
		This is not how I would normally pattern a sphere, where I would allow the seams to follow great-circles on the surface.
		In this case the seams are outside the sphere and the centre-lines of the panels follow great-circles.
	bruno.postle.net /neatstuff/image_patterner/ip-slicer (902 words)

	Platonic Solids
		The inradius is the radius of the sphere inscribed in a given polyhedron.
		The circumradius is the radius of the circumscribed sphere.
		All Platonic solids have an inscribed sphere tangent to every face and a circumscribed sphere through every vertex.
	whistleralley.com /polyhedra/platonic.htm (381 words)

	[No title] (Site not responding. Last check: 2007-09-11)
		In case of a hexahedron (a box), this attribute represents the radius of the inscribed sphere.
		For the other polyhedra, it is the radius of the circumscribed sphere.
		Where R is the radius of the outer spherem r the radius of the inner sphere, A the surface area and V the volume:
	www.sciface.com /STATIC/DOC30/eng/plot_Polyhedron.html (630 words)

	[No title]
		It is not uncommon for Gregory to use insights centered around these two key words when discussing the most fundamental aspect of human experience, the vast yet ultimately circumscribed sphere of knowledge.
		At the same time, realization that we are inherently circumscribed can be an opportunity of opening ourselves to the possibility that something may lay outside what we know and experience.
		Neither is it limited (peras), nor can it be circumscribed (perigraphesthai) in its growth towards the good; however, its present state of goodness, even if especially great and perfect, is only the beginning (archen) of a more transcendent (huperkeimenou), better stage.
	www.bhsu.edu /artssciences/asfaculty/dsalomon/nyssa/boundaries.htm (7656 words)

	Tetrahedron
		The centre of a tetrahedron is the intersection of two space heights (1,2,3).
		It is centre of gravity, centre of the sphere through the four corners, and centre of the largest sphere, which still fits inside the tetrahedron (4).
		The number of the spheres in one layer is 1,3,6,10..., generally n(n+1)/2.
	www.mathematische-basteleien.de /tetrahedron.htm (475 words)

	Hyper Spheres Circumscribed and Inscribed
		Each vertex must be at the same distance -- the radius -- from the center R of this sphere.
		Each face must be at the same distance -- the radius -- from the centre R of this sphere.
		The distance between the centers of these two spheres -- the circumscribing and inscribing spheres -- is
	www.rism.com /Trig/hyper_spheres.htm (1492 words)

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