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Topic: Inscribed sphere

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Circumscribed sphere

Sphere

Platonic solid

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Tetrahedron

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	Archimedes on Spheres and Cylinders
		Specifically, he discovered and proved that the surface area of a region of a sphere sliced off by a plane equals the area of a circle whose radius is the straight-line distance from the central point of that region to the perimeter.
		Archimedes' theorem is that the surface area of the region of the sphere below the horizontal plane H is equal to the area of a circle of radius t.
		The incremental surface area of the cylinder slice between the planes A and B is 2pi r dy, and the surface area of the sphere between those planes is 2pi x dc.
	www.mathpages.com /home/kmath343.htm (0 words)

	inscribed - definition by dict.die.net
		Note: A line is inscribed in a circle, or in a sphere, when its two ends are in the circumference of the circle, or in the surface of the sphere.
		A triangle is inscribed in another triangle, when the three angles of the former are severally on the three sides of the latter.
		A sphere is inscribed in a polyhedron, when the sphere touches each boundary plane of the polyhedron.
	dict.die.net /inscribed (202 words)

	Kepler's model of the five platonic solids and the orbits of the planets - Astronomy.com Forums
		The sphere of Mercury is the insphere of an octahedron.
		The sphere of Mars is the circumsphere of this dodecahedron and the insphere of a tetrahedron.
		The sphere of Jupiter is the circumsphere of this tetrahedron and the insphere of a cube.
	www.astronomy.com /ASY/CS/forums/180626/ShowPost.aspx (1388 words)

	PlanetMath: tetrahedron
		Just as a triangle always can be inscribed in a unique circle, so too a tetrahedron can be inscribed in a unique sphere.
		To find the centre of this sphere, we may construct the perpendicular bisectors of the edges of the tetrahedron.
		These six planes will meet in the centre of the sphere which passes through the vertices of the tetrahedron.
	planetmath.org /encyclopedia/Tetrahedron.html (416 words)

	Min-Energy Configurations of Electrons On A Sphere
		The determination of the stable equilibrium configurations of N particles confined to the surface of a sphere and repelling each other by a specified force law (such as an inverse square force) is known as Thomson’s Problem, named after J. Thomson, who studied such configurations in relation to his “plum pudding” model of the atom.
		Therefore, if N positively charged particles are constrained to the surface of a sphere, and N is not equal to the number of vertices of a Platonic solid, then the particles must have an equilibrium configuration that is not perfectly symmetrical.
		N=11: The equilibrium configuration of 11 charged particles on the surface of a sphere has one particle at the north pole (level 0). At a depth of 0.484692 (level 1) down from the north pole are two particles on opposite sides.
	www.mathpages.com /home/kmath005/kmath005.htm (0 words)

	molecular close packing (Site not responding. Last check: )
		The structure of each sphere is along the lines of a icosidodecahedron encased in the space structure of a truncated icosahedron.
		The individual 6 orbits of the inner and outer spheres are not in conflict.
		This double unit Icosahedron would be 1/2 the the size of the one inscribed into a octahedron with an equivalent chord length.
	ourworld.compuserve.com /homepages/robert_conroy/molecula.htm (349 words)

	Reference.com/Encyclopedia/Inscribed sphere
		In geometry, an inscribed sphere of a polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's facets.
		All regular polyhedra have inscribed spheres, but some irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes.
		The radius of sphere inscribed in a polyhedron P is called the inradius of P.
	www.reference.com /browse/wiki/Inscribed_sphere (155 words)

	Kepler, Johannes (1571-1630)
		Born in Weil der Stadt, southwest Germany, Kepler studied at the university of Tübingen and, as a graduate, was tutored by Michael Maestlin who introduced him to the heliocentric concepts of Copernicus.
		Specifically, his theory states that if a sphere is drawn to touch the inside of the path of Saturn, and a cube is inscribed in the sphere, then the sphere inscribed in that cube is the sphere circumscribing the path of Jupiter.
		Then if a regular tetrahedron is drawn in the sphere inscribing the path of Jupiter, the insphere of the tetrahedron is the sphere circumscribing the path of Mars, and so inward, putting the regular dodecahedron between Mars and Earth, the regular icosahedron between Earth and Venus, and the regular octahedron between Venus and Mercury.
	www.daviddarling.info /encyclopedia/K/KeplerJ.html (882 words)

	Geodesic Domes and Charts of the Heavens
		Although the concept of the sky as a sphere may have occurred as early as 2,000 BC in China, it is recorded that in the 6th century BC.
		On the sphere are inscribed not only constellations, but circles representing the elliptic boundaries of the zodiac, and the major parallel circles.
		One such device, the Gottorp Armillary Sphere, built in 1653 by Andreas Busch was a marvel of craftsmanship and art, mechanized to show movement of the sun and with six silver angels representing the known planets.
	telacommunications.com /geodome.htm (1361 words)

	Table of Contents (Site not responding. Last check: )
		to sphere; axis of sphere bisects and is normal to cylinder axis.
		Sphere to scalene triangle in plane perpendicular to sphere axis with one vertex on axis; sphere does not intersect plane of triangle.
		Sphere to interior surface of coaxial right circular cylinder; sphere outside end of cylinder, and sphere radius less than cylinder radius.
	www.me.utexas.edu /~howell/tablecon.html (3457 words)

	Chapter Inrunning <i>to</i> Insecurity of I by Webster's Dictionary (1913 Edition)
		A line is inscribed in a circle, or in a sphere, when its two ends are in the circumference of the circle, or in the surface of the sphere.
		A triangle is inscribed in another triangle, when the three angles of the former are severally on the three sides of the latter.
		A sphere is inscribed in a polyhedron, when the sphere touches each boundary plane of the polyhedron.
	www.bibliomania.com /2/3/257/1200/23038/3.html (304 words)

	HIPASS galaxy catalogue (HICAT) animations
		The celestial sphere is inscribed with a 15° equatorial graticule in grey, with IAU constellation boundaries in magenta labelled with their standard three-letter abbreviation.
		The supergalactic equator is marked on the celestial sphere in yellow with the galactic equator in green.
		The distribution on the observer's celestial sphere is then projected onto the map plane using a gnomonic projection; it can be shown that this sequence of operations is equivalent to computing a zenithal perspective projection of the Earth-bound galaxy distribution with the observer at the point of projection.
	www.atnf.csiro.au /people/mcalabre/animations/index.html (2320 words)

	Archimedes on Spheres and Cylinders
		This establishes the fact that the number pi, which we originally defined as the ratio of the circumference to the diameter of a circle, is also the ratio of the circle's area to the area of a square whose edges equal the circle's radius.
		Specifically, he discovered and proved that the surface area of a region of a sphere sliced off by a plane equals the area of a circle whose radius is the straight-line distance from the central point of that region to the perimeter.
		Archimedes' theorem is that the surface area of the region of the sphere below the horizontal plane H is equal to the area of a circle of radius t.
	mathpages.com /home/kmath343.htm (2740 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.
	mcraefamily.com /MathHelp/GeometrySolidTetrahedron.htm (1055 words)

	Welcome to the Transcendata Europe Website
		The medial surface of a solid is formed by the locus of an inscribed sphere of maximal diameter as it rolls around the interior of the solid.
		The medial axis of a surface is the 2D equivalent of the medial surface and is formed by the locus of an inscribed disc of maximal diameter as it rolls around the interior of the surface.
		The position of the touching points of the inscribed sphere are the closest points on the object boundary to a given point on the medial surface, and these co-ordinates are useful in many applications and can be given in Cartesian form, or even in parameter or element metric spaces as desired.
	www.fegs.co.uk /motech.html (4764 words)

	Inscribed sphere (Site not responding. Last check: )
		In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces.
		It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere.
		For example the regular small stellated dodecahedron has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron.
	en.askmore.net /Inscribed_sphere.htm (297 words)

	Filename: RC2029.TXT Source: +quot;Life between Death + Rebirth+quot; by Rudolf Steiner [P
		He grows out into the spheres, but the spheres of the dead are not separate from each other as in the case of men on earth.
		Something is inscribed by him in all the spheres, in the Mercury sphere, the Venus sphere, the Sun sphere, the Mars sphere, the Jupiter sphere, the Saturn sphere and even beyond.
		While passing through the Mars sphere he has inscribed there a quality of his character through the fact that he acquired in that sphere a certain element of aggressiveness that was not previously in him.
	www.skepticfiles.org /cp002/rc2029.htm (6093 words)

	N-Dimensional Volumes
		This is a description of a geometric means to calculate the volume of a n-ball inscribed in a n dimensional hypercube.
		Here is a relationship between spheres and solid balls using cones: By definition, a cone over an object (not intersecting the origin) consists of all the lines connecting that object to the origin.
		Since the surface area of the region from phi1 to phi2 in spherical coordinates is 2pi(cos(phi1)-cos(phi2)), which is also equal to the surface area of the portion of the cylinder of radius one which surrounds this region.
	www.geom.uiuc.edu /docs/forum/ndvolumes (838 words)

	Circumscribed sphere - Wikipedia, the free encyclopedia
		In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices.
		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.
		All regular polyhedra have circumscribed spheres, but some irregular polyhedra do not have all vertices lieing on a common sphere, although it is still possible to define the smallest containing sphere for such shapes.
	en.wikipedia.org /wiki/Circumscribed_sphere (168 words)

	The math of stuff
		In that sphere is inscribed a tetrahedron; and Mars moves on that figure's inscribed sphere.
		The dodecahedron inscribed in the Mars-orbit sphere has the Earth-orbit sphere as its inscribed sphere, in which the inscribed icosahedron has the Venus-orbit sphere inscribed.
		Finally, the octahedron inscribed in the Venus-orbit sphere has itself an inscribed sphere, on which the orbit of Mercury lies.
	www.maa.org /devlin/devlin_11_01.html (1410 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)

	Tiling the Sphere with Congruent Triangles
		Not every tiling of the sphere arises from a polyhedron; it is possible that the vertices of a spherical tile may not be coplanar.
		However, in the special case of triangular tiles, meeting edge to edge, there is a bijection between monohedral tilings of the sphere and inscribed polyhedra with congruent triangular faces.
		This is one of the various reasons why it was reasonable for Sommerville [5] (in 1923) and Davies [1] (in 1967) to restrict their attention to tiles that met edge-to-edge.
	www.mi.sanu.ac.yu /vismath/bridges2005/dawson/index.html (2381 words)

	[No title]
		Here in we read that the surface area of any sphere is 4 times the area of its greatest circle (S=4Pi r²) and that the volume of a sphere is 2/3 of the volume of the cylinder in which it is inscribed (V=4/3 Pi r³).
		The approach to this problem devised by Archimedes, which consists of inscribing and circumscribing regular polygons with large numbers of sides, was the one followed by all those who subsequently dealt with the problem of determining Pi until the development of series expansions in the late 17th century.
		Following Archimedes' wish a sphere inscribed in a cylinder, proudly showing his discovery that the volume of the sphere equals 2/3 of the volume of the cylinder, is marked on his tomb.
	mathsforeurope.digibel.be /Archimedes.htm (2821 words)

	dandelinspheres.py
		#the circle inscribed in the triangle of the 3 points
		# the sphere inscribed in the given planes
		# the sphere exscribed in the given planes, opposite to the point of
	pw1.netcom.com /~ajs/scripts/dandelinspheres.html (194 words)

	Sphericity error evaluation: theoretical derivation and algorithm development. \| Professional, Scientific, and ... (Site not responding. Last check: )
		In the MRS criterion, two concentric spheres at minimum radial separation must be found such that they contain all points on the actual spherical surface.
		The center of this sphere is then used to find the smallest circumscribed and largest inscribed spheres.
		Two concentric spheres at minimum radial separation are to be found that contain all the points in the set.
	www.allbusiness.com /professional-scientific/architectural-engineering/777498-1.html (741 words)

	Chapter Thirty Two
		The altitudes of which being equal to the perpendicular from the centre of the body to the centre of the base, or the same as the Radius of the Sphere inscribed in the body.
		Likewise for the Triangle of the Octahedron, and the Square of the Cube.
		The Radius of the inscribed Sphere is sought.
	www-groups.dcs.st-and.ac.uk /history/Miscellaneous/Briggs/Chapters/Ch32.html (732 words)

	Regular polyhedra inscribed in a sphere or a cube (Site not responding. Last check: )
		In book XIII of his Elements Euclid discusses the construction of the regular polyhedra inscribed in a given sphere.
		* stands for one of the polyhedra and expresses a relation between the diameter of the given sphere** and the side (the edge) of the polyhedron.
		For four polyhedra Euclid starts with the given sphere, or with its diameter and gives a construction of the side of the polyhedron.
	cage.rug.ac.be /~hs/polyhedra/polyhedra.html (183 words)

	Formula Derivations for Polyhedra (Site not responding. Last check: )
		The inradius of a solid is the radius of the inscribed sphere.
		Point T is the center of the base, which is also the point where the inscribed sphere is tangent to that face.
		The circumradius is the radius of the circumscribed sphere.
	whistleralley.com /polyhedra/derivations.htm (1093 words)

	puz57
		In his work, Measurement of the Circle, he approximated the value of pi by inscribing and circumscribing a circle with a 96-sided regular polygon.
		Archimedes showed that the volume of an inscribed sphere is two-thirds the volume of the cylinder circumscribing it.
		He requested that this formula and the corresponding diagram be inscribed on his tomb.
	www.logicville.com /puz57.htm (174 words)

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