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Topic: Alexander horned sphere

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	Sphere Summary
		The radius of a sphere is a line segment whose one endpoint lies on the sphere and whose other endpoint is the center.
		For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes 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.bookrags.com /Sphere (1421 words)

	NSDL Metadata Record -- Antoine's Horned Sphere -- from MathWorld
		The outer complement of Antoine's horned sphere is not simply connected.
		Antoine's horned sphere is inequivalent to Alexander's horned sphere since the complement in \mathbb{R}^3 of the bad points for Alexander's horned sphere is simply connected.
		Alexander, J. "An Example of a Simply-Connected Surface Bounding a Region which is not Simply-Connected." Proc.
	nsdl.org /mr/699010 (104 words)

	Alexander’s Horned Sphere
		An example of what, in topology, is called a "wild" structure; it is named after the Princeton mathematician James Waddell Alexander (1888-1971) who first described it in the early 1920s.
		The Horned Sphere is topologically equivalent to the simply-connected surface of an ordinary hollow sphere but bounds a region that is not simply-connected.
		The Horned Sphere can be embedded in the plane by reducing the interlock angle between ring pairs from 90° to 0°, then weaving the rings together in an over-under pattern.
	www.daviddarling.info /encyclopedia/A/Alexanders_Horned_Sphere.html (214 words)

	Alexander horned sphere at AllExperts
		The horned sphere was introduced by James Waddell Alexander in 1924 as a counterexample to his previous claim of a three-dimensional Jordan-SchÃ¶nflies theorem.
		Alexander's horned sphere is a particular embedding of the 2-sphere into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R
		Although the horned ball is not a manifold, RH Bing showed that its double is in fact the 3-sphere.
	en.allexperts.com /e/a/al/alexander_horned_sphere.htm (395 words)

	NSDL Metadata Record -- Alexander's Horned Sphere -- from MathWorld
		The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere.
		It is homeomorphic with the ball \mathbb{B}^3, and its boundary is therefore a sphere.
		The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated.
	nsdl.org /mr/697559 (145 words)

	The Official Alexander Sphere Appreciation Page (Site not responding. Last check: 2007-10-11)
		Alexander's horned sphere is an example of a convoluted, intertwined surface for which it is difficult to define an inside and outside.
		Although this may be hard to visualize, Alexander's horned sphere is homeomorphic to a ball.
		The boundary is, therefore, homeomorphic to a sphere.
	sprott.physics.wisc.edu /pickover/pc/hornedsphere.html (212 words)

	Alexander biography
		Alexander studied mathematics and physics at Princeton, where he was a student of Veblen, obtaining a B.S. degree in 1910 and an M.S. degree in 1911.
		Alexander had virtually become a recluse after he retired in 1951 and the McCarthy era resulted in his disappearance from public life.
		Alexander's work around this time went a long way to put the intuitive ideas of Poincaré on a more rigorous foundation.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Alexander.html (804 words)

	Jordan Curve Theorem (Site not responding. Last check: 2007-10-11)
		Brouwer was unable to prove the analog of the Jordan-Schönflies theorem, that the inside and outside of such an imbedded sphere are homeomorphic to the inside and outside of the standard sphere in Euclidean space (ie.
		Alexander announced that he had a proof of this result.
		The outside of the horned sphere is not simply connected, unlike the outside of the standard sphere: a loop around one of the horns of the horned sphere can't be continuously deformed to a point without passing through the horned sphere.
	www.math.ohio-state.edu /~fiedorow/math655/Jordan.html (470 words)

	No Title (Site not responding. Last check: 2007-10-11)
		During the 1950's, the study of wild 2-dimensional spheres in 3-dimensional space emerged as a central theme in geometric topology.
		However, embeddings of the 2-sphere (the analogue of the circle) in 3-dimensional space may have the closure of the bounded component not be homeomorphic to a 3-dimensional ball (the analogue of the 2-dimensional disk).
		A natural question is whether necessary and sufficient conditions exist for an embedding of a 2-dimensional sphere to be tame (i.e., to have the bounded component be homeomorphic to the 3-dimensional ball).
	www.uwm.edu /~gb/COLLOQUIA/01-09-28 (235 words)

	Alexander's Horned Sphere (Site not responding. Last check: 2007-10-11)
		Alexander's Horned Sphere is homeomorphic to a ball.
		Alexander's Horned Sphere is, among other things, an example of why the Jordan Curve Theorem cannot be expanded to higher dimensions.
		If it were possible, you would be deforming something that is not simply connected (the compliment of the horned sphere) into something that is simply connected (the compliment of the ball).
	www.stat.duke.edu /~jel2/home/background_pics/horned_sphere.html (174 words)

	Alexander (print-only)
		In 1928 he discovered the Alexander polynomial which is much used in knot theory.
		Also around 1935 Alexander discovered cohomology theory, at essentially the same time as Kolmogorov, and the theory was announced in the 1936 Moscow Conference.
		Among the many honours bestowed on Alexander was his election to the American Academy of Sciences in 1930.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Alexander.html (801 words)

	If It Looks Like a Sphere...: Science News Online, June 14, 2003 (Site not responding. Last check: 2007-10-11)
		In contrast, a coffee cup is not a sphere but instead a distorted version of a doughnut, and a pretzel can be considered a doughnut with three holes instead of one.
		Even though spheres and tori sit in three-dimensional space, mathematicians focus on their surfaces and so view them as two-dimensional, unlike solid balls and filled-in doughnuts, which are three-dimensional.
		For instance, just as the sphere is the two-dimensional boundary of the three-dimensional ball, mathematicians have defined the hypersphere as the three-dimensional boundary of the four-dimensional ball—a space that's hard to visualize but that can nevertheless be analyzed mathematically.
	www.sciencenews.org /20030614/bob10.asp (2652 words)

	- Gideon Weisz - Sculptor
		This sculpture has 5 levels of loops (horns), and is only a subset of the true infinitely Horned Sphere.
		This means that it can be continuously deformed (in the 4th dimension) into a solid sphere.
		Unlike a standard sphere, however, the surface is not simply connected.
	gideonweisz.com (431 words)

	[No title]
		However, a loop of string around one of the main horns cannot be removed in a finite number of moves; so it's embedded "wildly" in space.
		That is, the 3-sphere "is" a sphere of circles in the same way that a torus is a circle of circles.
		In this representation, the circle that passses through the point of the sphere that we have chosen to be "at infinity" becomes a straight line, while the "opposite" circle encircles it symmetrically.
	cs.stmarys.ca /~dawson/images2.html (1078 words)

	Cameron's Art Page
		I gave a paper on automatic font decoration with celtic knotwork at RIDT'98 in St Malo, France (here are the colour illustrations).
		This picture is a visualisation of Alexander's horned sphere embedded in the plane, from the article "Rep-tiles with woven horns" published in the journal Computers and Graphics.
		The following picture is a closeup of the busy part of a traditional horned sphere.
	members.optusnet.com.au /cameronb/art/art-1.htm (269 words)

	Algebraic Topology: Knots, Links, Braids
		Thus, one usually restricts knots to be tamely embedded, e.g., as a simple closed polygonal curve, and we'll do so as well.
		Example (Alexander's horned sphere) A homeomorphic image of the sphere S(2) in E(3) such that the complement of the image is not simply connected.
		Example (Antoine's necklace) A homeomorphic image of the Cantor set (which is compact and totally disconnected) in E(3) such that the complement of the image is not simply connected.
	www.win.tue.nl /~aeb/at/algtop-5.html (1380 words)

	Ultra Fractal: Showcase (Site not responding. Last check: 2007-10-11)
		Alexander's horned sphere: a famous object in topology.
		There is an extensive explanation for this in Mathworld, which I humbly admit that I do not understand.
		One does not, however, need to understand the deep mathematical meaning in order to draw it!
	www.ultrafractal.com /showcase/jos/alexanders-horn.html (42 words)

	My Interests (Site not responding. Last check: 2007-10-11)
		3-dimensional space (or in general, knotted and linked n-dimensional spheres in (n+2)-dimensional space.) Knot theory is one of the most active areas of mathematics today.
		Yggdrasil, the tree of Norse mythology whose branches lead to heaven, is realized here as Alexander's horned sphere.
		When it comes to college sports, of course I root for my alma mater, the TCU Horned Frogs, but I also keep up with the schools in my home state: Oklahoma Sooners and Oklahoma State Cowboys.
	www.math.rochester.edu /people/faculty/amheap/interests.html (1539 words)

	Palmer Hall Renovation (Site not responding. Last check: 2007-10-11)
		Earlier about 1980, the department commissioned an art student to paint murals on two walls.
		One of these murals was painted over in the remodel, but the department insisted on preserving the mural depicting Alexander's Horned Sphere.
		As of 1999 it was located on an interior wall of the visiting professor office.
	www.coloradocollege.edu /Dept/ma/History/Topics/PalRen.html (281 words)

	Matt Day's Math Art (Site not responding. Last check: 2007-10-11)
		A model of the classifying space of the baumslag-solitar group (1,2):
		The sphere of radius 2pi in the 3-d Heisenberg group, viewed in the standard matrix coordinates.
		There is another coordinate system in which the action of the point stabilizers is by rotations; in that system this would look like a cigar.
	www.math.uchicago.edu /~mattday/mathart.html (62 words)

	Introduction to Topology MTH 425 (Site not responding. Last check: 2007-10-11)
		Here is a strange mathematical object whose topological significance you will understand later.
		It is called the "Alexander horned sphere." (You will also know something about the significance of the "Klein bottle" pictured above.)
		Study guide and review session for final exam
	www.math.uri.edu /~pakula/mth425.htm (268 words)

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