Three-dimensional space

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Topic: Three-dimensional space

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	Kant
		Its construction is made possible not merely by the self-consistency of the concept 'three-dimensional sphere', but by perceptual space being what it is. The a priori construction of a physical three-dimensional sphere must not be confused with the physical construction of, say, a sphere of wood or metal.
		Yet the possibility of the physical construction is based on the possibility of the a priori construction—the metal sphere on the possibility of a sphere in space—just as the impossibility of the physical construction of a fifteen-dimensional sphere is based on the impossibility of the corresponding a priori construction.
		For example, the concept of a fifteen-dimensional sphere, which is quite self-consistent, cannot be constructed—even though we can (and must) postulate objects for it if we are to make the statement that in a 'space' of at least fifteen dimensions 'there exist' at least two such spheres with no 'point' in common.
	members.aol.com /kasis4/Nisos/Kant.html

	sciforums.com - Four spatial dimension "Shadows" theory
		Shadows postulates the quantized energy associated with photons is the result of the curvature caused by the peaks and valleys of a “matterenergy” wave moving on in the surface of three dimensional space with respect to the fourth spatial dimension.
		Imagine the two-dimensional “surface” of a three dimensional sphere to be analogous to the “spherical surface” of a dimensional sphere with respect to the fourth spatial dimension.
		Chapter three will also show “c” or the velocity of light must be factored out of the equation E=mc^2 when defining the “static” or “non moving” dynamic resonate relationships between mass and energy component of space because the individual mattercules of space are not moving with respect to the speed of light.
	www.sciforums.com /printthread.php?t=13330&pp=40

	THREE DIMENSIONAL DATA PRESENTATIONS
		Three variables are presented using position in the space, all variables are presented using the lengths of the rays of the star.
		Planets I. The simplest presentation of multivariate data is a presentation using planets: three selected variables are shown using positions in the space.
		After that VRML was used in many areas: data organization, three dimensional “maps”, modeling, mathematics, chemistry, medicine...
	vlado.fmf.uni-lj.si /vrml/dsi.96/defengl.htm

	SYMMETRY OF THREE-DIMENSIONAL QUASICRYSTALS
		Projection of a regular lattice of higher dimension into the real physical space of lower dimension is a very useful method to construct a model structure of general lattices unifying usual lattices and quasilattices; the well-known examples are one-dimensional quasilattice as a projection from two-dimension, 2D-PT from five-dimension and three-dimensional quasilattice from six-dimension.
		As a three- dimensional point configuration, they are regardes as 22 vertices of a 30-hedron with bottleneck, among whose 30 faces 10 are equilateral triangles, the other 10 are another kind of equilateral triangles and the last 10 are trapezia.
		The radii along three kinds of symmetry axes and to the edges are tabulated in Table 8.
	www.mi.sanu.ac.yu /vismath/ogawa/3da.htm (2119 words)

	Three-Dimensional Sketching
		The basic requirements of the create, select, and edit operations don't really change when we jump to three dimensions, and the kinds of visual status feedback provided by the Macintosh drawing tools would be just as invaluable in a three-dimensional model space as they are on a two-dimensional screen.
		The modeling space is characterized by a noun-verb syntax direct -manipulation interface, using direct mouse control of a crosshair in three dimensions and objects with click-selectable handles to achieve facile editing in three dimensions.
		There is a further difficulty in defining a three dimensional sketch interface by metaphor.
	www.designlaboratory.com /faculty/matthews.kevin/research/mp3.02b.html (10470 words)

	Three Dimensional
		A line in 3 dimensional space can also be parametrized using
		This gives the equation of a PLANE in three space.
		To use 3D contour plotting, the following command must be executed first.
	www.ma.iup.edu /projects/CalcDEMma/vectors/vectors4.html (211 words)

	Alltag
		Having the assignment to design in space beyond our atmosphere and the familiar contact with the three conventional dimensions, the man's imagination is first confronted by the fact that all coordinates (x,y,z) are equal and by the realization that there is at least one more dimension.
		These table volumes can be grouped in space as desired to form bigger but also smaller arrangements.
		The utilizations are activated by the rotation of the sphere toward the gravity axis.
	www.archphoto.it /IMAGES/alltag/alltag.htm (211 words)

	If It Looks Like a Sphere...: Science News Online, June 14, 2003
		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.
		In the late 1970s, mathematician William Thurston, now at the University of California, Davis, envisioned a way to tame the menagerie of three-dimensional spaces—an idea that gave mathematicians a roadmap for proving the Poincaré conjecture.
	www.sciencenews.org /articles/20030614/bob10.asp (211 words)

	Spheres in Disguise: Solid proof offered for famous conjecture: Science News Online, April 26, 2003
		The traditional, or two-dimensional, sphere is the set of all points in three-dimensional space that are a given distance from a fixed center.
		French mathematician Henri Poincaré conjectured 99 years ago that, just as in the case of surfaces, any closed three-dimensional space in which loops can be tightened to a single point is really a three-dimensional sphere.
		A Russian mathematician may have finally cracked one of the most famous problems in mathematics: the Poincaré conjecture, a question about the shapes of three-dimensional spaces.
	www.math.uic.edu /~agol/scinewspoincare.htm (211 words)

	Brief History
		Chess Cubed did NOT have star spaces or specially marked king spaces (in fact no other version after "Space Chess" did.) Chess cubed actually extrapolated 3D versions of the 2D moves of the chess men.
		As far as I can determine the 8x8x3 playing field was first introduced in 1969 by the Pacific Games Company for a 3-D version they called "Space Chess".
		Chess in the Third Dimension had poor rules which did nor extrapol;ate 3d moves for the pieces.
	www.3dchessfederation.com /MilleniumHistory.htm (634 words)

	Three Dimensional 8x8xN Chess Game
		One interesting point of this new chess game is that each pieces position in the 3D space constitutes a surface in the 3D space.
		One of the oldest 3D chess games was invented by the Russian mathematician Ervand Georges Kogbetlianz in 1918, and we can consider that most modern invented 3D chess games are simply variants of this game.
		For this reason, I named my 3D chess game an 8x8xN 3D chess game, because the levels will vary from 0 to a desired level number N, (where N is an integer) along the z-axis direction.
	www.chessvariants.com /3d.dir/88n3d.htm (1291 words)

	Three-dimensional figures - Cylinders, cones and spheres - First Glance
		In this unit we'll study three types of space figures that are not polyhedrons.
		The sphere is a space figure having all its points an equal distance from the center point.
		Move your mouse cursor over the objects to learn more.
	www.math.com /school/subject3/lessons/S3U4L4GL.html (1291 words)

	Warm Up Exercises
		In a four dimensional sphere, three dimensional space is "bent" so that it folds back upon itself.
		The three dimensions that we experience might be curved in such a way that our three-dimensional space would eventually "join back up with itself".
		To understand the idea of a hypersphere, first consider that a normal, three dimensional sphere can be thought of as being constructed from a flat surface which is curved in such a way that it bends back and eventually joins itself.
	www.chrismaloney.com /projects/book/Exercises.html (1291 words)

	3-Dimensional Geometry
		In this section, we will explore the geometry of three-dimensional space.
		The 3D coordinate space has three axes, the x axis, going toward and away; the y axis, going left and right; and the z axis, going up and down.
		Many three-dimensional properties are very easy to figure out given the two dimensional properties.
	www.horacemann.pvt.k12.ny.us /academics/math/pcbch/vect/3d.html (1291 words)

	Clinton Conley
		Now, even though space is three dimensional, elevation off of the Earth’s surface is often of minimal importance to navigators, especially those in times before air travel was prevalent.
		Spatial geometry, particularly relations between the three-dimensional sphere and the two-dimensional sheet of paper, is integral to understanding the layout of the planet.
		A projection, formally speaking, is a function from a higher dimensional space to a lower dimensional one.
	www.cco.caltech.edu /~sciwrite/journal03/A-L/conley.html (1291 words)

	Shapes And Implementations In Three-Dimensional Geometry - Mucke (ResearchIndex)
		For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three- dimensional space.
		Shapes And Implementations In Three-Dimensional Geometry - Mucke (ResearchIndex)
		Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the "shape" of the set.
	citeseer.ist.psu.edu /514237.html (1291 words)

	Viewing Four-dimensional Objects in Three Dimensions
		A second way to view three dimensional polytopes in two dimensions is by means of a projection.
		This method of viewing higher dimensional objects as well as others is one way people can understand the shape of higher dimensional space.
		The sphere explains to the square the existence of higher dimensional objects like itself, and ways in which the square can understand the form of such objects.
	www.geom.uiuc.edu /docs/forum/polytope (977 words)

	Dimension article - Dimension Latin degrees freedom measurements shape Physical dimensions - What-Means.com
		For example, the space in which we live appears to be 3-dimensional.
		We can move up-or-down, north-or-south, or east-or-west, and movement in any other direction can be expressed in terms of just these three.
		Moving down is the same as moving up a negative amount.
	www.what-means.com /encyclopedia/Three-dimensional (596 words)

	An Unshellable Triangulation
		In three dimensional space a simplex is just a tetrahedron.
		Unfortunately, to appreciate this example interactive manipulation to show the three dimensional structure is almost essential.
		The higher dimensional analog of a triangle is a simplex.
	www.math.utah.edu /~alfeld/Rudin.html (536 words)

	BioMed Central Full text Visualising very large phylogenetic trees in three dimensional hyperbolic space
		We introduce the idea of visualising phylogenetic trees in three dimensional hyperbolic space with the Walrus graph visualisation tool and have developed a conversion tool that enables the conversion of standard phylogenetic tree formats to Walrus' format.
		These tools lay out trees in a two dimensional Euclidean space and are useful for visualising trees of up to a few hundred nodes.
		Some software tools, like Hypertree [6], have increased the number of visualisable nodes using 2D hyperbolic space to provide a "focus+context" view, where a subset of the data can be viewed at higher resolution with the remaining contextual data still in view (see the Deep Green project for an example with plant phylogenies [7]).
	www.biomedcentral.com /1471-2105/5/48 (1609 words)

	MPR Page
		M.C Escher's drawings were of mind boggling structures that logically would be impossible in three dimensional space.
		Three dimensional models based on the works of M.C. Escher.
		Some of these drawings were based on the work of British mathematician Sir Roger Penrose.
	www.msn.fullfeed.com /~jpdesign/MPR.html (366 words)

	Three Dimensional N*N-Queens Problems
		QP solution replicated infinitely many times in three dimensions, it scatters points very uniformly in space at a density of 1/N. Other variants on the 2DNQP have been studied [3].
		If we name the three visible faces S, S' and S", then either they are all linear Latin squares or none of them are.
		We call three squares related in this way perpendicular Latin squares (since the use of orthogonal has been preempted).
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Recn/Queens3D (1527 words)

	Meta
		Therefore, the geometry and the dynamics of our three-dimensional Metagalaxy (identify with the circumference) are formed by the global geometry of four-dimensional spherical volume (identify with the sphere).
		In a definite period of time, while accepting the energy from the electromagnetic waves, elements of the left sphere are organizing in their space as meridians coming from the equators to the poles.
		A particle without mass is a photon, which has a zero radius of the globe and a zero surface of the sphere, that is not a compactification, but it is the oscillations of four-dimensional elements.
	personal.primorye.ru /markvol/Meta.html (1527 words)

	Three-dimensional figures - Space figures - First Glance
		A sphere is a space figure having all its points an equal distance from the center point.
		Polyhedrons are space figures with flat surfaces, called faces, which are made of polygons.
		In this unit, we'll study the polyhedron, the cylinder, the cone, and the sphere.
	www.math.com /school/subject3/lessons/S3U4L1GL.html (1527 words)

	The Integral Sphere: A Mathematical Mandala of Reality, by Thomas J McFarlane
		It should be noted that a single four-dimensional integral sphere maps to a three-dimensional space of infinitely nested concentric spheres (corresponding to "spheres of latitude" on the four-dimensional integral sphere).
		The analog of circles of longitude in the three-dimensional spherical mandala are "spheres of longitude" on the four-dimensional spherical mandala.
		First of all, the AQAL planar mandala is not presented as mathematically equivalent to a single higher-dimensional sphere.
	www.integralscience.org /sphere.html (1527 words)

	Three-Dimensional Chess
		One problem presented by three-dimensional chess is that of being able to easily reach all the levels of the board, and yet having sufficient space between them to reach in to each one.
		Unfortunately, the three sides are in a relationship such that one is, in effect, before, and the other is after, the third player, so the array is not symmetric with only two players.
		Also, the top and bottom three levels are small rectangles; the green area simply shows where these rectangles stand in relation to the central 5 x 5 square of the five middle levels, so that alignment is unambiguous.
	www.hypermaths.org /quadibloc/chess/ch05.htm (1746 words)

	homogeneity
		This is a two-dimensional universe that is curved into a three-dimensional sphere, with a finite amount of matter, distributed in a lumpy manner.
		This is a two-dimensional universe that is curved into a three-dimensional sphere, with a finite amount of matter, smoothly distributed.
		Space itself is expanding by the continuous enlargement of the sphere.
	www.alienryderflex.com /homogeneity (1746 words)

	Scientific American: Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions
		For 100 years mathematicians have been trying to prove a conjecture that was first proposed by Henri Poincaré relating to an object known as the three-dimensional sphere, or 3-sphere.
		But in 1982 Michael H. Freedman, now at Microsoft Research, succeeded in proving a 4-dimensional version of the Poincaré conjecture (above three dimensions there are subtly different ways to formulate the conjecture).
		It is interesting to speculate that it is not a coincidence that our universe of space and time is four dimensional, the most mathematically complicated case.
	www.sciam.com /print_version.cfm?articleID=0003848D-1C61-10C7-9C6183414B7F0000 (1746 words)

	riding5
		In all three possibilities for the shape of the universe if you back up time you find that at some finite time in the past (currently estimated to be about 15 billion years ago) all of three-space was concentrated at a point.
		The three dimensional generalization of the sphere is called the three-sphere.
		In this case if the radius of these three spheres is getting bigger (if the universe is expanding) then eventually the growth of the radius will stop and the spheres will start to get smaller (the universe will start contracting).
	home.pacbell.net /stevepur/physics/riding/riding.5.html (1746 words)

	Three-dimensional figures - Space figures - In Depth
		Three-dimensional geometry, or space geometry, is used to describe the buildings we live and work in, the tools we work with, and the objects we create.
		There are many other space figures--an endless number, in fact.
		All of the points of a sphere are at the same distance from its center.
	www.mr.math.com /school/subject3/lessons/S3U4L1DP.html (1746 words)

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