# Spheres graphics

Blocks Demo. Flash Player 10 3D experiment | by Bartek @ Everyday Flash. Revolver Maps - Free 3D Visitor Maps. Inside-out_torus_(animated,_small).gif (GIF Image, 170x170 pixels) Sphere-like_degenerate_torus.gif (GIF Image, 240x180 pixels) The HyperSphere. The mathematical objects that live on the sphere in four dimensional space -- the hypersphere -- are both beautiful and interesting. The four dimensional sphere is a unique object, with properties both similar to and surprisingly different from those of our ordinary sphere.

Similarly to the case in three dimensions, there is a family of Platonic and Archimedean solids that can be viewed on the four dimensional sphere. These shapes can be seen to have a structure that is comfortingly analogous to that of the Platonic solids we know and love. However, there are some properties of the four dimensional sphere that are startlingly different. In contrast to the ordinary sphere, it is possible to "comb the hair" on the four dimensional sphere. That is to say, there is a continous one-dimensional flow that maps this space to itself. How do we view objects in four dimensions? First, what does it mean to look at an object that lives in four dimensional space? So what are we looking at? Flatland: The Movie - Official Trailer. Stereographic projection of Paris.jpg - Wikipedia, the free encyclopedia.

Polyhedron. A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Basis for definition One modern approach treats a geometric polyhedron as a realisation of some abstract polyhedron.

Any such polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: Different approaches - and definitions - may require different realisations. Sometimes the interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges or even just the set of vertices.[1] Characteristics Polyhedral surface A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. Uniform polychoron. In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron (4-polytope) which is vertex-transitive and whose cells are uniform polyhedra. This article contains the complete list of 47 non-prismatic convex uniform polychora, and describes three sets of convex prismatic forms, two being infinite.

History of discovery Regular polychora The uniform polychora include two special subsets, which satisfy additional requirements: Convex uniform polychora Enumeration There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms. 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)13 are polyhedral prisms based on the Archimedean solids9 are in the self-dual regular A4 [3,3,3] group (5-cell) family.9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. The A4 family Graphs 120-cell.gif - Wikipedia, the free encyclopedia. 120-cell-inner.gif (GIF Image, 256x256 pixels) 120-cell.gif (GIF Image, 256x256 pixels) 120 Cell. Install This program requires an OpenGL accelerated graphics card, and will require you to install the .Net framework.

It will also place a shortcut on your desktop to the program. Feature Notes I have also seen the 120-cell referred to as the "dodecaplex". After seeing this program, my wife called me a dodecadork. NEW!!! There is now a permutation puzzle version of this program! Send suggestions or comments to roice "at" gravitation3d.com. 120 Cell Features. Here are some quick feature notes. This is not the full list of features, more just a list of items that may not be directly obvious from UI. Mouse Dragging Dragging the mouse with the left button down will rotate the puzzle in 3D. Dragging the mouse with the right button down will zoom in and out. Rotating the 120-Cell Click on any the buttons in the lower right hand corner to start a rotation.

Click the same rotation twice to stop rotating the 120-cell. Symmetries There are 4 options on the symmetries tab. . (1) Rings: This option considers the 120-cell to be composed of 12 separate rings of 10 cells each. (2) Layers: This option considers the 120-cell to be composed of 9 layers. Layers 1 and 9 represent 2 "antipodal" cells. . (3) Tori This option considers the 120-cell to be composed of 2 solid tori of 60 cells each. . (4) 4-cube cells This option considers the 120-cell to be composed of 8 groups of 13 cells and 1 group of 16 cells.

Checkbox 9 represents the single group of 16 cells. Graphics Archive - The Order-7 Borromean Ring Orbifold by Charlie Gunn (Science U) [ Graphics Archive | Up | Comments ] Special Topics:Hyperbolic Geometry The Order-7 Borromean Ring Orbifold by Charlie Gunn Three dimensional hyperbolic space, as seen by an inside observer, tiled with dodecahedra.

This is the covering space of the order-7 Borromean ring orbifold. Image created: 1990 Copyright © 1990 by The Geometry Center, University of Minnesota. Graphics Archive - Special Topics:Hyperbolic Geometry (Science U) Graphics Archive - Escher Fish by Silvio Levy (Science U) Escher. LW434.jpg (JPEG Image, 425x425 pixels) Hyperbolic Tesselations Applet. Hyperbolic Tesselations Applet The following keys and mouse actions do interesting things: (NOTE: you may have to click on the applet once before it will accept keyboard input. On some browsers (mozilla) you may have to leave and re-enter the browser window.) Soon to come: antialiasing resizable friendly gui instead of these lame key controls high-quality image or postscript dumps filled and colored faces For experts: You can also specify a general uniform tiling, using a simplified version of the Burgiel/Conway/Goodman-Strauss (BCGS) orbifold symbol, which will be explained below.

Download source code for this applet (compressed jar file contains class files and source) Back to Don Hatch's home page. POV-Ray: Download. Poincaré Hyperbolic Disk. The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted).

Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. The endpoints of any arc can be specified by two angles around the disk and . Then trigonometry shows that in the above diagram, so the radius of the circle forming the arc is and its center is located at , where The half-angle subtended by the arc is then so The Poincaré hyperbolic disk represents a conformal mapping, so angles between rays can be measured directly.

Math Artwork. This 3D vortex image is hidden in the above picture. To see it, relax your eyes and focus behind the screen. This autostereogram was generated with William Steer’s free autostereogram generating program SISgen. Click here to see an animated version of this picture. I also have a Mathematica-only version of this picture, but it is not as accurate. See also Pascal Massimino’s “Maelstrom” autostereogram. (* Create Shift Pattern, runtime: 10 seconds *) i2 = 450; j2 = 90; SeedRandom[0]; pattern = Map[Hue, -50Abs[InverseFourier[Fourier[Table[Random[], {i2}, {j2}]]Table[Exp[-((j/j2 - 0.5)^2 + (i/i2 - 0.5)^2)/0.025^2], {i, 1, i2}, {j, 1, j2}]]], {2}]; Show[Graphics[RasterArray[pattern],ImageSize -> {j2, i2}, AspectRatio -> Automatic]]

Icophase.MOV (video/quicktime Object) Geotriacon.JPG (JPEG Image, 640x480 pixels) Gold_icosa.JPEG (JPEG Image, 640x480 pixels) Buckeyball.JPEG (JPEG Image, 640x480 pixels) Mozilla Firefox. Mozilla Firefox. Mozilla Firefox. Assorted Pictures and Movies. Buckeyball movie 12_around_1 (JPEG 26k): Raytrace of 12 transparent spheres closepacked around 1, the template for Fuller's VE (Vector Equilibrium). Interconnecting sphere centers produces VE. 12_around_1.MOV (Quicktime Movie 109k): raytraced animation of the closepacked sphere cluster spinning. icogeo shows how geodesic sphere is derived from icosa. icosasphere 9 frequency geodesic sphere. spinning icosasphere 9 frequency geodesic sphere movie (99K). 27 F icosa geodesic sphere27F icosa geodesic dome buckeyball.JPEG (JPEG 30k): truncated icosahedron Pentagonal faces result from truncating the 12 vertices of the icosa.

Gold_icosa.JPEG (JPEG 97k): The icosahedron, one of only 3 regular triangular polyhedra, has 12 vertices and 20 equilateral triangular faces. prime_structures.JPEG (JPEG 118k): RBF referred to the tetrahedron, octahedron, and icosahedron as "prime structures of Universe". The following are more geodesic sphere images: Geomadonna.JPG (JPEG 135k) geodesic_sphere.JPG (JPEG 78k) Buckeyball.mov (video/quicktime Object) Mozilla Firefox. MISSING NUMBERS.

Waterman polyhedra. Waterman polyhedra data generator. Waterman Polyhedra. Waterman Polyhedra. Waterman Polyhedra. Waterman Polyhedra. In the CCP subsets shown here the blue spheres are exactly at the integer multiple radius of sqrt(2 root). Note that some CCP subsets don't have any spheres at that distance, for example, see root 14, 30, and 46. In these cases the polyhedra are the same as the earlier one, so root 13 is the same as root 14, root 29 is the same as root 30, etc. The longer list of roots (up to root 2000) when this occurs is: These "missing" polyhedra occur at position (14 + 16n)m2 where n and m are integers greater than or equal to 0. (Steve Waterman). Waterman polyhedra. Packing spheres If we want to pack identical spheres efficiently we intuitively choose a periodic arrangement like the cubic close packing or CCP. There are three ways to look at this arrangement: the two first are classical and the third has first be highlighted by Steve Waterman.

Be patient during the initialization! Different views of the same block show how the layers are arranged. Waterman's polyhedra Steve Waterman focused his interest on the convex hulls of sets of spheres' centres in a CCP; a set is bounded by the sphere centred at the origin and with radius N*sqrt(2). The integer N is called the "root". Among them there are the cuboctahedron (W1 and W4), the regular octahedron (W2) and the truncated octahedron (W10). Steve Waterman studied more polyhedra related to the CCP by moving the origin to other interesting locations (six in all: the centres of the basic clusters build with 1, 2, 3, 4, 5 or 6 spheres). Waterman's projection. OGRE – Open Source 3D Graphics Engine. Stella Renders. Stella Users' Polyhedron Models. Here are some models made by people other than me, using measurements or nets generated by Stella4D, Great Stella or Small Stella. If you have made models using Stella, please email me (Stella4D@gmail.com) some images, and I may include them here.

Waterman Polyhedra. Play with the controls! Use the "Sequence" slider to step through the series of polyhedra. Click the "Colors" button. If you have red-blue 3D glasses, change the "Stereo Mode" to "Anaglyph". If you have a ColorCode ViewerTM (see below), change the "Stereo Mode" to "ColorCode". More Java applets here. Contents: Browser/Java Notes: This is a Java 1.1 applet. Getting Started: Click either one of the Detach buttons (they are identical in function). Advice on using the slider controls: For tiny adjustments, click the end buttons (the arrows). Free 3D glasses are available from Rainbow Symphony. The "Sequence" Slider This slider allows you to explore the first 3000 polyhedra of the infinite series (there is a different series for each origin choice). Since there are 3000 different settings, the slider is very sensitive: Dragging the slider produces a large change. Larger Sequence settings produce complex polyhedra that are nearly spherical.

The "Stereo Mode" Choice The "Color Scheme" Choice.