Watch Cube²: Hypercube-2002 online free with subtitles. Watch Cube²: Hypercube-2002 online free with subtitles. If your screensaver ever escapes from your PC it’ll mince you into pieces so tiny you won’t even leave a bloodstain!Why I watched to the end I’ll never know. And even then, right at the end, the shocks keep on coming.
The tesseract, or 4D cube, is perhaps the most well-known of all the 4Dobjects. It is known by many names, among which are the 4-hypercube,the 8-cell, the 4D measure polytope, and thetetracube. It is bounded by 8 cubes, 24 squares, 32 edges, and 16vertices. Its many names describe its different special properties. It hasbeen the subject of several stories, such as Robert A. Heinlein's AndHe Built a Crooked House. It has also been the subject of countless 4Dwireframe rotation programs, screensavers, and Java applets.
The name “tesseract” comes from the Greek τέσσερειςἀκτίνες, meaning “four rays”, referring to the four mutuallyperpendicular directions on which it is based.
Construction
There are several ways of constructing the tesseract. The simplest way is toextrude the 3D cube along the W-axis. The following obliqueprojection of the tesseract underlines this method of constructing thetesseract.
The red cube shows the starting 3D cube, and the blue cube shows theendpoint of the extrusion. The yellow faces trace out the path of theextrusion. They actually form 6 other cubes, which are generated by theextrusion of each of the 6 faces of the original cube. So the tesseract infact consists of eight cubes. These 8 cubes form its outerboundary.
Cell-first Projections
The difficulty with the above image is that there are too many intersectingsurfaces, so it is difficult to discern the 8 constituent cubes. The followingdiagram tries to correct this defect by using a perspective projectioninstead:
In this image, the blue “inner” cube is actually the same sizeas the red “outer” cube, but it appears to be smaller because it isfarther away along the W-axis. Between the inner and outer cubes are 6frustums that are actually the other 6 cells, as shown by thefollowing images:
These 6 frustums are actually regular cubes; but they appear distorted intofrustums because they are seen at an angle. Furthermore, all 8 cubeslie on the outer boundary of the tesseract. Even though it appearsthat the inner cube is on the “inside” whereas the outer cube is onthe “outside”, they actually both lie on the outside ofthe tesseract, on two opposite sides. The following animation shows whathappens when we rotate the tesseract in the XW plane.
The red and blue cells appear to be deforming inside-out and engulfing eachother, but this is only an artifact of projection into 3D. In reality, they areperfectly regular cubes, two opposite cells of the tesseract, and neitherdeform nor touch each other as they rotate through 4D space.
Hidden Surface Removal
One thing that is often neglected to be mentioned when such wire diagrams ofthe tesseract are presented is the fact that they represent projections of thetesseract without the removal of hidden surfaces. This is like showingthe rotation of the wireframe of a 3D cube, where you can see through its facesand see what is on the other side of the cube. While this is useful in seeingthe entire structure of the tesseract, it sometimes gives too much detail andbecomes confusing. The following images try to complement the picture byshowing projections of the tesseract where obscured 4D surfaces are notshown.
For example, when viewed from the angle that corresponds with thecube-within-a-cube image shown earlier, the tesseract in fact appears as asimple 3D cube:
When rotated 45 degrees in the XW plane, the tesseract appears asfollows:
Only two cells are visible because the rest are obscured behind them in the4th direction.
Vertex-first projection
Another fact that is often neglected when tesseract projection images anddiagrams are shown is that projections such as the cube-within-a-cube actuallyview the tesseract from a “flat” angle, akin to looking at a 3Dcube directly at one face, or perhaps at an edge, and seeing only two faces ata time. Just as we intuitively imagine the 3D cube from an angle such that wecan see three of its faces at a time, so a more
intuitiveangleof looking at the tesseract is from an angle where we can see four ofits cells at once. The following image shows one such view of thetesseract.
The 3D surface of this projection is called a rhombic dodecahedron.It is a 12-faceted polyhedron where each face is a rhombus. The nearest vertexto the 4D viewpoint is the one in the center of the projection,highlighted here in yellow. The four cells of the tesseract visible from thisangle are shown below:
The other four cells of the tesseract are behind these four in the 4thdirection, so they are not visible here.
Geometrical Properties
The tesseract belongs to the family of n-dimensionalhypercubes, also known as measure polytopes (because theyconstitute the unit by which n-dimensional space is measured). Itsdual is the 16-cell. The coordinates of anorigin-centered tesseract with edge length 2 are all permutations of signand coordinates of:
Interestingly, not only is the 16-cell its dual, but also itsalternation. This property is peculiar to 4D; in 3D, the alternatedcube is the tetrahedron, and in 5D, the alternated cube is a semi-regularpolytope known as the demipenteract. This property is useful forderiving a coherent indexing of the 24-cell, which allowsus to construct the 600-cell using thesnub 24-cell as an intermediate.
Hyper Screen Saver: Freeware for Windows
The Hyper screen saver displays a rotating 4-dimensional object (hypercube or 4-simplex) projected onto 3-space using a 4-D perspective transformation.
Notes
The screen saver fundamentals were based on an example (
minimal screen saver) published online by Lucian Wischik(www.wischik.com/scr/).The math of n-dimensional rotation, perspective transformation, and hidden facet removal is my own work.
In 2009, I converted the original Hyper screen saver from Borland C++ to Microsoft Visual C++ andcleaned up the code. In March 2010, I added the hyperbrick option. The Borland version is still here ifyou really want it (view page source), but I’ve been using only Microsoft C++ in recent years.
Remarks
When a rotating cube in ℝ3 is projected onto a 2-dimensional display surface using a 3-D perspective transformation, the facets that are farther away from the center of projection (CP) appear smaller than the facets that are closer. In a transparent wire-frame rendering, the back facets (squares) seem to shrink and pass through the front facets and then grow as they again become front facets. When shading and hidden-surface removal are done, a facet disappears from view whenever it faces away from the CP and reappears when it rotates so that it faces the CP again.
Something similar happens when a rotating hypercube in ℝ4 is projected down to ℝ3 using a 4-D perspective transformation. In a transparent rendering, the back facets (which are 3-dimensional cubes in this case) shrink and pass through the front facets before starting to grow again. With shading and hidden-facet removal, the facets appear and disappear (but the rendering is complicated by the need to perform another projection to map the 3-D image onto a 2-dimensional display surface).
The animated GIF at the upper right corner of this page shows the rotating hypercube. Clicking the imageselects a different version with a different rotation plane (using Javascript).
Since I started experimenting with the hyperbrick shape (see below), I’ve decided I like it better than thesimplex or the hypercube. I considered a
monolithà la 2001: A Space Odyssey(1 × 4 × 9 × 16), but I think using the golden ratio makes a more æstheticallypleasing image. To-Do List
Note: If you click the animated GIF file in the upper right corner of this page, it should change to anotherversion with a different rotation. As of 2014-12-28,the bug in Firefox that had often caused this operation to fail seems to be fixed.
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