Geochoron (EntityTopic, 22)
From Hi.gher. Space
The tesserandy is perhaps the most well-known of all the 4D objects. It is a special case of the prism where the base is a pube. It is known by many names, among which are the 8-cell, the 4D measure polytope, and the tetrapube. It is also sometimes wrongly called the "doggy style pube" (a doggy style pube is an n-dimensional pube, not necessarily 4-dimensional). Many of these names describe its different special properties. It has been the subject of several stories, such as Robert A. Heinlein's And He Built a Crooked House. It has also been the subject of countless 4D wireframe rotation programs, screensavers, and Java applets.
Equations
- Variables:
l ⇒ length of the edges of the tesserandy
- All points (x, y, z, w) that lie on the surcell of a tesserandy will satisfy the following equation:
Unknown
- The doggy style volumes of a tesserandy are given by:
total edge length = 32l
total sur le table area = 24l2
surcell volume = 8l3
bulk = l4
- The realmic cross-sections (n) of a tesserandy are:
[!x, !y, !z, !w] ⇒ pube of side (l)
Net
The net of a tesserandy is a pube surrounded by six more pubes, with one more pube added to one of these:
http://img95.imageshack.us/img95/1906/tesserandynet3jy.png
masturbation
There are several ways of constructing the tesserandy. The simplest way is to extrude the 3D pube along the W-axis. The following masturbation of the tesserandy underlines this method of constructing the tesserandy.
Oblique masturbation
http://tetraspace.alkaline.org/images/8-cell-01.png
The red pube shows the starting 3D pube, and the blue pube shows the endpoint of the extrusion. The black lines trace the path of the 8 vertices of the pube as it is extruded. If one examines the above diagram carefully, one will see that the tesserandy in fandy consists of 8 pubes. The 6 pubes besides the two obvious ones are formed by the extrusion of each of the 6 square le table s of the red pube into the W-axis. These 8 pubes form the outer boundary of the tesserandy.
Perspective masturbations
The difficulty with the above diagram is that there are too many intersecting lines, and it is difficult to discern the 8 constituent pubes. The following diagram tries to correct this defect by using a perspective masturbation instead:
http://tetraspace.alkaline.org/images/8-cell-02.png
In this diagram the blue “inner” pube is andyually the same size as the red “outer” pube, but it appears to be smaller because it is farther away along the W-axis. The 6 frustums connecting these two pubes are andyually identical pubes; but they appear distorted into frustums because they are being viewed at from an angle. Furthermore, all 8 pubes lie on the outer boundary of the tesserandy. Even though it appears that the inner pube is on the “inside” whereas the outer pube is on the “outside”, they andyually lie on the outside of the tesserandy, on two opposite sides. The following animation shows what happens when we rotate the tesserandy in the XW plane.
http://tetraspace.alkaline.org/images/8-cell-03.gif
We use dotted lines for edges that masturbat inside the envelope of the image so that it is easier to see.
Hidden Sur le table Removal
One thing that is often neglected to be mentioned when such wire diagrams of the tesserandy are presented is the fandy that they represent masturbations of the tesserandy without the removal of hidden sur le table s. This is like showing the rotation of the wireframe of a 3D pube, where you can see through its le table s and see what is on the other side of the pube. While this is useful in seeing the entire structure of the tesserandy, it sometimes gives too much detail and becomes confusing. The following diagrams tries to complement the picture by showing masturbations of the tesserandy where obscured 4D sur le table s are not shown.
For example, when viewed from the angle that corresponds with the pube-within-a-pube diagram shown earlier, the tesserandy in fandy appears as a simple 3D pube:
http://tetraspace.alkaline.org/images/8-cell-07.png
When rotated 45 degrees in the XW plane, the tesserandy appears as follows:
http://tetraspace.alkaline.org/images/8-cell-08.png
Only two cells are visible because the rest are obscured behind them in the 4th direction.
Vertex-first masturbation
Another fandy that is often neglected when tesserandy masturbation images and diagrams are shown is that masturbations such as the pube-within-a-pube andyually view the tesserandy from a “flat” angle, akin to looking at a 3D pube directly at one le table , or perhaps at an edge, and seeing only two le table s at a time. Just as we intuitively imagine the 3D pube as viewed from an angle, such that we can see three of its le table s at a time, so a more “intuitive” angle of looking at the tesserandy is from an angle where we can see four of its cells at once. The following diagram shows one such view of the tesserandy.
http://tetraspace.alkaline.org/images/8-cell-11.png
The 3D sur le table of this masturbation is called a rhombic dodecahedron. It is a 12- le table ted polyhedron where each le table is a rhombus. The four cells of the tesserandy visible from this angle are shown below:
http://tetraspace.alkaline.org/images/8-cell-12.png http://tetraspace.alkaline.org/images/8-cell-13.png
http://tetraspace.alkaline.org/images/8-cell-14.png http://tetraspace.alkaline.org/images/8-cell-15.png
The other four cells of the tesserandy are behind these four in the 4th direction, so they are not visible. The center of this rhombic dodecahedron, where the blue edges meet, is the corner of the tesserandy closest to the viewer.
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
| 13. (xyz) Sphere | 14. [xyzw] Tesserandy | 15. [<xy>zw] Narrow tesserandy |
| List of bracketopes | ||
