Aerochoron (EntityTopic, 15)

From Hi.gher. Space

Revision as of 19:07, 14 March 2008

The 16-cell is the dual of the tesseract. It is equivalent to the tesseract with vertices and cells exchanged. This is analogous with how the octahedron is the dual of the cube.

Geometry

Equations

• Variables:

l ⇒ length of the edges of the hexadecachoron

• All points (x, y, z, w) that lie on the surcell of a hexadecachoron will satisfy the following equation:

Unknown

• The hypervolumes of a hexadecachoron are given by:

total edge length = 24l
total surface area = 8sqrt(3)l²
surcell volume = 4sqrt(2)l³3^-1
bulk = Unknown

• The realmic cross-sections (n) of a hexadecachoron are:

Unknown

Projection

Vertex-first Projection

The vertex-first projection of the 16-cell is perhaps easiest to understand. It has an octahedral envelope, with the closest vertex lying at the center of the octahedron, with 8 tetrahedral cells surrounding it. The other 8 cells lie “behind” these 8 cells in the W direction, and meet at the opposite vertex.

http://tetraspace.alkaline.org/images/16-cell-01.png

Cell-first Projection

The cell-first projection of the 16-cell has a cubical envelope. This projection is interesting in that all the edges of the 16-cell project onto the edges of the cube and the diagonals on the faces of the cube, forming a wireframe of the cube with each face crossed. It corresponds with the two possible ways to inscribe a tetrahedron inside a cube such that the edges of the tetrahedron lie on the faces of the cube.

http://tetraspace.alkaline.org/images/16-cell-02.png

The blue edges outline the cell closest to the viewer. Between this cell and the cubical envelope are 4 tetrahedral volumes that correspond with the 4 cells surrounding this blue cell. There are 6 cells that project onto the 6 faces of the cubical envelope (not shown here because they are being viewed edge-on). These 6 cells connect with the other side of the 16-cell, which contains the remaining 5 cells in a dual arrangement to the cells seen here. The following figure shows this opposite cell:

http://tetraspace.alkaline.org/images/16-cell-03.png

4D is unique in that the cell-first projection of its hypercube, which is a 3D cube, has the same envelope as the cell-first projection of its cross polytope. Only 2D has the same phenomenon, although in 2D the hypercube and the cross polytope are in fact the same object (the square).

Construction

The 16-cell is so called because it consists of 16 tetrahedra joined 4 to an edge. This method of thinking about the 16-cell, however, can be difficult to visualize. It is relatively easy to understand how 4 tetrahedra sharing a common edge can be “bent” into 4D so that they each share a face with two other tetrahedra. But it is not quite so simple to imagine how 4 sets of tetrahedra attached thus can be assembled into a 16-cell.

Another method of constructing the 16-cell, which may be a bit easier to understand, is to begin with a 3D octahedron and tapering it in both the positive and negative directions along the W-axis. Consider the first case. The process of tapering is simply to stack progressively smaller octahedra on top of each other along the W-axis, until they have vanished to a single point. The tapered object is the trace formed by this process. As we taper the octahedron along the positive W-axis, each of its 8 triangular faces also grow smaller, stacking on top of the larger triangles behind it. At the end of the taper, the triangles vanish into a common apex. Thus, the tapering process forms 8 tetrahedral cells bent into the positive W-axis. This forms half of the 16-cell, with the apex being the +W vertex of the 16-cell. Now we repeat the same process in the negative W-axis, and we get another 8 tetrahedral cells in exactly the same formation, except that they are now bent in the negative W-axis, and form another apex on the opposite point, which is the -W vertex of the 16-cell. Joining these two halves together at their octahedral base, we obtain the entire 16-cell. The remaining 6 vertices of the 16-cell are precisely the 6 vertices of the octahedron we started out with.

With this understanding of the 16-cell, it is easy to see why it is the 4D equivalent of the octahedron. Just as the 2D square diamond is a 1D line segment tapered in the positive and negative Y-axis, and the octahedron is the 2D square diamond tapered in the positive and negative Z-axis, the 16-cell is the octahedron tapered in the positive and negative W-axis.