The Magic Encyclopedia ™

The Hypercube Layout
(by Aale de Winkel)

The reader might be familiar with the layout of the faces of a cube as a big T. In this article I show the idea works for higher dimensional hypercubes as well. Just as the ends of the T's horizontal can be rotated down the vertical one can place squares anywhere as long as edges correspond to the adjecent square. Below is an attempt to keep the n-dimensional layout in tact in the n+1 dimensional layout. The wikipedia article Hypercube shows the relation:
Ek,n = 2n-k (nk) or
Ek,n = 2 Ek,n-1 + Ek-1,n-1 with E0,0 = 1
as the amount of k dimensional subhypercube (ie. vertex, edge, face, cube, tesseract,....) in the n dimensional hypercube

The Hypercube Layout
In order to define things more precisely, one might view the discussion on a order 2 n-dimensional hypercube
the 2n corners are numbered 0 to 2n-1. Starting with the point '0' each dimension k adds 2k-1
to each existing point, the entire figure thus gets copied, adding the edges x-2k+x

tesseract-penteract.JPG
Note: the reader be warned not to take the penteract as pictured above, it is merely a helpful 3-dimensional analog of the 'real' thing
showing an interpretation of the axes ; the various corners can be numbered as indicated ; the position in hyperspace won't be as
shown in any 2 dimensional picture
also: k-agonals x - 2k-1-x ; x=[0..2k-1] ; k=1..n
square layout of the square
The series start with the point and line but besides these two also the square is relatively trivial the 4 points simply
connect with the edges 0-1, 2-3 (the line and its copy) the new edges 0-2 and 1-3 are the dimensional additions
square.jpg
cube layout of the cube
The also familiar cube start with the square 0123 above and its copy the square 4567 the 22=4 added
the edges 0-4, 1-5, 2-6 and 3-7. Together with the square edges the faces 0246, 1357, 2367 and 0145 are added
cube.jpg
tesseract layout of the tesseract
Going from the cube to the tesseract a 23=8 is added to each corner, though thus the square 0123 is
connected to the square 89AB, one need to connect the original top face with the copies backface to be able to
place all the new faces in a connecting manner into a layout picture
tesseract.jpg
penteract layout of the penteract
mirroring the tesseract into its new copy one can trace the tesseract layouts central row to form the new faces
between the tesseract and its copy. The 16 remaining squares could be placed entirely on the original tesseract
leaving its copy entirely bare of new layout faces, which is of cause merely a placement choice
tesseract-penteract.jpg
This list might be continued further. Just as with the familiar cube one can change layouts by placing squares at other places
as long as edges correspond with the adjecent edge. As interpretion of things in more than 3 dimension is based on its math folding
the central vertical and horizontal together one can experience a kind of beam inside the penteract, tracing the beam in the picture
above one sees how this beam bents. The real 'beam' probably forms an even weirder path though hyperspace.