Cybernetics of Epistemology - Understanding Understanding: Essays on Cybernetics and Cognition - page 239

FIGURE 6. Step by step construction of a 4-D cube (Tesser-
act) by folding seven 3-D cubes (cubes) out of the source
space (space) around their boundaries (edges). Compare
Fig. 6a with Salvador Dali's painting “Crucifixion (Corpus
Hypercubus)” in the New York metropolitan Museum as,
for instance, reproduced in. (17)
(iii) For completion, fold the lid cube around its connected boundary so
that all unconnected boundaries fall into place.
In Figures 4a and 4b, injunction (i) is carried out for the construction of
a two-dimensional “cube” (square) from four one-dimensional “cubes”
(line segments); 4c and 4d realize injunction (ii); and the last step of con-
structing this structure by closing the lid is depicted in 4d and 4e.
For the construction of three- and four-dimensional cubes, the three nec-
essary steps are sketched in Figures 5 and 6. It is clear that perpendicular
projections of such structures into the spaces from where they come do not
offer anything new (perpendicular projections of a 3D cube onto a plane
will show only a square.) Not until these structures are tilted and obliquely
projected do their higher manifolds become apparent.
This can be seen in Figure 7, where a Tesseract, with its base cube paral-
lel to our space is obliquely projected into 3-space. (To be appreciated by
a reader of this report, this projection, in turn, is projected onto the two-
dimensionality of the plane of the paper.)
Figures 8a, b, and c respectively show the stereoscopic pairs of Tesseract
(hypercube), the handed block from a Soma cube (a three-dimensional
handed figure which can be made to change handedness by a four-space
rotation), and a representation of a Klein bottle as two Mobius strips with
corresponding points connected through four-space.
These objects can be appreciated in their three-dimensionality without a
stereopticon when they are fixated with crossed eyes: fixate your index
finger held between eye and plane of the paper at about 12 inches away.
Observe the convergence of the two stereo pictures into one, then transfer
your gaze from finger to paper, maintaining the crossed eyes. A 3D struc-
ture will float before your very eyes!
The way in which our experiment was conducted, either the person in
charge of the experiment or the test-person can control, via two manipula-