How to Build a Hypercube |
By Brandon A. Joyce and Richard L. Davis
How to Build a Hypercube |
By Brandon A. Joyce and Richard L. Davis
|
In endeavoring to fold a true 4-dimensional hypercube, we must first, as might be expected, concern ourselves with questions of mathematical definition. The foremost among them being, What precisely is a Hypercube?
With our mathematical definitions fixed and clearly explicated, we may, at last, get on with the actual construction of our Hypercube. In this exercise, we shall limit ourselves to but four dimensions, only later to generalize our Method to the fifth, sixth, and even seventh dimensions. Much ink has been spilled and much time spent, on the consideration of Hypercubes. Notwithstanding, we regret to say that no progress has been made, among the community of scientific men, in producing a real Hypercube. That is because Natural Philosophy, so long mired in the muck of empiricism, has made little to no progress in the last three to four centuries since the publication of Boyle’s decadent A Free Enquiry into the Vulgarly Received Notion of Nature. It is our task then, here and elsewhere, to place Natural Philosophy back on the secure path of rational speculation and self-evident ideas. Once the thinking man is freed of his positivistic shackles, and no longer led astray by the seduction of language and the coarseness of his senses, only then is he prepared to rightly conceive of the fourth dimension. Many of our contemporaries have failed in this regard. I beg our audience to consider what would prompt such great men of distinction to dismiss Hypercubes as an outright impossibility? Again the senses. What, then, is to save us from such dissimilation? How are we to guarantee for ourselves the success that other great geometers have won so definitively and demonstrably throughout the ages? The answer is, of course, by recourse to Method. It seems obvious that if we, in the manner Euclid or Descartes, were to advance only by rigorous Method, placing aside all other commonplaces and never once straying from direct implication, we cannot help but to achieve our intended result. We know full well that the great majority of scientific men will not believe that such results can be practically and immediately realized, but this tract shall prove otherwise; indeed provide photographical evidence of the Hypercube’s properly-folded end-state. After considerable reflection, we determined that the most sure-footed Method available to us was the Method of Folding, first pioneered, though never executed, by R. G. Hollingdale. Observe: Below we have a three dimensional cube unfolded into two dimensional space, a procedure easily completed by a young schoolchild.  It follows that, to successfully fold a Hypercube, we should adopt the selfsame Method; beginning with it unfolded into three dimensions, as shown diagrammatically below.  However, folding is no more complicated than bringing end to proper end, as indicated in the following schematic.  Faces must also be aligned correctly  Sturdy materials are gathered and assembled.  
Initial construction of the unfolded Hypercube.  
 
 Assiduously following the schematic, folding begins.
      
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and has Schläfli symbol 
.
, namely 
, where
is a binomial coefficient. The number of nodes in the n-Hypercube is therefore 2 to the n, the number of edges is 
, the number of squares is 
, the number of cubes is 
, et cetera.
