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?

Simply stated, a Hypercube is an n-dimensional regular polytope with mutually perpendicular sides. Moreover, it is denoted and has Schläfli symbol .

It may also interest the reader to know, that the number of k-cubes contained in an n-cube can be easily found from the coefficients of , 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.

The implications have been left to the reader as an exercise.

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.

The unfolded Hypercube (and the authors pictured, right):

The authors demonstrate the startling nature of the fourth dimension.

Assiduously following the schematic, folding begins.

And finally.....

A hypercube in its properly-folded end-state.



Born, M. Problems of Atomic Dynamics. Cambridge, MA: MIT Press, 1926.

Carr, J.R. and Palmer, J. A. Revisiting the accurate calculation of block-sample covariances using gauss quadrature. Mathematical Geology, 25(5):507-524, 1993.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 123, 1973.

Christensen, R. Plane answers to complex questions: the theory of linear models. Springer-Verlag, New York, second edition, 1996.
Czepa, A. Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918, 140 pages.

Dewdney, A. K. "Computer Recreations: A Program for Rotating Hypercubes Induces Four-Dimensional Dementia." Sci. Amer. 254, 14-23, Mar. 1986.

Fischer, G. (Ed.). Plates 3-4 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 4-5, 1986.

Gardner, M. "Hypercubes." Ch. 4 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 41-54, 1977.

Ingleton, A. W. (1977). Transversal matroids and related structures. In Higher combinatorics (ed. Aigner, M.), pp. 117-131. Reidel, Dordrecht.

Inukai, T. and Weinburg, L. (1981). Whitney connectivity of matroids. SIAM J. Alg. Disc. Methods 2, 108-120.

Jaeger, F., Vertigan D. L. and Welsh, D. J. A. (1990). On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Phil. Soc. 108, 35-53.

Jensen, P. M. (1978). Binary fundamental matroids. In Algebraic methods in graph theory (eds. Lovasz, L. and Sos, V. T.), Colloq. Math. Soc. Janos Bolyai 25, pp. 281-296. North-Holland, Amsterdam.

Kahn, J. (1988). On the uniqueness of matroid representations over GF(4). Bull. London Math. Soc. 20, 5-10.

Kelly, D. and Rota, G. -C. (1973). Some problems in combinatorial geometry. In A survey of combinatorial theory (eds. Srivastava, J. N. et al), pp. 309-312. North-Holland, Amsterdam.

Kingan, S. R. (1994). Structural results for binary matroids. Ph. D. Thesis, Louisiana State University.

Kingan, S. R. (1996). Binary matroids without prisms, prism duals and cubes. Discrete Math. 152, 211-224.

Kingan, S. R. (1996). On binary matroids with a K33-minor. In Matroid theory: Proceedings of the 1995 AMS-IMS-SIAM Joint Summer Research Conference (eds. Bonin, J., Oxley, J. G. and Servatius, B.), American Mathematical Society, Providence, RI.

Kingan, S. R. and Oxley, J. G. (1996). On the matroids in which all hyperplanes are binary. Discrete Math. 160, 265-271.

Kingan, S. R. (1997). A generalization of a graph result of D. W. Hall. Discrete Math. 173, 129-135.

Kingan, S. R. (1999). On the intersections of circuits and cocircuits in binary matroids. Discrete Math. 195, 157-165.

Kirkpatrick, P. B. (1975). On homologies in finite combinatorial geometries. Bull. Austral. Math. Soc. 13, no. 1, 85-99.

Klee, V. (1971). The greedy algorithm for finitary and cofinitary matroids. In Combinatorics (Proc. Sympos. Pure Math., Vol XIX, Univ. California, Los Angeles, CA, 1968), pp. 137-152. American Mathematical Society, Providence, RI.

Kuratowski, K. (1930). Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 271-283.

Löfgren, L. (1959). Irredundant and redundant boolean branch-networks. IRE Transactions on Circuit Theory CT-6, Special Supplement 158-175.

Lageweg, B. J. (1973). An algorithm for a maximum weighted common partial transversal. Mathematisch Centrum, Afdeling Mathematische Besliskunde, BW 25/73. Mathematisch Centrum, Amsterdam.

Lang, S. (1965). Algebra. Addison-Wesley, Reading, MA.

Las Vergnas, M. (1970). Sur un théoréme de Rado. C. R. Acad. Sci. Paris Sér. A-B 270, A733-A735.

Las Vergnas, M. (1970). Sur la dualité en théorie des matroides. C.R. Acad. Sci. Paris Sér. A-B 270, A804-A806.

Las Vergnas, M. (1970). Sur les systemes de representants distincts d'une famille d'ensembles. C. R. Acad. Sci. Paris Ser. A-B 270, A501-A503.

Las Vergnas, M. (1971). Sur la dualité en théorie des matroïdes. In Théorie des matroïdes (Rencontre Franco-Britannique, Brest, 1970), Lecture Notes in Math., Vol. 211, pp. 67-85. Springer, Berlin.

Las Vergnas, M. (1975). Matroides orientables. (English summary) C. R. Acad. Sci. Paris Sér. A-B 280, A61-A64.

Las Vergnas, M. (1975). Sur les extensions principales d'un matroide. C. R. Acad. Sci. Paris Sér. A-B 280, A187-A190.

Lesieur, L. (1970). Géométries combinatories. Enseignement Math. (2) 16, 185-193.

Lewin, M. (1970). Essential coverings of matrices. Proc. Camb. Phil. Soc. 67, 263-267.

Li, Weixuan (1983). On matroids of the greatest W-connectivity. J. Combin. Theory Ser. B 35, 20-27.

Lindström, B. (1973). On the vector representation of induced matroids. Bull. London Math. Soc. 5, 85-90.

Lindström, B. (1983). The non-Pappus matroid is algebraic. Ars Combinatoria 16B, 95-96.

Lindström, B. (1984). On binary identically self-dual matroids. European J. Combin. 5, 55-58.

Lindström, B. (1984). A simple non-algebraic matroid of rank three. Utilitas Math. 25, 95-97.

Lindström, B. (1985). A desarguesian theorem for algebraic combinatorial geometries. Combinatorica. 5, 237-239.

Lindström, B. (1985). On the algebraic characteristic set for a class of matroids. Proc. Amer. Math. Soc. 95, 147-151.

Lindström, B. (1985). On the algebraic representations of dual matroids. Dept. of Math., Univ. of Stockholm, Reports, No. 5.

Lindström, B. (1985). More on algebraic representations of matroids. Dept. of Math., Univ. of Stockholm, Reports, No. 10.

Lindström, B. (1986). A non-linear algebraic matroid with infinite characteristic set. Discrete Math. 59, 319-320.

Lindström, B. (1986). The non-Papus matroid is algebraic over any finite field. Utilitas Math. 30, 53-55.

Lindström, B. (1987). A class of non-algebraic matroids of rank three. Geom. Dedicata 23, 255-258.

Lindström, B. (1987). A reduction of algebraic representation of matroids. Proc. Amer. Math. Soc. 100, 388-389.

Lindström, B. (1987). An elementary proof in matroid theory using Tutte's coordinatization theorem. Utilitas Math. 31, 189-190.

Lindström, B. (1988). Matroids, algebraic and non-algebraic. In Algebraic, extremal and metric combinatorics (1986) (eds. Deza, M. -M. et al), London Math. Soc. Lecture Notes, 131, pp. 166-174. Cambridge University press, Cambridge.

Lindström, B. (1988). A generalization of the Ingleton-Main lemma and a class of non-algebraic matroids. Combinatorica 8, 87-90. Lindström, B. (1989). Matroids algebraic over F(t) are algebraic over F. Combinatorica 9, 107-109.

Lomonosov, M. V. (1974). A Bernoulli scheme with closure. (Russian) Problemy Peredaci Informacii 10, no. 1, 91-101.

Lorea, M. (1975). Hypergraphs et matroides. Colloque sur las Théorie des Graphes (Paris, 1974). Cahiers Center Études Recherche Opér. 17, no. 2-3-4, 289-291.

Lovász, L. (1972). A brief survey of matroid theory. Mat. Lapok 22, 249-267.

Lovász, L. (1977). Matroids and geometric graphs. In Combinatorial surveys: Proceedings of the sixth British combinatorial conference (ed. Cameron, P. J.), pp. 45-86. Academic Press, London.

Lovász, L. and Plummer, M. D (1986). Matching theory. North-Holland, Amsterdam.

Lovász, L. and Recski, A. (1973). On the sum of matroids. Acta Math. Acad. Sci. Hungar. 24, 329-333.

Lucas, D. (1974). Properties of rank preserving weak maps. Bull. Amer. Math. Soc. 80, 127-131.

Lucas, D. (1975). Weak maps of combinatorial geometries. Trans. Amer. Math. Soc. 206, 247-279.

Pappas, T. "How Many Dimensions Are There?" The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 204-205, 1989. Skiena, S. "Hypercubes." §4.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 148-150, 1990.

Sloane, N. J. A. Sequences A000079/M1129, A001787/M3444, A001788/M4161, A001789/M4522, and A091159 in "The On-Line Encyclopedia of Integer Sequences." Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004.

Trott, M. "The Mathematica Guidebooks Additional Material: Hypercube Projections."

Turney, P. D. "Unfolding the Tesseract." J. Recr. Math. 17, No. 1, 1-16, 1984-85.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 113-114 and 210, 1991.

Wilker, J. B. "An Extremum Problem for Hypercubes." J. Geom. 55, 174-181, 1996. Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979.

produced at the
The South Philadelphia Athenæum