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Operational Constraints on Dimension of Space Imply both Spacetime and Timespace

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Since polynomials of higher than fourth degree, which is the algebraic counterpart of generic geometric dimension, are insolvable in general, then presumably no more than just four mutually orthogonal geometric dimensions can be placed within a single geometric space if it is expected to be fully operational. Hence a hierarchical notion of dimension is needed in order to ensure that at least virtual orthogonality is respected, which in turn implies presence of certain hierarchically organized multispatial structures. It is shown that the operational constraint on physical spaces implies of necessity presence of both: 4-dimensional (4D) spacetime and a certain 4D timespace.


International Letters of Chemistry, Physics and Astronomy (Volume 36)
J. Czajko, "Operational Constraints on Dimension of Space Imply both Spacetime and Timespace", International Letters of Chemistry, Physics and Astronomy, Vol. 36, pp. 220-235, 2014
Online since:
July 2014

[1] Kendall M.G., A course in the geometry of n dimensions. New York: Hafner Publishing, 1961, p.2.

[2] Brouwer L.E.J., Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten II. Amsterdam, 1919, p. 3f.


[3] Hummel J.A., Vector geometry. Reading, MA: Addison-Wesley, 1965, p.95.

[4] Audin M., Geometry. Berlin: Springer, 2003, p.7.

[5] Schwarz J.H., Spacetime duality in string theory. [pp.69-87 in: Schwarz J.H. (Ed. ) Elementary particles and the universe. Cambridge: Cambridge Univ. Press, 1991].


[6] Witten E., Lectures on Quantum Field Theory. [pp.1225-1424 in: Deligne P. et al. Quantum Fields and Strings: A Course for Mathematicians II. Providence, RI: Am. Math. Soc., 1999, pp. 1225ff.

[7] Witten E., Phys. Today April 1996, 24-30.

[8] Czajko J., Chaos Solit. Fract. 20 (2004) 683-700.

[9] Czajko J., Chaos, Solit. Fract. 11 (2000) 2001-(2016).

[10] Czajko J., Stud. Math. Sci. 7(2) (2013) 25-39.

[11] Czajko J., Stud. Ma th. Sci. 7(2) (2013) 40-54.

[12] Czajko J., International Letters of Chemistry, Physics and Astronomy 11(2) (2014) 89-105.

[13] Czajko J., International Letters of Chemistry, Physics and Astronomy 13(1) (2014) 32-41.

[14] Czajko J., Chaos, Solit. Fract. 12 (2001) 951-967.

[15] Czajko J., Chaos, Solit. Fract. 19 (2004) 479-502.

[16] Thom R. Modelès mathématiques de la morphogenèse. Paris: Christian Burgois Éditeur, 1980, p.91.

[17] Thom R., Quid des stratifications canoniques. [pp.375-381 in: Brasselet J. -P. (Ed. ) Singularities. Lille 1991. Cambridge: Cambridge Univ. Press, 1994, see p.376].


[18] Toth G., Glimpses of algebra and geometry. New York: Springer, 1998, p.284.

[19] Efimov N.W., Höhere Geometrie. Berlin: VEB Deutscher Verlag der Wissenschaften, 1960, p.224.

[20] Cartan E., Leçons sur la géométrie des espaces de Riemann. Paris: Gauthier-Villars, 1963, p.11, 14ff.

[21] Cartan E., Théorie des groupes finis et continus et la géométrie différentielle. Traitées par la méthode du repère mobile. Paris: Gauthier-Villars, 1937, p. 9ff.


[22] Kaplan W., Elements of ordinary differential equations. Reading, MA: Addison-Wesley, 1964, p.169.

[23] Frazer R.A., Duncan W.J. & Collar A.R., Elementary matrices and some applications to dynamics and differential equations. Cambridge: Cambridge Univ. Press, 1960, p.64.

[24] Schneider H. & Barker D.P., Matrices and linear algebra. New York: Dover, 1989, p.217.

[25] Pettofrezzo A.J., Matrices and transformations. Englewood Cliffs, NJ: Prentice-Hall, 1966, p.97.

[26] Thrall R.M. & Tornheim L., Vector spaces and matrices. New York: Wiley, 1963, p.257.

[27] Stewart F.M., Introduction to linear algebra. Princeton, NJ: Van Nostrand, 1963, p.215.

[28] Halmos P.R., Austr. Math. Soc. 25 (1982) 161, see p.166.

[29] Gerdt V.P., Int. J. Mod. Phys. C 4(2) (1993) 279-286.

[30] Swallow J. Exploratory Galois theory. Cambridge: Cambridge Univ. Press, 2004, p.171.

[31] Bézout E., General theory of algebraic equations. Princeton, NJ: Princeton Univ. Press, 2006, p. 7ff.

[32] Tignol J. -P., Galois' theory of algebraic equations. New York: Longman, 1988, p.274.

[33] Jammer M., Concepts of space. The history of theories of space in physics. New York: Dover, 1993, p.154, 173, 177.

[34] Mansouri F., Witten L., Can isometries tell us about the extra dimensions? pp.509-511.

[35] Gullberg J., Mathematics. From the birth of numbers. New York: W.W. Norton, 1996, p.300.

[36] Weisstein E.W., CRC concise encyclopedia of mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2003, p.17.


[37] Cooke R., Classical algebra. Its nature, origins and uses. Hoboken, NJ: Wiley, 2008, p. 118f.

[38] Abel N.H., A demonstration of the impossibility of the algebraic resolution of general equations whose degree exceeds four. [pp.271-6 in Stahl S., Introductory modern algebra. A historical approach. New York: Wiley Interscience, 1997. ].

[39] Fraleigh J.B., A first course in abstract algebra. Reading, MA: Addison-Wesley, 1982, pp. 437ff.

[40] Cox D.A., Galois theory. Hoboken, NJ: Wiley, 2004, p.200.

[41] Dobbs D. & Hanks R., A modern course on the theory of equations. Washington, NJ: Polygonal Publishing, 1992, p.106.

[42] Houzel C., The work of Niels Henrik Abel. [pp.21-177 in: Laudal, O.A. & Piene, R. (Eds. ) The legacy of Niels Henrik Abel. The Abel bicentennial, Oslo, 2002. Berlin: Springer, 2004, p.50].


[43] Strang G., Linear algebra and its applications. San Diego: Harcourt Brace Jovanovich, Publishers, 1988, p.251.

[44] Joyner D., Adventures in group theory. Baltimore: The Johns Hopkins Univ. Press, 2002, p.72.

[45] Livio M., The equation that couldn't be solved. New York: Simon & Schuster, 2005, p.285.

[46] Halburd R., Nonlinearity 12 (1999) 931-938.

[47] Serret J. -A., Cours d'algèbre supérieure 2. Paris, 1928, p.512.

[48] Postnikov M.M., Fundamentals of Galois' theory. Delhi: Hindustan Publishing, 1961, p.137.

[49] Nicholson W.K., Introduction to abstract algebra. New York: Wiley, 1999, p.79, 512 & 111.

[50] Zeidler E. (Ed. ) Oxford user's guide to mathematics. Oxford: Oxford Univ. Press, 2004, p.676.

[51] Pontrjagin L., Topological groups. Princeton: Princeton Univ. Press, 1946, p. 171f.

[52] Hungerford T.W., Algebra. New York: Springer, 1974, p.116.

[53] Lagrange J.L., Zusätze zu Eulers Elemente der Algebra. Unbestimmte Analysis. Leipzig, 1898, p.81.

[54] Poincaré H., The value of science. New York: Dover, 1958, p.54.

[55] Duffin R.J., PNAS USA 78 (1981) 4661-4662.

[56] Stahl S., Introductory modern algebra. A historic approach. New York: Wiley, 1997, p.124.

[57] Netto E., The theory of substitutions and its applications to algebra. Bronx, MY: Chelsea, 1964, p.178, 275.

[58] Infeld L., Whom the Gods love. The story of Evariste Galois. New York: Whittlesey House, 1948, p.58.

[59] Weber H., Lehrbuch der Algebra I. New York: Chelsea, p.670.

[60] Marcus M. & Minc H., Introduction to linear algebra. New York: Macmillan, 1969, p.74.

[61] Hurewicz W., J. reine angew. Math. 169 (1933) 71.

[62] Hestenes D., J. Math. Phys. 16(3) (1975) 556-572 see p.557.

[63] Malone J.J., Proc. Edinb. Math. Soc. 18 (1973) 235 see p.238.

[64] Uspensky J.V., Theory of equations. New York: McGraw-Hill, 1948, p.82.

[65] Yano K., Les espaces a connexion projective et la géométrie projective des « paths.


[66] Perlis S., Theory of matrices. New York: Dover, 1991, p.179.

[67] Franklin J.N., Matrix theory. Mineola, NY: Dover, 2000, p.75.

[68] MacDuffee C.C., Vectors and matrices. Menasha, WI: The Math. Assoc. of America, 1949, p.23.

[69] Tetra B.C., Basic linear algebra. New York: Harper & Row, 1971, p.75.

[70] Dehn E., Algebraic equations. An introduction to the theories of Lagrange and Galois. New York: Columbia Univ. Press, 1938, p.155.

[71] Mal'cev A.I., Foundations of linear algebra. San Francisco: W.H. Freeman, 1963, p.276.

[72] Hladik J., Spinors in physics. New York: Springer-Verlag, 1999, p. 135ff.

[73] Grassmann H., Gesammlte mathematische und physikalische Abhandlungen I/I: Die Ausdehnunglehre von 1844 und die geometrische Analyse. New York: Johnson Reprint Corp., 1972, p. 13ff.

[74] Kline M., Mathematical thought from ancient to modern times II. New York: Oxford Univ. Press, 1990, p.784.

[75] Pettofrezzo A.J., Vectors and their applications. Englewood Cliffs, NJ: Prentice-Hall, 1966, p.70.

[76] Grassmann H., Die Ausdehnungslehre von 1844 oder die lineale Ausdehnungslehre. Leipzig, 1878, p.73.

[77] Lorenz K. (Ed. ) Konstruktionen versus Positionen. Beiträge zur Diskussion um die Konstruktive Wissenschaftstheorie. Berlin: Walter de Gruyter, 1979, p.495.

[78] Czajko J., Chaos Solit. Fract. 13 (2002) 17-23.

[79] Lang S., Algebraic structures. Reading, MA: Addison-Weslley, 1967, p.13.

[80] Da Costa R.C.T., Eur. J. Phys. 7 (1986) 269-273.

[81] Da Costa R.C.T., Phys. Rev. A 25 (1982) 2893-2900.

[82] Poincaré H., Mathematics and science. Last essays. New York: Dover, 1963, p.17, 28.

[83] Halmos P.R., Austr. Math. Soc. 25 (1982) 161, see p.166.

[84] Konopleva N.P., Kaluza-Klein or fibre bundles? [pp.248-251 in: Barut A.O. et al. (Eds. ) Quantum systems. New trends and methods. Singapore: World Scientific, 1995, see p.249].

[85] Borštnik N. M., Nielsen H.B., Phys. Lett. B486 (2000) 314-321.

[86] Dedekind, R., J. reine angew. Math. 50 (1855) 272.

[87] Kolata G., Science 217 (1982) 432-433.

[88] Hausdorff F., Zwischen Chaos und Kosmos oder vom Ende der Mataphysik. Baden- Baden: Agis Verlag, 1976, p.103.

[89] Balaban A.T., Harary F., J. Chem. Docum. 11 (1971) 258.

[90] Artin E., Galois theory. Notre Dame, IN: 1959, p.4. ( Received 21 June 2014; accepted 29 June 2014 ).

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