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Area in Grassmann geometry

  • Desmond Fearnley-Sander
  • Tim Stokes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1360)

Abstract

In this survey paper we give the basic properties of Grassmann algebras, present a generalised theory of area from a Grassmann algebra perspective, present a version for Grassmann algebras of the Buchberger algorithm, and give examples of computation and deduction in Grassmann geometry.

Keywords

Grassmann Algebra Geometry Theorem Oriented Area Mechanical Theorem Prove Buchberger Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Apel, J., A relationship between Gröbner bases of ideals and vector modules of G-algebras, Contemporary Mathematics 131 (1992), 195–204.Google Scholar
  2. 2.
    Buchberger, B., Gröbner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory, ed. Bose, N. K., Reidel, 1985, 184–232.Google Scholar
  3. 3.
    Chou, S., Proving elementary geometry theorems using Wu's algorithm, in Automated Theorem Proving: After 25 Years, ed. Bledsoe, W. W., and Loveland, D. W., Am. Math. Soc., 1984, 243–286.Google Scholar
  4. 4.
    Chou, S., Mechanical Geometry Theorem Proving, Reidel, 1988.Google Scholar
  5. 5.
    Chou, S., Gao, X., and Zhang, J., Automated geometry theorem proving using vector calculation, in Proc. ISSAC (Kiev, 1993), 284–291.Google Scholar
  6. 6.
    Chou, S., Gao, X., and Zhang, J., Automated production of traditional proofs for constructive geometry theorems, in Proc. 8th IEEE Symbolic Logic in Computer Science (1993), 48–56.Google Scholar
  7. 7.
    Chou, S., Gao, X., and Zhang, J., Automated production of traditional proofs in solid geometry, J. Ant. Reas. 14, 257–291.Google Scholar
  8. 8.
    Coxeter, H. S. M., and Greitzer, G., Geometry Revisited, Math. Ass. Am., 1967.Google Scholar
  9. 9.
    Doubilet, P., Rota, G., and Stein, J., On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory, Studies in Applied Math. 57 (1974), 185–216.Google Scholar
  10. 10.
    Fearnley-Sander, D., Hermann Grassmann and the creation of linear algebra, Am. Math. Monthly 86 (1979), 809–817.Google Scholar
  11. 11.
    Fearnley-Sander, D., Affine geometry and exterior algebra, Houston J. Math. 6 (1980), 53–58.Google Scholar
  12. 12.
    Fearnley-Sander, D., Hermann Grassmann and the prehistory of universal algebra, Am. Math. Monthly 89 (1982), 161–166.Google Scholar
  13. 13.
    Fearnley-Sander, D., The Idea of a Diagram, in Resolution of Equations in Algebraic Structures, ed. Ait-Kaçi, H., and Nivat, M., Academic Press 1989, 127–150.Google Scholar
  14. 14.
    Feynman, R. P., Leighton, R. B., and Sands, M., Lectures on Physics, Addison-Wesley, 1963.Google Scholar
  15. 15.
    Grassmann, H. G., Gesammelte Mathematische and Physikalische Werke, ed. Engel, F., 3 vols. in 6 parts, Leipzig, 1894–1911.Google Scholar
  16. 16.
    Hestenes, D., New Foundations for Classical Mechanics, Kluwer, 1986.Google Scholar
  17. 17.
    Hestenes, D., and Ziegler, R., Projective geometry with Clifford algebra, Acta. Appl. Math. 23 (1991).Google Scholar
  18. 18.
    Hong, H., Wang, D., and Winkler, F. (eds.), Algebraic approaches to geometric reasoning, Ann. Math. and AI 13 (1, 2) (1995).Google Scholar
  19. 19.
    Hungerford, T. W., Algebra, Holt, Rinehart and Winston, 1974.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Desmond Fearnley-Sander
    • 1
  • Tim Stokes
    • 2
  1. 1.Department of MathematicsUniversity of TasmaniaHobartAustralia
  2. 2.School of Mathematical and Physical SciencesMurdoch UniversityMurdochAustralia

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