Advertisement

Annali di Matematica Pura ed Applicata

, Volume 53, Issue 1, pp 45–55 | Cite as

Isodynamic & isogonic simplexes

  • Sahib Ram Mandan
Article

Summary

A simplex in an n-space and its tangential simplex formed of the n+1 tangent hyperplanes of its circumhypersphere (0) at its vertices are polar reciprocal of each other w. r. t. (0). The n+1 joins of their corresponding vertices, in general, do not concur [3, p.41, Ex.7;4;18;24;25]. But when n=2, the3 joins of the corresponding vertices of a triangle and its tangential triangle always concur at itsLemoine point L as its3 symmedians which are the isogonal conjugates of its medians w. r. it [7] such that its circumcircle coincides with the polar conic of L w. r. t. it [9]. For n=3, the4 joins of the corresponding vertices of a tetrahedron and its tangential tetrahedron concur at itsLemoine point L, if and only if it is isodynamic [5], as its4 Lemoinians [17] (called symmedians byCourt [6] which join its vertices to theLemoine points of its opposite faces (but not as the isogonal conjugates of its medians w. r. t. it) such that its circumsphere coincides with the polar quadric [16] of L w. r. t. it. The purpose of this paper is to develope analogonsly the theory of an isodynamic simplex, in an n-space (n>3), which is in rerspective with its tangential simplex, called isogonic under the circumstances. Their relationship as cevian & anticevian simplexes, and their association with S-configurations and related cevian quadrics are also pointed out.

Keywords

Polar Conic Opposite Face Tangent Hyperplane Polar Quadric Isogonal Conjugate 
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.

References

  1. [1]
    H. F. Baker,Principles of geometry, vol. 4, Cambridge, 1925.Google Scholar
  2. [2]
    -- --,Principles of geometry, vol. 2, Cambridge, 1930.Google Scholar
  3. [3]
    -- --,Principles of geometry, vol. 3, Cambridge, 1934.Google Scholar
  4. [4]
    —— ——,Polarities for the nodes of a Segre cubic primal in space of four dimensions, « Proc. Camb. Phil. Soc. » vol. 32 (1936), pp. 507–520.zbMATHCrossRefGoogle Scholar
  5. [5]
    N. A. Court,Sur le tétraedre isodynamique, « Mathesis » vol. 49 (1935), pp. 345–351zbMATHGoogle Scholar
  6. [6]
    -- --,Modern pure solid geometry, New York, 1935.Google Scholar
  7. [7]
    -- --,College geometry, New York, 1952.Google Scholar
  8. [8]
    —— ——,Sur les tétraedres circonscrits per les aretes a une quadrìque, « Mathesis », vol. 63 (1954), pp. 12–18.zbMATHMathSciNetGoogle Scholar
  9. [9]
    H. S. M. Coxeter,The real projective plane, Cambridge, 1955.Google Scholar
  10. [10]
    S. R. Mandan,Properties of mutually self-polar tetrahedra, « Bul. Cal. Math. Soc. », vol. 33 (1941), pp. 147–155.zbMATHMathSciNetGoogle Scholar
  11. [11]
    —— ——,Umbilical projection in four dimensional space, Proc. Ind. Acad. Sci. », vol. A 28 (1948), pp. 166–172.MathSciNetGoogle Scholar
  12. [12]
    —— ——,An S-configuration in Euclidean & elliptic n-space, « Can. J. Math. », vol. 10 (1958), pp. 489–501.zbMATHMathSciNetGoogle Scholar
  13. [13]
    —— ——,Harmonic inversion, « Math. Mag. » vol. 32 (1959), pp. 71–78.MathSciNetCrossRefGoogle Scholar
  14. [14]
    -- --,On four intersect ng spheres, « Jour. Ind. Math. Sec. », vol. 24 (1959).Google Scholar
  15. [15]
    —— ——,Semi-orthocentric & orthogonal simplexes in a 4-space, « Bul. Cal. Math. Soc. », vol. 52 (1960) pp. 21–29.MathSciNetGoogle Scholar
  16. [16]
    —— ——,Cevian simplexes, « Proc. Amer. Math. Soc. », vol. 11 (1960), pp. 837–845.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    —— ——,Semi-isodynamic & isogonic tetrahedra, « Rend. Mat. e delle sue Appl. », vol. 19 (1960), pp. 401–415.MathSciNetGoogle Scholar
  18. [18]
    —— ——,Polarity for a quadric in an n-space, « Rev. Faculty of Sciences of University of Istanbul », vol. 24 (1960), pp. 21–38.zbMATHMathSciNetGoogle Scholar
  19. [19]
    -- --,Polarity for a simplex, « Rend. Circ. Mat. Pal », (to appear).Google Scholar
  20. [20]
    -- --,Semi-inverse simplexes, (to appear).Google Scholar
  21. [21]
    -- --,On n+1 intersecting hyperspheres in an n-space, « Jour. Australian Math. Soc. », (to appear).Google Scholar
  22. [22]
    -- --,Altitudes of a simplex in an n-space, Ibid.Google Scholar
  23. [23]
    -- --,Orthogonal hyperspheres, « Acta Math. Acad. Sci. Hungang ». (to appear).Google Scholar
  24. [24]
    -- --,A porism on 2n+5 hyperspheres in an n-space, ibid.Google Scholar
  25. [25]
    J. A. Todd &H. S. M. Coxeter,Solution of an advanced Problem of S. Beatty, « Amer. Math. Mo. », vol. 51 (1944) pp. 599–600.CrossRefGoogle Scholar

Copyright information

© Swets & Zeitlinger B. V. 1961

Authors and Affiliations

  • Sahib Ram Mandan
    • 1
  1. 1.KharagpurIndia

Personalised recommendations