A Locus with 25920 Linear Self-Transformations

Preface; Introduction; 1. The fundamental notation; 2. The equation of the Burkhardt primal; 3. Similarity, or equal standing, of the forty-five nodes, and of the twenty-seven pentahedra; 4. The Jacobian planes of the primal; 5. The k-lines of the primal; 6. The Burkhardt primal is rational; 7. The particular character of the forty-five nodes, and the linear transformation of the primal into itself by projection from the nodes; 8. The forty Steiner threefold spaces, or primes, belonging to the primal; 9. The plane common to two Steiner solids; 10. The enumeration of the twenty-seven Jordan pentahedra, and of the forty-five nodes, from the nodes in pairs of polar k-lines; 11. The reason for calling the Steiner tetrahedra by this name; 12. The enumeration of the twenty-seven pentahedra from nine nodes of the Burkhardt primal; 13. The equation of the Burkhardt primal in terms of a Steiner solid and four association primes; 14. Explicit formulae for the rationalization of the Burkhardt primal; 15. The equation of the Burkhardt primal referred to the prime faces of a Jordan pentahedron; 16. The thirty-six double sixes of Jordan pentahedra, and the associated quadrics; 17. The linear transformations of the Burkhardt primal into itself; 18. Five subgroups of the group 23.34.40 transformations; 19. The expression of the fundamental transformations B, C, D, S as transformations of x1,...,x6. The expression of B, C, D, S in terms of nodal projections; 20. The application of the substitutions of x1,...,x6 to the twelve pentahedra {A}, {B},..., {F0}; 21. The transformation of the family {A} by means of Burkhardt's transformations; 22. Derivation of the Burkhardt primal from a quadratic; Appendix, note 1. The generation of desmic systems of tetrahedra in ordinary space; Appendix, note 2. On the group of substitutions of the lines of a cubic surface in ordinary space; Index of notations.