On Computer Modeling of Finite-generated Free Projective Planes
Keywords:
free projective planes, finite geometries, combinatorial design
Abstract
This paper treats computer modeling of the process of constructing free projective planes — more precisely, to algorithmically finding their successive incidence matrices; and also to considering some numerical characteristics of these matrices. Matrix and bilinear forms approaches are used to study the growth rate of the number of new elements (points, lines) during step-by-step process of constructing projective plane starting with the Hall Pi^4 configuration. It appears that the number of new elements grows asymptotically as a double exponent (linear on log(log) scale.) Rough estimate from above also gives double exponential growth rate.
References
1. W.W. Sawyer, Prelude to Mathematics, Penguin Books, 1957.
2. “Combinatorial design” in Wikipedia [online]; https://en.wikipedia.org/wiki/Combinatorial_design
3. M. Hall, “Protective planes”, Trans. Amer. Math. Soc., no. 54, pp. 229–277, 1943.
4. M. Hall, The Theory of Groups, NY, 1959.
5. L. I. Kopejkina, “Decomposition of Protective Planes”, Bull. Acad. Sci. USSR Ser. Math. Izvestia Akad. Nauk SSSR, no. 9, pp. 495–526, 1945.
6. R. Hartshorne, Protective Planes. Lecture Notes Harvard University, NY, 1967.
7. L. C. Siebenmann, “A Characterization of Free Projective Planes”, Pac. J. of Math., vol. 15, no. 1, pp. 293–298, 1965.
8. R. Sandler, “The Collineation Groups of Free Planes”, Trans. Amer. Math. Soc., no. 107, pp. 129–139, 1963.
9. Hang Kim Ki, F. W. Roush, “A Universal Algebra Approach to Free Projective Planes”, Aequationes Mathematicae University of Waterloo, no. 18, pp. 389–400, 1978.
10. A. I. Shirshov and A. A. Nikitin, Algrbraic Theory of Projective Planes, Novosibirsk, USSR: 1987.
11. H. J. Ryser, Combinatorial Mathematics, The Mathematical Association of America, 1963.
12. F. Karteszi, Introduction to Finite Geometries, Budapest, Hungary: Akademia Klado, 1976.
13. M. M. Postnikov, Lectures in Geometry, Semester 1: Analytic Geometry, Moscow, USSR: Mir Publishers, 1982.
14. M. Berger, Geometry I, Springer, 2009.
15. E. Casas-Alvero, Analytic Projective Geometry, European Mathematical Society, 2014.
2. “Combinatorial design” in Wikipedia [online]; https://en.wikipedia.org/wiki/Combinatorial_design
3. M. Hall, “Protective planes”, Trans. Amer. Math. Soc., no. 54, pp. 229–277, 1943.
4. M. Hall, The Theory of Groups, NY, 1959.
5. L. I. Kopejkina, “Decomposition of Protective Planes”, Bull. Acad. Sci. USSR Ser. Math. Izvestia Akad. Nauk SSSR, no. 9, pp. 495–526, 1945.
6. R. Hartshorne, Protective Planes. Lecture Notes Harvard University, NY, 1967.
7. L. C. Siebenmann, “A Characterization of Free Projective Planes”, Pac. J. of Math., vol. 15, no. 1, pp. 293–298, 1965.
8. R. Sandler, “The Collineation Groups of Free Planes”, Trans. Amer. Math. Soc., no. 107, pp. 129–139, 1963.
9. Hang Kim Ki, F. W. Roush, “A Universal Algebra Approach to Free Projective Planes”, Aequationes Mathematicae University of Waterloo, no. 18, pp. 389–400, 1978.
10. A. I. Shirshov and A. A. Nikitin, Algrbraic Theory of Projective Planes, Novosibirsk, USSR: 1987.
11. H. J. Ryser, Combinatorial Mathematics, The Mathematical Association of America, 1963.
12. F. Karteszi, Introduction to Finite Geometries, Budapest, Hungary: Akademia Klado, 1976.
13. M. M. Postnikov, Lectures in Geometry, Semester 1: Analytic Geometry, Moscow, USSR: Mir Publishers, 1982.
14. M. Berger, Geometry I, Springer, 2009.
15. E. Casas-Alvero, Analytic Projective Geometry, European Mathematical Society, 2014.
Published
2017-07-20
How to Cite
Гогин, Н. Д., & Мюлляри, А. А. (2017). On Computer Modeling of Finite-generated Free Projective Planes. Computer Tools in Education, (4), 14-28. Retrieved from http://cte.eltech.ru/ojs/index.php/kio/article/view/1501
Issue
Section
Algorithmic mathematics and mathematical modelling
This work is licensed under a Creative Commons Attribution 4.0 International License.
