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Affine planes: An example of research on geometric structures

Published online by Cambridge University Press:  03 November 2016

Günter Pickert
Günter Pickert*
Affiliation:
Mathematisches Institut, Justus Liebig-Universität, 6300 Giessen, Federal Republic of Germany
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Extract

At the end of the nineteenth century David Hilbert made the decisive step in the foundations of geometry (for the historical development see [1]): a road was opened to a theory of geometric structures. In spite of the fact that one spoke (and speaks still today!) strangely enough of ‘implicitly’ defining the geometrical concepts through the axioms, it became clear in the sequel that the axiom system of Hilbert constitutes an ‘explicit’ definition: not of course for ‘point’, ‘line’, etc. but of the concept ‘euclidean geometry’ (or ‘euclidean space’).

Type
Research Article
Information
The Mathematical Gazette , Volume 57 , Issue 402 , December 1973 , pp. 278 - 291
Copyright
Copyright © Mathematical Association 1973

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References

1. Freudenthal, H., The main trends in the foundations of geometry in the 19th century. Logic, Methodology and Philosophy of Science, Proc. of the 1960 International Congress (Stanford, 1962).Google Scholar
2. Dembowski, P., Finite geometries (Berlin, Heidelberg, New York, 1968).Google Scholar
3. Bumcrot, R. J., Modern projective geometry (New York, 1968).Google Scholar
4. Hughes, D. R. and Piper, F. C., Projective planes (New York, Heidelberg, Berlin, 1973).Google Scholar
5. André, J., Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z. 60, 136–168 (1954).Google Scholar
6. Spencer, J. C. D., On the Lenz-Barlotti classification of projective planes, Q. J. Math. 11, 241–257 (1960).Google Scholar
7. Yaqub, J. C. D. S., On projective planes of Lenz-Barlotti Class 16, Math. Z. 95, 60–70 (1967).Google Scholar
8. Yaqub, J. C. D. S., On planes of Lenz-Barlotti Class 16 and planes of Lenz Class III, Math. Z. 129, 109–136 (1972).Google Scholar
9. Scherk, P. and Lingenberg, R., Rudiments ofaffine geometry (Toronto, 1974?).Google Scholar
10. Mann, H. B., Analysis and design of experiments (New York, 1949).Google Scholar
11. Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes, Can. J. Math. 1, 88–93 (1949).CrossRefGoogle Scholar
12. Ostrom, T. G., Finite translation planes (Berlin, Heidelberg, New York, 1970).Google Scholar
13. Artin, E., Coordinates in affine geometry, Notre Dame Math. Coll. (1940). Reprinted in The collected papers of Emil Artin, pp. 505–510 (1965).Google Scholar
14. Artin, E., Geometric algebra (New York, 1957).Google Scholar
15. Bruck, R. H. and Bose, R. C., The construction of translation planes from projective planes from projective spaces, J. of Algebra 1, 85–102 (1964).Google Scholar