A NOTE ON PAPPIAN AFINNE PLANES

Year 2003, Issue: 004, 133 - 142, 15.09.2003

Pınar Anapa İbrahim Günaltılı

Abstract

In (Schmidt and SteinitZ,-1996); an affine plane with Jixed basis ~I ,t2,0}

is coordinated. Then, a ternary operation T on R which is a set of points

on I which is dependent on the coordinate system 11,/2, t is deJined. In

addition, two different binary operation denoted by +,. on R using ternary

operation T. After then, it is showed that (R,+,.) is a division ring. In

this paper, Jirst of all we examined the relation between (R, T) ternary ring

and Desargues postulate in aJine plane. After then, we showed that (R,+,.)

is Jield in case affine plane satisfies Pappus Theorem. This results appeared

in theJirst author's Msc thesis.

Keywords

AJine plane, Desargues Postulat, Pappus Theorem

A NOTE ON PAPPIAN AFINNE PLANES

Abstract

( Schmidt ve Ralph,-1996 ) da {a, tl , t 2} tabanina bagl, olarak bir ajin

diizlem koordinatlanmtsur. Daha sonra 11,12, t koordinat sisitemine baglt

olarak I dogrusu uzerindeki noktalartn kumesi R olmak iizere R kiimesi

uzerinde bir T ucli! islem tanimlanarak; (R,+,.) run bir bolumlu halka

oldugu gosterilmistir. Bu makalede ilk olarak afin diizlemde (R, T) iiclii

halkasi ile Desargues Postulatt arastndaki ilgi incelendi. Daha sonra, ajin

duzlemin Pappus Teoremini saglamasi durumunda (R,+,.) nin bir cisim

oldugu gosterildi. Bu sonuclar ilk yazarin Master tezinde gorulebilir.

Keywords

Pappus Teoremi, Afin duzlem, Dezarg Postulatt

References

• [1] Anapa, P. The coordinatization of affine planes and its ternary ring, Fen Bilimleri Enstitusu ,Msc Thesis, 1998.
• [2] Dembowski, P. Finite Geometries, Springer-Verlag New York Inc.1968.
• [3] Stefan E. Schmidt and Ralph Steinitz . The Coordinatization of Affine Planes by Rings, Geo.Ded. 62.299-317,1996.