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Summary: Identifying Models of the Octave Projective Plane
Daniel Allcock
15 August 1995
allcock@math.berkeley.edu
Department of Mathematics,
University of California,
Berkeley, CA 94720.
1991 mathematics subject classification: 51A35 (17C40)
Published in Geometriae Dedicata 65(1997) 215--217.
Abstract.
We provide a convenient identification between two models of the projective plane over the alterna
tive field of octaves: Aslaksen's coordinate approach and the classic approach via Jordan algebras.
We do this by modifying a 1949 lemma of P. Jordan.
The Octave Plane
The projective plane OP 2 over the alternative field O of octaves (also called Cayley numbers)
may be viewed from several perspectives. Two particularly attractive models are the elegant
coordinatization due to H. Aslaksen using `restricted homogeneous coordinates' [1], and the model
developed extensively by H. Freudenthal, in which the points of OP 2 are identified with a set of
idempotents in J, a certain Jordan algebra [2]. What is missing is a convenient means to pass
between these two languages. This paper makes the observation that a lemma due to P. Jordan
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