Coxeter- Theorem 5.32
- Author:
- AndrewLyon
- Topic:
- Geometry
5.32 An involution is determined by any two of its pairs.
Following from theorem 5.31, it is convenient to denote the involution AA1B projective to A1AB1 by
(AA1)(BB1)
or (A1A)(BB1), or (BB1)(AA1), and so forth. This notation reminas valid when B1 coincides with B; in other words, the involution AA1B projective to A1AB, for which B is invariant, may be denoted by
(AA1)(BB).
If (AD)(BE)(CF), as in Figure 2.4A (reproduced and provided), we can combine the projectivity AECF projective to BDCF of 5.11 with the involution (BD)(CF) to obtain AECF projective to BDCF which is projective to DBFC which shows there there is a projectivity in which AECF projective to DBFC. Since this interchanges C and F, it is an involution, namely
(BE)(CF) or (CF)(AD) or (AD)(BE); all equivalent statements.
Thus the quadrangular relation (AD)(BE)(CF) is eqquivalent to the statement that the projectivity ABC projective to DEF is an involution, or that
ABCDEF projective to DEFABC.