On Fri, 28 Jul 2000, Barry Wolk wrote:

> On June 20, in message 1049 I said:

>

> > Theorem 1: If the quadrangle PoPaPbPc is desmic, then it has

> > an ex-mate, and this ex-mate is also desmic

[then a mistake, corrected in:]

> Theorem 1.4 revised: The ex-mate of a desmic quadrangle always lies

> on the cubic determined by that quadrangle....

>

> This is getting more and more interesting. It proves that, in the

> group tables for cubic curves that Steve has been posting, the

> ex-mate of each row will be another row of the same table. I now

> suspect that, for the special case of a desmic quadrangle, this

> ex-mate construction is a generalization of some other construction

> that is already known for cubic curves. It would be nice to remove

> the requirement that the cubic passes through A,B and C.

... after which I snipped some stuff that suggests both Steve

and Barry are reinventing some wheels.

The rows of these group tables are all things of the form

P, P+x, P+y, P+z

where x,y,z are the three elements of order two in the group.

Also, three points P,Q,R of the cubic are in line just if

P+Q+R is the particular element I like to call "line",

and it then follows that their three rows form a desmic system

(provided they're distinct). In particular, the "desmic mate" of

row P is always another row, namely that of

line - P.

I hope that sets the record straight!

I'm very intrigued indeed by Barry's result that the ex-mate of

a row is always another row. What is the relation between the

corresponding group elements? The fact that the ex-mate relation is

symmetric suggests that it has the form

P + Q might be a certain constant, which I'll call "wolk".

If this is the case, then P + Q + (line - wolk) = line, so

that the line PQ passes through a fixed point

R = line - wolk

on the cubic. Is this in fact the case, and if so, what is this

fixed point R? Is it isogonally symmetric to D, the fourth

point of the row A B C D ?

John Conway

PS : could you remind me of the definition of ex-mate - I'm here

without any notes. JHC