From "On the Superlines of a Quadric Surface
in Five-Dimensional Space," by Arthur Cayley —
"… we have on the quadric two systems
of superlines; a superline of the one kind
answering in ordinary space to the lines
which pass through a given point, and
a superline of the other kind answering
to the lines which lie in a given plane."
(Observe that, as regards the five-dimensional
geometry, this is no distinction of nature
between the two kinds of superlines, they are
simply correlative to each other, like the two
systems of generating lines of a quadric in
ordinary space.)"
[Italics added.]
— Quarterly Journal of Pure and Applied
Mathematics, vol. XII (1873), pp. 176-180.
In the picture below, the four-colored bricks numbered 1 through 7 at left
form a Cayley superline "answering to the lines which lie in a given plane."
The above lines lie in the "given plane" that appears in the left half
of a 4×4 array in the Miracle Octad Generator (MOG) of R. T. Curtis.
They are seven lines in PG(3,2) on the same plane in PG(3,2).
By using the MOG, you can also find seven lines in PG(3,2) on
the same point , and the corresponding seven Klein-quadric points.
Cayley's remark on "correlative" kinds of superlines
suggests a related illustration of MOG use . . .
The MOG can be used to find the seven points in the Klein quadric
corresponding to the seven lines in the PG(3,2) plane that appears in
the top half of a 4×4 array . . .
The top-half "given plane" in the above picture corresponds, under the
diamond theorem's projective correlation,* to the "given point" represented
by the line diagram bisecting a 4×4 array horizontally.
* V. Harish, N. R. Kumar and N. R. Raajan, "New visual secret sharing scheme
for gray-level images using diamond theorem correlation pattern structure,"
2016 International Conference on Circuit, Power and Computing Technologies
(ICCPCT), Nagercoil, India, 2016, pp. 1-5, doi: 10.1109/ICCPCT.2016.7530155.
