### (POG)(RYB)

that 7-space has seven-way symmetry is obvious
(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.
here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.
we then chose the “secondaries” line
{Green, Purple, Orange} and permuted;
the “primaries” permute accordingly;
voila.
*************************************************
a 2-way symmetry can be displayed by “swapping”
each primary with its “opposite”:
(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect
in any of the other colors;
for each (there are three)
“line” through a given color,
interchange the positions for
the other two colors.
the lines-on-blue are
{bgy, bmo, bpr}, so
“reflection in the Blue” has
(GY)(MO)(PR)
as its permutation-notation.)

1. https://vlorbik.wordpress.com/2014/09/10/graphical-representations-for-the-simple-group-of-order-168/

so the drawing project becomes
“draw all 24 rainbow-stars having
Mud at the top”.

this’ll then be the “identity coset”
in the ${{PSL_2({\Bbb F}_7)}\over{\langle\rho\rangle}}$
at hand, where \rho is a reprentative
(in PSL(2,7)) of “(MRBGPYO)”
(our seven-way permutation).

“rotate”—that’s why they call it \rho—
all twenty-four of these all seven ways.
that’s our G (\iso PSL(2,7) \iso GL(2,3)).

when i drew all 168
(in a 3-way symmetric version),
i messed it up somehow.

this time i’m publishing when i get to 24.

2. done long ago by somebody else—roy g.~biv style.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)