Photo on 11-24-15 at 6.14 PM

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;
a 2-way symmetry can be displayed by “swapping”
each primary with its “opposite”:

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
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.

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