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Chess Geometry
I can see what you mean now. It is the old image of the hexagon with its
diagonals drawn in that flips between representing transparent cubes facing
in oppositte directions, as follows:
 --
/\/\
----
\/\/
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The Tetrahedral geometry corresponds to one Bishop binding of a cubic
geometry. The asterisks on the 9x9x9 cubic board below indicate the plane
corresponding to a hex board of side 5. If you colour the different cubic
cells based on the four Unicorn bindings, you will find that this
corresponds to colouring the hex cells based on the four Dabbaba bindings.
Your projection will indeed project each of the unmarked cells onto one of
the marked ones.

Level 1         Level 2         Level 3
----*----       -----*---       ------*--
---*-----       ----*----       -----*---
--*------       ---*-----       ----*----
-*-------       --*------       ---*-----
*--------       -*-------       --*------
---------       *--------       -*-------
---------       ---------       *--------
---------       ---------       ---------
---------       ---------       ---------


Level 4         Level 5         Level 6
-------*-       --------*       ---------
------*--       -------*-       --------*
-----*---       ------*--       -------*-
----*----       -----*---       ------*--
---*-----       ----*----       -----*---
--*------       ---*-----       ----*----
-*-------       --*------       ---*-----
*--------       -*-------       --*------
---------       *--------       -*-------


Level 7         Level 8         Level 9
---------       ---------       ---------
---------       ---------       ---------
--------*       ---------       ---------
-------*-       --------*       ---------
------*--       -------*-       --------*
-----*---       ------*--       -------*-
----*----       -----*---       ------*--
---*-----       ----*----       -----*---
--*------       ---*-----       ----*----