Date: Thu, 17 Jul 80 22:28:00 -0400 (EDT)
From: Bernard S. Greenberg <Greenberg@MIT-Multics >
~~~ Subject: The Higher Crosses

For the amusement of the experienced cubemeister, and the
education of those desirous of learning the Art, I have here
produced a Description of the Methods, as I have used,
for the production of the Cruces Plummeri & Christmani.
These are the elegant configigurations of lid Crosses
of which we spoke earlier today.

**********************************************************************
I herein describe the algorithms for construction of
the higher (Plummer, Christman) crosses from a solved position.
>From an unsolved position, it is faster to "solve" directly to
the desired configuration; by following the steps below,
the eager cubist may learn exactly what these configurations are.

Of the words and phrases I use:

I call the faces front, back, right, left, top, bottom. A face
has 9 cubies, viz., 4 corner cubies, 4 edge cubies, and its
center cubies. Separating 2 opposite faces, is a "center
slice", being of 4 center cubies and 4 edge cubies. As I
hold the cube, I call three center slices: floor-parallel,
body-parallel, body-slicing. For instance, the body-parallel
and body-slicing centerslices meet in the front face.

I name "double-swap" the transform which is performed as follows:
Double-swap (front, back) ;parameter-faces
Turn body-slicing centerslice 180.
Turn bottom face 180.
Turn body-slicing centerslice 180.
Turn bottom face 180.

Observe well what it has done, viz. swapped the two cubies of the
turned centerslice on the front with those of the back. You
will use it as needed during the following shenanigans:

```----------------------------------------------------------------------
To achieve Christman's (DPC at MIT-MC) Cross, the simpler of the two:
```

Rotate the body-slicing centerslice 180. Rotate the floor-parallel
centerslice 90 either way (your choice).
Stare hard at what you have. The CORNER CUBIES and CENTER CUBIES
are in their final position for the Crux Christmani; all further
hacking will be simply to move the EDGE CUBIES, IN PAIRS, into
place. To achieve ANY Crux Plummeri or Crux Christmani configuration,
learn how to do the initial rotations (see below for the CP)
so that you get the center cubies to corners you want, and hack from
there.

I will now describe the edge-cubie moves for the CC given that
the centerslices have been aligned to orient the center cubies
as needed:

Among the six faces you now have, find one of the two that
have a solid stripe between two sides of the same color, i.e.,
x y x
x y x
x y x
and align it like so, so that the stripe is vertical, and this
face is the front.

Note that the edge cubies of the y y y stripe want to be exchanged
with the two x-showing edge cubies, i.e.,
x x x
y y y
x x x
(Remember that the goal is x y x/y y y/x y x)
You can tell that they want to be inthe horiz. positions by their
non-showing faces, which you will observe match the center-cubies
on the right and left sides.
To do this:
1. Perform doubleswap on front-back.
2. Rotate the FRONT so that when you do (3), the two cubies
we just moved to the back will come to such place so that
when we undo this step (see 4), they will be in the right
place. This will be either 90 deg. left or right.
3. Perform doubleswap on front-back.
4. Undo step 2, i.e., turn FRONT 90 deg the "other" way.
Whehter you blew (2) or not, you will now find you have
(x x x/y y y/x x x) on front. If you understood 2 and DIDNT
blow it, you will have the sides of the y y edge cubies matching
the side centers (if you blew it, doubleswaps on the side faces
can fix you up).

You will see the floor-parallel centerslice begin to form a band.
We will now finish that band. The two appropriate cubies (to go
in the two rear positions of the floor-parallel centerslice are
now on the front plane, the x x cubies of the last step. Note
that a simple doubleswap on front-back would move them to the
back face, but the WRONG two places on the back face. Easy.
So, turn the back face 90 degrees and do the doubleswap,
and unturn the back. Choose which 90 such that these two
cubies wind up in the right place.

You will now find you have solid bands and solid crosses
galore. The front and back should have solid crosses, and the
floor-parallel slice should now be a solid band.
Look at the top of the cube. Make it the front.
Orient it so that it is (a b a/c c c/a b a). Do a front-back
doubleswap, and now look at the remaining face pair we
havent been thinking about. Do the appropriate doubleswap on
them to get solid crosses, and then you should have the Crux Christmani.
Study well what you have: three pairs of alternated crosses.

```----------------------------------------------------------------------
```

The Crux Plummeri (after DCP at MIT-MC who first came up with it,
altho by solving-to) is exactly equal to doing the entire
above transformation twice, at 90 degrees. The following, however,
is a direct route from solved that is more intuitive.

Take the cube, turn the body-slicing centerslice up 90 deg.
Turn the floor-parallel centerslice 90 deg clockwise as seen from
the top. Note well the configuration of corners versus centers;
it is the final one. Note that you will have two triplets
of trebly-interleaved colors: that is the characteristic of the CP.

Look at the TOP or BOTTOM. Let's say the TOP. Make it the front.
Orient it so that you see
x y x
x y x
x y x

Only the top or bottom look like this; this is what you have to
remember to look for after aligning centers to taste.
We're gonna rotate the y y y band into the horiz position.
Do this exactly as for the CC above, producing
(x x x/ y y y/x x x)
Next goal is again to complete the solid band of the floor-parallel
centerslice by doubleswapping front/back so that
the x x edgecubies,w hich would complete that band, go to
the back. Of course, we must temporarily rotate the
back during this doubleswap, so that they go to the side
positions ofthe back when swapped. Do so, completing the
solid color-band of the floor-parallel slice.

Now consider the top and bottom. You note that exactly one appropriate
doubleswap between top and bottom would give us solid
crosses on both. Do it.

Take what had been the top just now, and call that the front.
Note that there are solid crosses on front and back, and the
body-parallel plane is correct and complete.
Think about the front: it looks like
a b a
b b b
a b a

Although it looks right fromt the front, the two vertical b-edge
cubies want to be the two horizontal b-edge cubies, as a cursory
inspectionof the top bottom and sides of the cube will show.
This is true of the back, as well.

Tofix up the FRONT do this:

1. Doubleswap front/back
2. Rotate the FRONT (temporarily) 90 degrees sothat the two
vertical b-edge cubies are gonna come to the right place,
3. And doubleswap front/back
4. Undo 2.
5. Doubleswap front/back.

Now you see all is right save the back. It wants the same
thing done to it. Do it for it; Do this same thing
just doNe in the last 5 steps for the back (viewing it as the
temporary front).

It is done. Consider it.

An exquisite variant ont he CP is obtained by taking on of the
trebly-bound sides and rotating the centers via the well-known
center-cubie rotating algorithm. As the centers are rotated
left or right, either a sextuple checkerboard or a stunning
triply-rotated canon of centers , edges, and corners appears.
The checkerboard is amusing insofar as it appears to a
novice cubist to be the Pons Asinorum 6tuple checkerboard
made by 6 twists (described earlier today), but cannot be
fixed (solved, or produced) without the consummate hair
of the CP that only true cubemeisters can execute.

The application of the Pons Asinorum checkerboard transform
to the CP (as well as the CC) produces interesting and
suprising results.

```----------------------------------------------------------------------
The Higher Crosses are fascinating insofar as they appear to be
very simple edge-cube hacks, but are in fact quite "far"
from home; the CP being exactly twice as "hairy" (far)
as the CC (discovered by ALAN) is in itself a source of
wonderment.
```