From:

~~~ Subject:

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.

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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.