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Date: Wed, 8 Apr 1998 07:55:06 -0700
Subject: A workable 6x6x6 cube design (probably)
Message-Id: <19980408.075506.7150.0.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)

I have found that the 6x6x6 cube can only be made practical if the
outer rows of cubies are slightly larger (about 3mm or 1/8 inch).
If the rows are all the same size then some cross-sections of pieces
(e.g. the corner pieces) are less than 3 sq-mm, and other pieces are
extremely thin (0.6mm in some places).

If the plastic is black (or white) and the stickers are all the same
size then the inequality in the size of the cubies will be effectively
masked.  The stickers would have to be spaced evenly.  The cube will
look as if it has a small 'border' but the perception will be that the
cubies are the same size.

This design is actually almost as strong as the 4x4x4 cube.  It contains
an internal frame plus 256 movable pieces of ten different types.  No
cross section of a piece is smaller than 7 sq-mm (the 4x4x4 has center
pieces with 9 sq-mm cross section).  Two of the types of piece
(FACE EDGE PIECE, SPACER PIECE 2) come in two mirror image forms, so the
number of molds that would be needed to produce this is 14 (counting two
for the internal frame).  The internal mechanism would need to be greased
to allow it to turn smoothly, but it should be no worse than the 5x5x5.

The following is an exact geometric description of each piece.  To be
able to understand this you need to know how to use Cartesian and Polar
coordinates.  All pieces are intersections of planes, spheres, and
hyperboloids (which can probably be approximated as cones).

The SPACER PIECE 2 could probably be replaced by some sort of rectangular
but rounded blob-like thing, it does not need to be an exact shape and
the cube might turn more smoothly if it is rounded.  It also might then
be possible to make it symmetrical so they could be produced with a
single mold, which would slightly reduce production cost.

Comments, suggestions and quibbles are welcome.



LEGEND - x,y,z are Cartesian coordinates, r is distance from origin
         Dx, Dy, Dz is distance from x, y, z axis respectively

NO TOLERANCES - pieces must be shrunk away from all sides a little bit

DIMENSIONS assume that the size of an inner CUBIE is 100 and the size of
an outer CUBIE is 125, this allows the pieces to be much stronger than if
the cubies were all the same size.  The puzzle occupies the space such
that
-325<x<325, -325<y<325, -325<z<325.

Maximum vertical play needed (i.e. strain in the radial direction, mostly
on the inside) is about 12 units, or 1.5 mm on a 78mm cube.  If all parts
are designed with a tolerance of 2 units (0.24 mm) (e.g. by shrinking
each
part 1 unit (0.12 mm) away from all sides) then the maximum vertical play
needed is about 8 units (1.0 mm).  Extra space could be provided (shrink
inner face of frame away from center by another couple of units), if this
is too much.


PUZZLE CONTAINS:
    08 CORNER PIECE
    24 INNER EDGE PIECE
    24 OUTER EDGE PIECE
    48 FACE EDGE PIECE
    24 FACE CORNER PIECE
    24 CENTER PIECE
    24 INTERNAL EDGE PIECE
    08 INTERNAL CORNER PIECE
    24 SPACER PIECE 1
    48 SPACER PIECE 2
    01 INTERNAL FRAME



----------------------------------------------------------------------

CORNER PIECE consists of:

        all points such that 200<x<325, 200<y<325, 200<z<325
    AND all points such that x>175, y>175, z>175, 280<r<360

INNER EDGE PIECE consists of:

        all points such that 0<x<100, 200<y<325, 200<z<325, r>320
    AND all points such that 0<x<70, y>175, z>175, 280<r<360

OUTER EDGE PIECE consists of:

        all points such that 100<x<200, 200<y<325, 200<z<325, r>360
    AND all points such that 100<x<175, y>175, z>175, 320<r<360
    AND all points such that 70<x<175, y>175, z>175, 280<r<320

FACE EDGE PIECE consists of:

        all points such that 0<x<100, 100<y<200, 200<z<325, r>360
    AND all points such that 0<x<100, 100<y<175, z>175, 320<r<360
    AND all points such that 0<x<70, 70<y<175, z>175, 280<r<320

FACE CORNER PIECE consists of:

        all points such that 100<x<200, 100<y<200, 200<z<325, r>360
        all points such that 100<x<175, 100<y<175, z>175, 320<r<360
    AND all points such that 70<x<175, 70<y<175, z>175, 280<r<320

CENTER PIECE consists of:

        all points such that 0<x<100, 0<y<100, 200<z<325, r>360
    AND all points such that 0<x<70, 0<y<70, z>175, 280<r<360
    AND all points such that 0<x<120, 0<y<120, z>0, 240<r<280
    AND all points such that x>60, y>60, z>0, 200<r<240,
Dz<sqrt(z^2+60^2)
    AND all points such that x>30, y>30, z>0, 100<r<200,
Dz<sqrt(z^2+30^2)

INTERNAL EDGE PIECE consists of:

        all points such that 60<x<z, -60<y<60, z>0, 200<r<240
    AND all points such that 30<x<z, -30<y<30, z>0, 100<r<200

INTERNAL CORNER PIECE consists of:

        all points such that Dx>sqrt(x^2+60^2), Dy>sqrt(y^2+60^2),
                             Dz>sqrt(z^2+60^2), x>0, y>0, z>0,
                             200<r<240
    AND all points such that Dx>sqrt(x^2+30^2), Dy>sqrt(y^2+30^2),
                             Dz>sqrt(z^2+30^2), x>0, y>0, z>0,
                             100<r<200
    AND all points such that x>120, y>120, z>120, 240<r<280

SPACER PIECE 1 consists of:

        all points such that x>120, y>120, 0<z<120, 240<r<280

SPACER PIECE 2 consists of:

        all points such that 70<x<100, 175<y<200, z>175, 320<r<360

INTERNAL FRAME consists of

        all points such that -60<x<60, -60<y<60, 200<r<240
    AND all points such that -60<x<60, -60<z<60, 200<r<240
    AND all points such that -60<y<60, -60<z<60, 200<r<240
    AND all points such that Dx<30, r<200
    AND all points such that Dy<30, r<200
    AND all points such that Dz<30, r<200
    AND all points such that r<100

--Tenie Remmel (tenie1@juno.com)