Message #1341

From: "Galla, Matthew" <mgalla@trinity.edu>
Subject: Other 4D puzzles
Date: Sun, 23 Jan 2011 03:18:20 -0600

Hey everyone,

As I mentioned in my response about my solve of the 120Cell, I have been
looking into some other 4D puzzles and have worked out how several of these
puzzles should work and even discovered some interesting properties. Here is
a snipet from my 120Cell solve message I sent Roice discussing this subject:

"I am still hoping for more complicated 4D puzzles and am willing to do
whatever I can to help make them a reality. Coding a 4d space like you have
is quite intimidating, but perhaps I can try to build off a pre-existing one
with some guidance. I have already determined what the 4D analogue of the
FTO (face turning octahedron, invented some time last year if you have not
already seen it) would look like and how it would function as well as the 4D
analogue of the Skewb and Helicopter Cube (on that note I also have a
suggestion as to how to make the interface for 4D puzzles that are non-face
rotating, like the Skewb and Helicopter Cube). I have also made some
interesting discoveries like for example making a 4D puzzle out of a 3D
puzzle can make some additional internal cuts without altering the exterior
of a 3D face (true for all three puzzle I mentioned so far) and how a 4D
Skewb is not deepcut! (that is every cell looks like a Skewb and seems to
behave as such) The vertex turning deepcut hypercube has faces that
externally each look like a dino cube. Is there anything I can do to make
help make these a reality? After spending 150 hours on the 120Cell, I can
honestly say that about 146 of the hours all feel exactly the same and I am
dying to find a more interesting 4D puzzle to explore :)"

To expand a little on some of the things I mentioned above, the 4D FTO would
be a 24Cell with faces that look like an exploded version of this puzzle:
http://www.jaapsch.net/puzzles/octaface.htm
with one big difference, in addition to every cut on the 3D analogue of the
puzzle, the 4D version has and additional cut perpendicular to the vertices
of each face that line up with first cut down. :/ Sorry, I know that wasn’t
very well worded and I’m not sure how well sending a picture would work
through a yahoo group. Let me try again: these extra cuts would essential
cut off the vertex pieces of each cell. Removing the pieces that are
affected by this new unexpected cut will result in cells that have an
exterior that matches this puzzle:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=451
(If you can follow my inadequate descriptions above, the 4D FTO would have 6
distinct visible pieces, not just the 5 present on an exploded 3D FTO - the
extra comes from splitting each of the vertex pieces of the 3D Fto in half)

A similar phenomenon occurs on both the 4D helicopter cube (3D:
http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50)
and 4D Skewb (3D: http://www.jaapsch.net/puzzles/skewb.htm) [by analogue, I
mean each cell looks like the respective puzzle and moves in a similar
manner]. In each of these puzzles, the new cut clips off the corners.
Remembering that to truly express the 4D nature of these puzzles, each cell
must be "exploded", so what used to be he vertex pieces for each of these
puzzles have now been cut in half resulting in an internal piece that
behaves as one might have expected the single original piece to act and an
external piece that in addition to moving every time the internal piece
moves, can also be affected by a non-adjacent face.

As to a nice interface for non-face rotating 4D puzzles, my suggestion is to
display the wireframe of a 3D solid that displays all the symmetries implied
by the rotation between the faces and perform clicks not on the puzzle
itself, but only on this wireframe. For example, on a 4D Skewb, rotations
are made around the "corners" of each cell. These rotations are all
equivalent to some rotation on a face turning 16Cell. So, in the Hypercube
shape, we could display wireframes of tetrahedrons that "float" between the
appropriate corners of 4 hypercube cells. When the user clicks on a face of
this floating wirefram tetrahedron, both the tetrahedron and the pieces
affected by the corresponding "vertex twist" all rotate. Clicking on the
actual stickers of the puzzle does nothing; all rotations are executed by
clicking on these "rotation polyhedra". In the case of the 4D Helicopter
Cube, the appropriate wireframe shape would be a triangular prism -
rotations around both the triangle faces and the rectangular faces are
possible moves on the 4D Helicopter Cube, and each of these rotations can be
executed unambiguously by clicking on the appropriate face of the triangular
prism wireframe floating between the cells of the puzzle.

As to the deepcut comment, attempt to visualize a 4D Skewb puzzle, that is -
a hypercube consisting of exploded skewbs (with additional cuts clipping off
the corners). Now identify all the pieces affected by one particular
rotation and try to identify the move that is on the opposite side of the
puzzle. Identified correctly, this opposite move does not affect any of the
same pieces. However, not every piece is affected by these two moves! There
is a band of pieces remaining untouched, much like the slice of a 3x3x3 left
untouched by UD’. This means the puzzle is not deepcut! If we push the 3D
hyper cutting planes deeper into the 4D puzzle, we get cells that look like
Master Skewbs. Continuing to push, certain pieces of these Master Skewbs get
thinner and thinner until they vanish at the point when opposing hyperplanes
meet. This is the deepcut vertex turning 8Cell puzzle. Each cell looks like
an exploded Dino Cube. There is a distinct 4D 8Cell puzzle with cells that
look like dino cubes that is shallower cut. Although these puzzles are
visually identical, a single move on the shallower cut puzzle affects pieces
on only 4 cells while a single move on the deepcut puzzle affects pieces on
all 8 cells. Also of interest is the series of complicated looking puzzles
that appear at cut depths between the 4D Skewb and each of these dino cell
puzzles, although there are only 3 slices per axis in these puzzles (same
order as 3x3x3), each cell is an exploded Master Skewb!

Although I have explored several other ideas, the three puzzles (4D FTO, 4D
Skewb, 4D Hlicopter Cube) I have mentioned so far seem to be ideal
candidates for the next run of 4D puzzles, they implement some complex piece
interactions without becoming too large or too visually crowded.

These puzzles are of an incredible interest to me, because the interactions
of the pieces are so much more intricate than the 120Cell or any of the
simplex vertex puzzles possible in the current MC4D program! As I mentioned
in my message to Roice, I have a good idea of how each of these puzzles look
and function and would gladly assist anyone (Roice? haha) who wants to
attempt to program it. In the meantime, I will take a look at the code Roice
has provided me and try to do some work myself, but I highly doubt I will
have success without an experienced programmer’s help ;)

I would love to hear others’ thoughts on these!
-Matt Galla