Message #1342

From: schuma <>
Subject: Re: Other 4D puzzles
Date: Sun, 23 Jan 2011 18:44:22 -0000

Hi Matt,

Thank you for starting the discussions about other 4D puzzles.

Can you explain more about why the 4D analogue of the FTO is a 24-cell instead of a 16-cell? Although the faces of the 24-cell are octahedra, 24-cell is a self-dual polytope that is not a simplex. From this point of view, it has no 3D analog. In fact it has no analog in any dimension other than 4D. However, the 16-cell belongs to the family of cross-polytopes, which are the duals of hypercubes, and exist in any number of dimensions. ( In 3D, the cross-polytope is 16-cell. Therefore I think a natural extension of FTO is a cell-turning 16-cell, because they share more similarities.

For example, you may know that in 3D, the FTO can be regarded as a shape-mod of Rex Cube, a vertex turning cube ( If the 4D FTO is a shape-mod of the vertex turning hypercube, it should be a cell-turning 16-cell instead of a cell-turning 24-cell.

No matter calling it 4D FTO or else, I believe what you have described in the third paragraph is a cell-turning 24-cell. It should be an amazing puzzle to solve. I have special feeling about it because of its uniqueness in all the dimensions.


— In, "Galla, Matthew" <mgalla@…> wrote:
> 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:
> 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:
> (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:
> and 4D Skewb (3D: [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