# Message #1342

From: schuma <mananself@gmail.com>

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. (http://en.wikipedia.org/wiki/Cross-polytope). 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 (http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=12659). 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.

Nan

— In 4D_Cubing@yahoogroups.com, "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:

> 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

>