Message #831

From: Roice Nelson <>
Subject: Re: [MC4D] Introducing "MagicTile"
Date: Sun, 31 Jan 2010 17:52:47 -0600

Greetings again,

I wanted to make a minor clarification. I didn’t describe my desired
order-2 twist for a length-2 Megaminx well, and incorrectly wrote that this
twist would swap opposite faces. It would swap half the material for two
pairs of opposite faces. It would also completely swap a further two pairs
of faces (one pair is of adjacent faces, one is not, but neither are
opposite). Grab a Megaminx and picture an entire half of the puzzle being
rotated 180 degrees to see the twist. Hopefully the goal was clear enough
though… a length-2 Megaminx that wouldn’t suffer from the weirdness of the
Impossiball. It seems a worthy aesthetic goal for puzzles to have no
overlapping material when twisting, but I’m always curious of other opinions
on things like that.


On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson <> wrote:

> Thanks Matt!
> I agree with your thoughts about inverting the outermost face, and have
> added it to "the scary list". The same issue comes up in the 4D puzzles (at
> least in my implementation of Magic120Cell since I allow showing cells
> mirrored by the projection). I had originally reversed the twisting of
> those faces, but ended up reverting that when I discovered some bug entropy
> related to it. Also, I worried it could lead to confusion, like "hey,
> that’s not counterclockwise!". On the other hand, it makes the projection
> effects explicit. It might actually be nice to have an option for this,
> which I think I’ll do when I get to it.
> The two layer puzzles are an interesting topic for sure. I put them in the
> list even though they don’t currently do anything (except in the Megaminx
> family), because they look pretty. I was on the fence about enabling the
> Megaminx behavior as is, but did so solely because of the Impossiball. It
> is the only puzzle in the list right now which overlaps material when
> twisting. I actually feel there is a better length-2 analogy for the
> Megaminx than the Impossiball, which is twisting a slicing circle that is a
> "great circle" (cuts the unprojected puzzle sphere in half). The order of
> this twist is 2 instead of 5 and swaps opposite faces. It also twists
> without overlapping any material, which is why I prefer it. The reason I
> didn’t include this as the length-2 version is that this twist is
> edge-centered rather than face-centered, and I had made the (arbitrary)
> decision to restrict to the latter in the first version. I have to keep
> myself sane somehow :)
> The situation is similar in other puzzles. Check out the length-2 digonal
> and trigonal puzzles and note that the nicest twists there (which don’t
> overlap material) are vertex-centered. To answer your question about
> supporting all even length puzzles vs. just length-2, absolutely. The guts
> are capable, I just didn’t expose those as menu options until the behavior
> is better worked out. And I welcome further discussion of the specification
> of these puzzles! In particular, what would be a good way to specify
> edge/vertex twists?
> When you get to the infinite tilings, thinking about even-layers becomes
> even stranger, and there is analogizing work to be done :) In the hexagonal
> case, I think the right twist of the length-2 puzzle would be a translation
> of half of the puzzle! (do you agree?) Again, the reason I favor this is
> because no material overlaps. The problem I ran into was how to specify
> such twists elegantly. Try thinking about it and backing yourself into some
> corners :) One thing I can say is that a { clicked cell + direction }
> simply isn’t enough information to specify it. The same is true in the
> hyperbolic cases.
> Related to the last paragraph, it is interesting that outer twists for the
> infinite tilings don’t make sense regardless of whether the puzzle is even
> or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? The
> topology restricts the movement, which (I think) makes sense to me if I
> picture the hexagonal puzzle on a torus instead of unrolled as in MagicTile.
> Anyway, I’m perhaps getting a little off topic, but the reason to mention
> this is that we won’t be able to specify reorientations for the infinite
> tilings as a twist "with all slices down". However, I’d love to see
> reorientations (both view rotations and panning) done some other way in both
> the spherical/hyperbolic cases. To see a really nice example of panning
> hyperbolic space, check out Don’s hyperbolic tessellation applet<>!
> The are some big challenges to get this working in MagicTile, and
> performance is one of the largest, since so much needs to be drawn.
> In regards to your question about the tiling patterns, there is a simple
> procedure I used to get the current list, and it involved specifying two
> numbers. First was the number of reflections from a "fundamental" cell to
> an "orbit" cell (I actually called them masters/slaves in the code). Second
> was which polygon segment to do this reflection across. So for example,
> take a look at the 6-colored octagonal puzzle. This would be 2,4
> (equivalently 4,2). The white center cell is reflected twice across the 4th
> segment of each adjacent cell, and recursively thereafter. I wish I
> understood this all better, as I actually just had the program run through a
> loop to see which of the configurations "converged". Some end up fitting
> together and some don’t. Math Magic! Hope this helped clarify.
> has some papers with lists that would probably help expose the
> magic more. Btw, the patterns that work in the hexagonal case end up
> producing puzzles where the number of colors are perfect squares, go figure!
> And I wasn’t able to find working patterns for polygons with 11 and 13
> sides, though I wonder if they exist and I wasn’t recursing deeply enough.
> So there is plenty more discussion that could happen, but I don’t want to
> overload it right off the bat. One last thing though related to your final
> comment. I put a feature in the program just for you Matt! Under "Options
> -> Edit Settings…", play with the "Slicing Circles Expansion Factor".
> This is analogous to "deepening the cuts" on the original puzzles, which I
> know you’ve wanted in the 4D puzzles. It is a fully experimental setting
> and I know cases where it doesn’t work well, but often it does. Try 1.4 on
> a Megaminx for a more difficult puzzle!
> Also, if anyone feels some of this discussion shouldn’t be on the
> hypercubing mailing list, let me know. I’m a little worried some might feel
> the hyperpuzzling connection a little too tenuous.
> All the best,
> Roice
> On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla <> wrote:
>> Very nice, Roice.
>> Although the options are clearly limited, this program is a work of art.
>> The hyperbolic face patterns combined with the scrambled colors are
>> absolutely beautiful.
>> I played around with some puzzles I already understood, and it takes a
>> while to get used to the little quirks in your program, but well worth it! I
>> did notice that the outermost face for the cube and megaminx series is
>> controlled opposite to my intuition. If I am thinking in terms of macros and
>> try to apply a macro I know works near the center of the puzzle on the
>> outermost face, I find that I must invert every move on the outermost face.
>> A closer look reveals that this is because the outermost face is inverted.
>> Now this is just an idea, but have you considered inverting the movement for
>> just the outermost face? Although it my confuse some things visually, I
>> think it may be an overall improvement solving-wise.
>> Also, most of your 2-layer puzzles are currently not working (which I’m
>> sure you already know). Are you looking into correcting this function of the
>> program? If so, can we expect puzzles with an even number of layers >2? For
>> puzzles with even layers (excluding cube) the visual pieces will have to
>> pass under/over/through each other. This is an inevitable behavior if you
>> restrict the exterior shape of a puzzle (which your program does because it
>> forces it to be drawn on a hyperplane). However, as you demonstrated with
>> the two-layered megaminx (impossiball) this is clearly do-able.
>> I am also looking forward to an updatewhere we can reorient some of these
>> puzzles! This is allowable on the cubical and dodecahedral puzzles by
>> holding down every layer number, but I would love to watch some these
>> hyperplane tesselations shift!
>> My favorite thing about your program, however, is the identical puzzles
>> with different sticker patterns. I am very interested to know how you came
>> up with the different patterns of colors on say, {6,3}, as well as the other
>> puzzles with multiple color-pattern options.
>> All in all, an excellent program that opens up a world of puzzles I had
>> never considered before! Although, I should say that none of the puzzles in
>> your program are very hard ;)
>> Thank you for once again expanding the limits on twisty puzzles!
>> Matt Galla
>> PS How many moves counts as an official scramble so I can start submitting
>> my solves? :)
>> On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson <> wrote:
>>> Thanks to everyone for the thoughtful feedback on my question this week.
>>> I appreciate it, and it was good to get your perspectives.
>>> I think I’m ready enough to share a first pass of the new Rubik analogue
>>> I started playing with before the MC4D 4.0 fun, which I mentioned the
>>> possibility of here<>some time ago. While you might observe it doesn’t quite fall into the
>>> category of hyperpuzzles, it does in at least once sense mentioned below :D
>>> Here is the page with the download, pictures, and a video<>.
>>> To describe the analogue idea, I’ll just quote the beginning of the
>>> explanation on that page:
>>>> This program aims to support twisty puzzles based on regular polygonal
>>>> tilings <> having Schlafli
>>>> symbols <> of the form
>>>> {p,3} for any p>=2. That is, all regular tilings of polygons with two or
>>>> more sides, where three tiles (puzzle faces) meet at a vertex. The Rubik’s
>>>> cube is the special case where faces are squares (p=4). The other familiar
>>>> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you’ll
>>>> discover the last takes a slightly different form under this abstraction
>>>> (akin to Jing’s Pyraminx <>).
>>>> All the other puzzles are new as far as I know, and some may be surprising,
>>>> e.g. the puzzles based on digons <>
>>>> (p=2).
>>>> Each 2D tiling admits a particular constant curvature (homogenous)
>>>> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
>>>> p=6, and Hyperbolic for p>=7. Since you can’t "isometrically embed" the
>>>> entire hyperbolic plane in 3-space<>,
>>>> I have a connection to hyperpuzzling<> even
>>>> though I’m talking about 2D tilings!
>>> …
>>> I’ve actually only solved the 3x3x3 on it so far, and I wonder if it may
>>> be more fun to watch than play! I’ve been calling it MagicTile, though
>>> perhaps there could be something better? As with everything, it is a known
>>> work in progress (the length of the task list has grown to scary
>>> proportions). I have no plans for further development at the moment, though
>>> I’ll happily fix any glaring bugs.
>>> Enjoy!
>>> Roice
>>> P.S. This is the only "twisty puzzle" group I’m active in, so if any of
>>> you are also members of other groups and think they would be interested to
>>> hear about these new puzzles, I’ll appreciate the exposure :)