# Message #832

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Introducing "MagicTile"

Date: Mon, 01 Feb 2010 00:36:32 -0600

This is embarrassing, but I was just playing with MagicTile and need to make

another correction to something I claimed - I probably should let my emails

sit a while before sending them! My fingers are crossed that this is the

last spam I feel compelled to make on this.

I was off track about vertex-centered twists on the trigonal puzzle being

the right ones for a length-2 version. It actually appears there is no good

(non-trivial and non-overlapping) twist for a length-2 trigonal puzzle. The

twists based on great circles that would slice faces in half would have

order 1, meaning they would have to rotate 360 degrees before the puzzle

could fit back together (sort of a trivial twist - there were some of these

in MC4D too, which we ended up disallowing). I think the trivial order is

related to the fact that the center of these twists don’t correspond to a

face center, edge, or vertex.

To at least try to make a useful observation, for puzzles with Schlafli

symbol { p, q }, face-centered twists will have order p and vertex-centered

twists will have order q. Edge-centered twists will have order 2, since two

faces meet at each edge…

Have a nice week all,

Roice

On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson <roice3@gmail.com> wrote:

> 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.

>

> Cheers,

> Roice

>

>

> On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson <roice3@gmail.com> 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<http://www.plunk.org/~hatch/HyperbolicApplet/>!

>> 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.

>> Tilings.org 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 <mgalla@trinity.edu>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 <roice3@gmail.com> 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<http://games.groups.yahoo.com/group/4D_Cubing/message/541>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<http://www.gravitation3d.com/magictile>.

>>>> 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 <http://en.wikipedia.org/wiki/Regular_tessellation> having Schlafli

>>>>> symbols <http://en.wikipedia.org/wiki/Schlafli_symbol> 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 <http://www.youtube.com/watch?v=FuD3YwQTW2c>).

>>>>> All the other puzzles are new as far as I know, and some may be surprising,

>>>>> e.g. the puzzles based on digons <http://en.wikipedia.org/wiki/Digon>

>>>>> (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<http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html>,

>>>>> I have a connection to hyperpuzzling<http://games.groups.yahoo.com/group/4D_Cubing/> 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 :)

>>>>

>>>>

>>>

>>>

>>>

>>

>>

>>

>