LS0tLS1CRUdJTiBQR1AgU0lHTkVEIE1FU1NBR0UtLS0tLQ0KSGFzaDogU0hBMQ0KDQpPbiBU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LS0tLS1CRUdJTiBQR1AgU0lHTkVEIE1FU1NBR0UtLS0tLQ0KSGFzaDogU0hBMQ0KDQpPbiBT
dW4sIDI3IEZlYiAyMDExIDAwOjMzOjA2IC0wMDAwDQoiTWF0dGhldyIgPGRhbWllbnR1cnRs
ZUBob3RtYWlsLmNvLnVrPiB3cm90ZToNCg0KWy4uLl0NCj4gQW5vdGhlciBpcyB0byB1c2Ug
dGhlICJmdWRnaW5nIiBjb25jZXB0IChzZWUNCj4gaHR0cDovL3d3dy5zaGFwZXdheXMuY29t
L21vZGVsLzIwNDcyMS9mdXR0bWlueC5odG1sP2dpZD11ZzEzNjAzIGZvcg0KPiBvbmUgZXhh
bXBsZSwgdGhvdWdoIHRoZSBpZGVhIGlzIGEgbGl0dGxlIGRpZmZlcmVudCBoZXJlKSBzdGFy
dGVkIGJ5DQo+IE9zY2FyIHZhbiBEZXZlbnRlciB0byB0d2VhayB0aGUgZ2VvbWV0cnkgYW5k
IHJlbW92ZSBzbWFsbCBwaWVjZXMgaWYNCj4gc28gZGVzaXJlZCwgaWYgYW55IGlkZWFzIGZy
b20gdGhpcyBhcmUgaW1wbGVtZW50ZWQuDQoNClNwZWFraW5nIGZvciBteSBvd24gYWVzdGhl
dGljcywgSSBkb24ndCB0aGluayBmdWRnaW5nIGhhcyBtdWNoIG9mIGENCnBsYWNlIGluIHRo
ZSB3b3JsZCBvZiBjb21wdXRlciBwdXp6bGluZy4gIEN1dHMgdGhhdCByZXF1aXJlIHBpZWNl
cyB0bw0KYmUgZnVkZ2VkIHRvIGVsaW1pbmF0ZSB0aGUgaW5maW5pdGUgY2FzY2FkZSBvZiBw
aWVjZXMgYnVnIG1lLg0KDQpPbiB0aGUgdG9waWMgb2YgZnVkZ2luZyB0aG91Z2gsIHRoZXJl
IHdhcyBhIGxvbmcgcmFtYmxpbmcgZGlzY3Vzc2lvbg0KYWJvdXQgZnVkZ2luZywgd2hhdCBp
dCBpcywgZXRjIGhlcmU6DQoNCmh0dHA6Ly93d3cudHdpc3R5cHV6emxlcy5jb20vZm9ydW0v
dmlld3RvcGljLnBocD9mPTE1JnQ9MjAxMDUNCg0KTW9zdCBvZiB0aGUgaW50ZXJlc3Rpbmcg
ZGlzY3Vzc2lvbnMgcmV2b2x2ZSBhcm91bmQgdGhlIHBvc3RzIGJ5IHVzZXINCid3d3dtd3d3
JyAoQ2FybCkuICBXb3ZlbiBpbiB3aXRoIHRoZSBmdWRnaW5nIGRpc2N1c3Npb24gaXMgYSBk
ZWJhdGUNCmFib3V0IHdoYXQgaXQgbWVhbnMgZm9yIGEgcHV6emxlIHRvIGJlICJkZWVwLWN1
dCIgd2hpY2ggSSBkb24ndCB0aGluaw0Kd2FzIHJlc29sdmVkLg0KDQpUaGUgaW50ZXJlc3Rp
bmcgYW5pbWF0aW9ucyBkb24ndCBzdGFydCB1bnRpbCBhYm91dCBtaWQtd2F5IHRocm91Z2gg
cGFnZQ0KMi4NCg0KSW4gc2hvcnQgdGhvdWdoLCBmdWRnaW5nIGlzIGEgbWVzc3ksIHNlZW1p
bmdseSBub24tbWF0aGVtYXRpY2FsIHRyaWNrDQp0byBtYWtlIGN1dHMgdGhhdCBzaG91bGRu
J3Qgd29yayBhY3R1YWxseSB3b3JrLiAgQ29uc2lkZXJpbmcgaG93IG1hbnkNCmdyZWF0IHB1
enpsZXMgdGhlcmUgYXJlIHRoYXQgZG8gd29yaywgSSB0aGluayB3ZSBzaG91bGRuJ3QgZm9j
dXMgdG9vDQptdWNoIGVuZXJneSBvbiBmdWRnaW5nIHB1enpsZXMgdGhhdCB3b3VsZG4ndCB3
b3JrIHJpZ2h0IG90aGVyd2lzZS4NCg0KDQo+IE5vIGRvdWJ0IHBlb3BsZQ0KPiBsaWtlIEJy
YW5kb24gYW5kIE5hbiB3aWxsIGhhdmUgbm8gcHJvYmxlbXMgd2l0aCB0aGVzZSBzbWFsbCBw
aWVjZXMNCj4gdGhvdWdoLg0KDQpJJ20gZmxhdHRlcmVkLiAgSSB0aGluayBwZW9wbGUgbGlr
ZSBOYW4gYW5kIEFuZHJleSBhcmUgaW4gYSBjbGFzcyBvZg0KdGhlaXIgb3duLg0KDQpCZXN0
IHJlZ2FyZHMsDQoNCkJyYW5kb24NCg0KLS0tLS1CRUdJTiBQR1AgU0lHTkFUVVJFLS0tLS0N
ClZlcnNpb246IEdudVBHIHYyLjAuMTYgKEdOVS9MaW51eCkNCg0KaUVZRUFSRUNBQVlGQWsx
c0RqMEFDZ2tRcWFHUHpBc2w5NExrOHdDZ2pQanVmSmZuTURTMjNWbkdZM01UUFhqZw0KWXdr
QW9JZ212YUI4RXV1bFdNcTU2NGlMNCtqZ3BNYTENCj1FcmpDDQotLS0tLUVORCBQR1AgU0lH
TkFUVVJFLS0tLS0NCg==
--------------090108080604070601050508
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
On 2/28/2011 1:05 PM, Brandon Enright wrote:
> "Matthew"
>
> [...]
>> Another is to use the "fudging" concept (see
>> http://www.shapeways.com/model/204721/futtminx.html?gid=ug13603 for
>> one example, though the idea is a little different here) started by
>> Oscar van Deventer to tweak the geometry and remove small pieces if
>> so desired, if any ideas from this are implemented.
> Speaking for my own aesthetics, I don't think fudging has much of a
> place in the world of computer puzzling. Cuts that require pieces to
> be fudged to eliminate the infinite cascade of pieces bug me.
>
> On the topic of fudging though, there was a long rambling discussion
> about fudging, what it is, etc here:
>
> http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=20105
>
> Most of the interesting discussions revolve around the posts by user
> 'wwwmwww' (Carl). Woven in with the fudging discussion is a debate
> about what it means for a puzzle to be "deep-cut" which I don't think
> was resolved.
>
> The interesting animations don't start until about mid-way through page
> 2.
>
> In short though, fudging is a messy, seemingly non-mathematical trick
> to make cuts that shouldn't work actually work. Considering how many
> great puzzles there are that do work, I think we shouldn't focus too
> much energy on fudging puzzles that wouldn't work right otherwise.
I've been thinking about this for a while and agree that this is a
question of aesthetics, but I can see how different viewpoints can be
equally valid mathematically. Matthew's link shows a twisty Buckyball
which makes for a nice-looking puzzle. It took me a while to see what
was fudged about it. The pentagonal faces give no problems but the
hexagonal faces are only completely kosher for 120 degree turns. 60
degree turns of the hexagonal faces move some stickers from hexagonal
faces to and from pentagonal ones. Given my aesthetic preferences, I
would simply not allow those 60 degree twists and everything would be fine.
I guess I would call the two valid definitions "geometric" versus
"permutation" orientations. If we think only in terms of generating
puzzles by slicing uniform polytopes, then we're taking a geometric
orientation and find ourselves struggling over tiny inelegant pieces
that are sometimes generated. However if you are more interested in the
group theoretic aspects of twisty puzzles, then you don't need to worry
about the geometry at all. From that point of view, "fudged" puzzles are
just nice visual aids that embody the essence of a particular puzzle
group, and any fudging needed to build one is completely unimportant.
One thing that I think we can all appreciate are those puzzles which
happen to satisfy both points of view, but beyond that we just have to
pick one or the other perfectly reasonable sides. This exercise has
clarified some of my previous feelings of ambivalence that I wasn't
quite able to settle. My particular problem was with the subject of
piece intersections during twisting. Some people on this list are
bothered by twisting geometry that doesn't smoothly slide in either the
model space or the viewing space, but that issue never bothered me. On
that point I was only interested in the geometric result of twisting.
Now I see that in those cases I was interested in the resulting
permutations and therefore didn't care what happened during the twist,
however my geometric aesthetic governed my interest in particular puzzles.
On a related note, I recently received the face-turning octahedron
puzzle that I ordered from the link
that Nan posted, along with a floppy cube. The connection to this
discussion is that the plastic of the floppy cube bends strangely as
parts of the central cube gets pried open while twisting. It's a very
odd mechanism that may not stand up to a lot of use, but then the puzzle
is not at all floppy, possibly because of that very design choice. I
don't want it to break, but the bending doesn't bother me at all if that
is what allowed them to build a non-floppy 3x3x1 puzzle.
Just for completeness sake, I can report that the octahedron puzzle is
quite nice though it does take some care to keep some twists from
locking up. I really wish that the twists would snap into place like my
Chinese megaminx, but that's not a big problem. The most surprising
thing I found about this puzzle was just how quickly it becomes
scrambled. Even a single twist from the pristine position is not
immediately clear that it is only one twist, and two intersecting twists
are not nearly as obvious how to untwist as similar twists on the
original Rubik's cube. It is a very handsome puzzle and makes me
appreciate the humble octahedron all the more.
-Melinda
--------------090108080604070601050508
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
http-equiv="Content-Type">
On 2/28/2011 1:05 PM, Brandon Enright wrote:
"Matthew" <damienturtle@hotmail.co.uk> wrote:
[...]
Another is to use the "fudging" concept (see
http://www.shapeways.com/model/204721/futtminx.html?gid=ug13603 for
one example, though the idea is a little different here) started by
Oscar van Deventer to tweak the geometry and remove small pieces if
so desired, if any ideas from this are implemented.
Speaking for my own aesthetics, I don't think fudging has much of a
place in the world of computer puzzling. Cuts that require pieces to
be fudged to eliminate the infinite cascade of pieces bug me.
On the topic of fudging though, there was a long rambling discussion
about fudging, what it is, etc here:
http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=20105
Most of the interesting discussions revolve around the posts by user
'wwwmwww' (Carl). Woven in with the fudging discussion is a debate
about what it means for a puzzle to be "deep-cut" which I don't think
was resolved.
The interesting animations don't start until about mid-way through page
2.
In short though, fudging is a messy, seemingly non-mathematical trick
to make cuts that shouldn't work actually work. Considering how many
great puzzles there are that do work, I think we shouldn't focus too
much energy on fudging puzzles that wouldn't work right otherwise.
I've been thinking about this for a while and agree that this is a
question of aesthetics, but I can see how different viewpoints can
be equally valid mathematically. Matthew's link shows a twisty
Buckyball which makes for a nice-looking puzzle. It took me a while
to see what was fudged about it. The pentagonal faces give no
problems but the hexagonal faces are only completely kosher for 120
degree turns. 60 degree turns of the hexagonal faces move some
stickers from hexagonal faces to and from pentagonal ones. Given my
aesthetic preferences, I would simply not allow those 60 degree
twists and everything would be fine.
I guess I would call the two valid definitions "geometric" versus
"permutation" orientations. If we think only in terms of generating
puzzles by slicing uniform polytopes, then we're taking a geometric
orientation and find ourselves struggling over tiny inelegant pieces
that are sometimes generated. However if you are more interested in
the group theoretic aspects of twisty puzzles, then you don't need
to worry about the geometry at all. From that point of view,
"fudged" puzzles are just nice visual aids that embody the essence
of a particular puzzle group, and any fudging needed to build one is
completely unimportant.
One thing that I think we can all appreciate are those puzzles which
happen to satisfy both points of view, but beyond that we just have
to pick one or the other perfectly reasonable sides. This exercise
has clarified some of my previous feelings of ambivalence that I
wasn't quite able to settle. My particular problem was with the
subject of piece intersections during twisting. Some people on this
list are bothered by twisting geometry that doesn't smoothly slide
in either the model space or the viewing space, but that issue never
bothered me. On that point I was only interested in the geometric
result of twisting. Now I see that in those cases I was interested
in the resulting permutations and therefore didn't care what
happened during the twist, however my geometric aesthetic governed
my interest in particular puzzles.
On a related note, I recently received the face-turning octahedron
puzzle that I ordered from the
that Nan posted, along with a floppy cube. The connection to this
discussion is that the plastic of the floppy cube bends strangely as
parts of the central cube gets pried open while twisting. It's a
very odd mechanism that may not stand up to a lot of use, but then
the puzzle is not at all floppy, possibly because of that very
design choice. I don't want it to break, but the bending doesn't
bother me at all if that is what allowed them to build a non-floppy
3x3x1 puzzle.
Just for completeness sake, I can report that the octahedron puzzle
is quite nice though it does take some care to keep some twists from
locking up. I really wish that the twists would snap into place like
my Chinese megaminx, but that's not a big problem. The most
surprising thing I found about this puzzle was just how quickly it
becomes scrambled. Even a single twist from the pristine position is
not immediately clear that it is only one twist, and two
intersecting twists are not nearly as obvious how to untwist as
similar twists on the original Rubik's cube. It is a very handsome
puzzle and makes me appreciate the humble octahedron all the more.
-Melinda
--------------090108080604070601050508--