Thread: "Interesting object"

From: "David Vanderschel" <DvdS@Austin.RR.com>
Date: Sun, 6 Feb 2011 18:48:17 -0600
Subject: Interesting object



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I just read the following article:
http://physicsworld.com/cws/article/indepth/44950

Hart's in-depth page on the construct is here:
http://www.georgehart.com/DC/index.html

Though I have not completely groked it yet, it struck me that there might b=
e yet another opportunity for a permutation puzzle here; so I am curious to=
see what insights some of the folks on the 4D_Cubing list might have. It =
would not surprise me if a connection can be found with some objects which =
have been discussed here.

Regards,
David V.

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=EF=BB=BF






I just read the following article:

href=3D"http://physicsworld.com/cws/article/indepth/44950">http://physicswo=
rld.com/cws/article/indepth/44950

 

Hart's in-depth page on the con=
struct is=20
here:

href=3D"http://www.georgehart.com/DC/index.html">http://www.georgehart.com/=
DC/index.html

 

Though I have not completely gr=
oked it=20
yet, it struck me that there might be yet another opportunity for a permuta=
tion=20
puzzle here; so I am curious to see what insights some of the folks on the=
=20
4D_Cubing list might have.  It would not surprise me if a connection c=
an be=20
found with some objects which have been discussed here.

 

Regards,

  David V.

 
L>

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From: Melinda Green <melinda@superliminal.com>
Date: Sun, 06 Feb 2011 18:20:19 -0800
Subject: Re: [MC4D] Interesting object



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Yes, that's definitely an interesting object, and yes, it does relate to=20
our particular interest. First, I think that George Hart is slightly=20
obscuring what I feel is the more natural way of describing the=20
polyhedron by having some edges crossing shared verticies as opposed to=20
terminating there. To simplify this unusual construction, just subdivide=20
each of those big triangles into four smaller ones and then the object=20
is much more easily described. That version also appears to be missing=20
from my collection of infinite regular polyhedra=20
. George Hart=20
helped me with these IRP's by copying an out-of-print book with a=20
collection of many figures containing many that I didn't already know=20
about. Even in its subdivided form, this polyhedron appears to be new to=20
me and not the book.

BTW, I know that George found my collection interesting because he once=20
copied my VRML files for the {5,5} which I had painfully worked out on=20
my own, and then he hosted it on his site without attribution, even=20
carefully removing my name from the comments. At least he took it down=20
when I confronted him. He's been a very nice and enthusiastic booster of=20
highly symmetric geometry and a generally nice and brilliant guy.

The way that his new surface relates to twisty puzzles is exactly the=20
same way that Roice implemented the twisty version of the {7,3} (duel of=20
the {3,7} ), also=20
known as Klein's Quartic . Any=20
of these sorts of finite hyperbolic polyhedra that live in infinitely=20
repeating 3-spaces (and probably many more that don't) can be turned=20
into similar twisty puzzles, especially ones in which 3 polygons meet at=20
each vertex. All of the possibilities that I know of would be the duels=20
of any of the polyhedra in the "Triangles" column of the table on my IRP=20
page . George's=20
gyrangle, once subdivided as I described above, can be seen as an=20
infinite {8,3}, and it's {3,8} duel could be made into a puzzle. Another=20
particularly beautiful {3,8} is this one=20
which has an=20
unusually high genus (five!). It is naturally modeled as a particular=20
cubic packing of snub cubes =20
meeting at their square faces, and with those faces removed. It was also=20
the hardest of all the IRP for me to model as there does not seem to be=20
a closed-form solution with which to compute the vertex coordinates. Don=20
helped me out with a method of computing them with an iterative function=20
to compute the coordinates to any required accuracy. I think that you=20
will agree from the screen shot that the 3D form is particularly=20
beautiful. I have no idea how difficult the resulting planar puzzle=20
might be but I'd definitely love to see it implemented. I'm looking at=20
you, Roice. :-)

Thanks for reporting on this new object, David. It's definitely=20
interesting and pertinent in several ways.
-Melinda

On 2/6/2011 4:48 PM, David Vanderschel wrote:
> =EF=BB=BF
>
> I just read the following article:
> http://physicsworld.com/cws/article/indepth/44950
> Hart's in-depth page on the construct is here:
> http://www.georgehart.com/DC/index.html
> Though I have not completely groked it yet, it struck me that there=20
> might be yet another opportunity for a permutation puzzle here; so I=20
> am curious to see what insights some of the folks on the 4D_Cubing=20
> list might have. It would not surprise me if a connection can be=20
> found with some objects which have been discussed here.
> Regards,
> David V.
>
>=20

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">


Yes, that's definitely an interesting object, and yes, it does
relate to our particular interest. First, I think that George Hart
is slightly obscuring what I feel is the more natural way of
describing the polyhedron by having some edges crossing shared
verticies as opposed to terminating there. To simplify this unusual
construction, just subdivide each of those big triangles into four
smaller ones and then the object is much more easily described. That
version also appears to be missing from my collection of href=3D"http://superliminal.com/geometry/infinite/infinite.htm">infin=
ite
regular polyhedra. George Hart helped me with these IRP's by
copying an out-of-print book with a collection of many figures
containing many that I didn't already know about. Even in its
subdivided form, this polyhedron appears to be new to me and not the
book.



BTW, I know that George found my collection interesting because he
once copied my VRML files for the {5,5} which I had painfully worked
out on my own, and then he hosted it on his site without
attribution, even carefully removing my name from the comments. At
least he took it down when I confronted him. He's been a very nice
and enthusiastic booster of highly symmetric geometry and a
generally nice and brilliant guy.



The way that his new surface relates to twisty puzzles is exactly
the same way that Roice implemented the twisty version of the {7,3}
(duel of the href=3D"http://superliminal.com/geometry/infinite/3_7a.htm">{3,7}=
),
also known as Klei=
n's
Quartic
. Any of these sorts of finite hyperbolic polyhedra
that live in infinitely repeating 3-spaces (and probably many more
that don't) can be turned into similar twisty puzzles, especially
ones in which 3 polygons meet at each vertex. All of the
possibilities that I know of would be the duels of any of the
polyhedra in the "Triangles" column of the table on my href=3D"http://superliminal.com/geometry/infinite/infinite.htm">IRP
page. George's gyrangle, once subdivided as I described above,
can be seen as an infinite {8,3}, and it's {3,8} duel could be made
into a puzzle. Another particularly beautiful {3,8} is href=3D"http://superliminal.com/geometry/infinite/3_8b.htm">this one<=
/a>
which has an unusually high genus (five!). It is naturally modeled
as a particular cubic packing of href=3D"http://en.wikipedia.org/wiki/Snub_cube">snub cubes
meeting at their square faces, and with those faces removed. It was
also the hardest of all the IRP for me to model as there does not
seem to be a closed-form solution with which to compute the vertex
coordinates. Don helped me out with a method of computing them with
an iterative function to compute the coordinates to any required
accuracy. I think that you will agree from the screen shot that the
3D form is particularly beautiful. I have no idea how difficult the
resulting planar puzzle might be but I'd definitely love to see it
implemented. I'm looking at you, Roice. :-)



Thanks for reporting on this new object, David. It's definitely
interesting and pertinent in several ways.

-Melinda



On 2/6/2011 4:48 PM, David Vanderschel wrote:

type=3D"cite">
=EF=BB=BF

pe">

=20=20=20=20=20=20
I just read the following article:

moz-do-not-send=3D"true"
href=3D"http://physicsworld.com/cws/article/indepth/44950">http=
://physicsworld.com/cws/article/indepth/44950

=C2=A0

Hart's in-depth page on
the construct is here:

moz-do-not-send=3D"true"
href=3D"http://www.georgehart.com/DC/index.html">http://www.geo=
rgehart.com/DC/index.html

=C2=A0

Though I have not
completely groked it yet, it struck me that there might be yet
another opportunity for a permutation puzzle here; so I am
curious to see what insights some of the folks on the
4D_Cubing list might have.=C2=A0 It would not surprise me if a
connection can be found with some objects which have been
discussed here.

=C2=A0

Regards,

=C2=A0 David V.<=
/div>
=C2=A0

=20=20=20=20=20=20





--------------070406040903020102070305--




From: Roice Nelson <roice3@gmail.com>
Date: Mon, 7 Feb 2011 23:02:37 -0600
Subject: Re: [MC4D] Interesting object



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Interesting... so this means there should be an alternate, genus 5 puzzle
based on {8,3}. And when I looked at genus 5 tilings at the tilings.org
table , indeed it suggests
there should be a 24-color version. It must have some tiling pattern which
does not fit into the simple rule I've used for the current puzzles. I'll
plan to investigate at some point, but it would be awesome if someone wante=
d
to see if they can figure out the pattern of 24 colors which fit together,
then explain it to me :) I made some blank pictures of an {8,3} tiling,
hopefully suitable for printing, to help out any who are interested in
tackling the problem. They are
here and
here . Thanks for
sharing the associated IRP Melinda - that arrangement of snub cubes really
is beautiful.

I'd love for MagicTile to handle the {3,7} and {3,8} triangle-faced puzzles
directly too, rather than just their duals, and have started investigating
how these might be sliced up. Andrey's amazing observation of edge
behavior on Alex's {4,4} puzzle made me wonder what else all the puzzles
with non-simplex vertex figures have in store for us!

By the way, anyone want to make a guess as to what this
puzzleis?
(hint: it is in conformal disguise)

Cheers,
Roice

P.S. I got to meet George at the Gathering for Gardner conference last
March. His daughter Vi was there as well, and has been making some waves
online of late. She has a unique site at www.vihart.com, which I'm
sure many in this group would enjoy.



On Sun, Feb 6, 2011 at 8:20 PM, Melinda Green wro=
te:

>
>
> Yes, that's definitely an interesting object, and yes, it does relate to
> our particular interest. First, I think that George Hart is slightly
> obscuring what I feel is the more natural way of describing the polyhedro=
n
> by having some edges crossing shared verticies as opposed to terminating
> there. To simplify this unusual construction, just subdivide each of thos=
e
> big triangles into four smaller ones and then the object is much more eas=
ily
> described. That version also appears to be missing from my collection of =
infinite
> regular polyhedra >.
> George Hart helped me with these IRP's by copying an out-of-print book wi=
th
> a collection of many figures containing many that I didn't already know
> about. Even in its subdivided form, this polyhedron appears to be new to =
me
> and not the book.
>
> BTW, I know that George found my collection interesting because he once
> copied my VRML files for the {5,5} which I had painfully worked out on my
> own, and then he hosted it on his site without attribution, even carefull=
y
> removing my name from the comments. At least he took it down when I
> confronted him. He's been a very nice and enthusiastic booster of highly
> symmetric geometry and a generally nice and brilliant guy.
>
> The way that his new surface relates to twisty puzzles is exactly the sam=
e
> way that Roice implemented the twisty version of the {7,3} (duel of the
> {3,7} ), also known a=
s
> Klein's Quartic . Any of these
> sorts of finite hyperbolic polyhedra that live in infinitely repeating
> 3-spaces (and probably many more that don't) can be turned into similar
> twisty puzzles, especially ones in which 3 polygons meet at each vertex. =
All
> of the possibilities that I know of would be the duels of any of the
> polyhedra in the "Triangles" column of the table on my IRP pageperliminal.com/geometry/infinite/infinite.htm>.
> George's gyrangle, once subdivided as I described above, can be seen as a=
n
> infinite {8,3}, and it's {3,8} duel could be made into a puzzle. Another
> particularly beautiful {3,8} is this one/infinite/3_8b.htm>which has an unusually high genus (five!). It is natural=
ly modeled as a
> particular cubic packing of snub cubescube>meeting at their square faces, and with those faces removed. It was al=
so the
> hardest of all the IRP for me to model as there does not seem to be a
> closed-form solution with which to compute the vertex coordinates. Don
> helped me out with a method of computing them with an iterative function =
to
> compute the coordinates to any required accuracy. I think that you will
> agree from the screen shot that the 3D form is particularly beautiful. I
> have no idea how difficult the resulting planar puzzle might be but I'd
> definitely love to see it implemented. I'm looking at you, Roice. :-)
>
> Thanks for reporting on this new object, David. It's definitely interesti=
ng
> and pertinent in several ways.
> -Melinda
>
>
> On 2/6/2011 4:48 PM, David Vanderschel wrote:
>
> =EF=BB=BF
> I just read the following article:
> http://physicsworld.com/cws/article/indepth/44950
>
> Hart's in-depth page on the construct is here:
> http://www.georgehart.com/DC/index.html
>
> Though I have not completely groked it yet, it struck me that there might
> be yet another opportunity for a permutation puzzle here; so I am curious=
to
> see what insights some of the folks on the 4D_Cubing list might have. It
> would not surprise me if a connection can be found with some objects whic=
h
> have been discussed here.
>
> Regards,
> David V.
>
>

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Content-Transfer-Encoding: quoted-printable

Interesting... so this means there should be an alternate, genus 5 puz=
zle based on {8,3}.=C2=A0 And when I looked at genus 5 tilings at the ef=3D"http://tilings.org/pubs/tileclasstables.pdf" target=3D"_blank">tiling=
s.org table
, indeed=C2=A0it suggests there should be a 24-color version=
. =C2=A0It must have some tiling pattern which does not fit into the simple=
rule I've used for the current puzzles.=C2=A0 I'll plan to investi=
gate at some point, but it=C2=A0would be awesome if someone wanted to see i=
f they can figure out the pattern of 24 colors which fit together, then exp=
lain it to me :)=C2=A0 I made some blank pictures of an {8,3} tiling, hopef=
ully suitable for printing, to help out any who are interested in tackling =
the problem. =C2=A0They are -3/8-3_tiling1.png">here=C2=A0and agictile/8-3/8-3_tiling2.png">here.=C2=A0 Thanks for sharing the associ=
ated IRP Melinda - that arrangement of snub cubes really is beautiful.>


=C2=A0

I'd love for MagicTile to handle the {3,7} and {3,8} triangle-face=
d puzzles directly=C2=A0too, rather than just their duals, and have started=
investigating how these might be sliced up.=C2=A0 Andrey's amazing obs=
ervation of edge behavior=C2=A0on Alex's {4,4} puzzle made me wonder wh=
at else=C2=A0all the puzzles with non-simplex vertex figures have in store =
for us!



=C2=A0

By the way, anyone want to make a guess as to what ww.gravitation3d.com/magictile/pics/what_am_i.png">this puzzle is?=C2=
=A0 (hint: it is in conformal disguise)

=C2=A0

Cheers,

Roice

=C2=A0

P.S. I got to meet George at the Gathering for Gardner conference last=
March.=C2=A0 His daughter Vi was there as well, and has been making some w=
aves online of late.=C2=A0 She has a unique site at ihart.com/" target=3D"_blank">www.vihart.com, which I'm sure=C2=A0m=
any in this group=C2=A0would enjoy.





=C2=A0

On Sun, Feb 6, 2011 at 8:20 PM, Melinda Green pan dir=3D"ltr"><blank">melinda@superliminal.com> wrote:

dding-left:1ex" class=3D"gmail_quote">


Yes, that's definitel=
y an interesting object, and yes, it does relate to our particular interest=
. First, I think that George Hart is slightly obscuring what I feel is the =
more natural way of describing the polyhedron by having some edges crossing=
shared verticies as opposed to terminating there. To simplify this unusual=
construction, just subdivide each of those big triangles into four smaller=
ones and then the object is much more easily described. That version also =
appears to be missing from my collection of com/geometry/infinite/infinite.htm" target=3D"_blank">infinite regular poly=
hedra
. George Hart helped me with these IRP's by copying an out-of-=
print book with a collection of many figures containing many that I didn=
9;t already know about. Even in its subdivided form, this polyhedron appear=
s to be new to me and not the book.



BTW, I know that George found my collection interesting because he once=
copied my VRML files for the {5,5} which I had painfully worked out on my =
own, and then he hosted it on his site without attribution, even carefully =
removing my name from the comments. At least he took it down when I confron=
ted him. He's been a very nice and enthusiastic booster of highly symme=
tric geometry and a generally nice and brilliant guy.



The way that his new surface relates to twisty puzzles is exactly the s=
ame way that Roice implemented the twisty version of the {7,3} (duel of the=
blank">{3,7}), also known as lein.html" target=3D"_blank">Klein's Quartic. Any of these sorts of=
finite hyperbolic polyhedra that live in infinitely repeating 3-spaces (an=
d probably many more that don't) can be turned into similar twisty puzz=
les, especially ones in which 3 polygons meet at each vertex. All of the po=
ssibilities that I know of would be the duels of any of the polyhedra in th=
e "Triangles" column of the table on my minal.com/geometry/infinite/infinite.htm" target=3D"_blank">IRP page. G=
eorge's gyrangle, once subdivided as I described above, can be seen as =
an infinite {8,3}, and it's {3,8} duel could be made into a puzzle. Ano=
ther particularly beautiful {3,8} is metry/infinite/3_8b.htm" target=3D"_blank">this one which has an unusua=
lly high genus (five!). It is naturally modeled as a particular cubic packi=
ng of =
snub cubes
meeting at their square faces, and with those faces removed.=
It was also the hardest of all the IRP for me to model as there does not s=
eem to be a closed-form solution with which to compute the vertex coordinat=
es. Don helped me out with a method of computing them with an iterative fun=
ction to compute the coordinates to any required accuracy. I think that you=
will agree from the screen shot that the 3D form is particularly beautiful=
. I have no idea how difficult the resulting planar puzzle might be but I&#=
39;d definitely love to see it implemented. I'm looking at you, Roice. =
:-)



Thanks for reporting on this new object, David. It's definitely int=
eresting and pertinent in several ways.
-Melinda=
=20




On 2/6/2011 4:48 PM, David Vanderschel wrote:=20
=EF=BB=BF=20
I just read the following article:


=C2=A0

Hart's in-depth page on t=
he construct is here:


=C2=A0

Though I have not completely =
groked it yet, it struck me that there might be yet another opportunity for=
a permutation puzzle here; so I am curious to see what insights some of th=
e folks on the 4D_Cubing list might have.=C2=A0 It would not surprise me if=
a connection can be found with some objects which have been discussed here=
.



=C2=A0

Regards,

=C2=A0 David V.
<=
/blockquote>


--0016e6da98b9d8548d049bbe4148--




From: "Andrey" <andreyastrelin@yahoo.com>
Date: Tue, 08 Feb 2011 09:05:23 -0000
Subject: Re: [MC4D] Interesting object



Roice,
Yes, their picture of the vertex shows that there are really 8 triagles m=
eet at the point, but they call it (10,3) lattice... It looks like the surf=
ace of the periodic subset of octahedral-tetrahedral lattice. May be you kn=
ow how to describe this subset?
My method of painting of (8,3) gives 28-color painting! Picture is here: =
http://astr73.narod.ru/pics/8x3_28.jpg Cells are indexed by frations a/b, w=
here a,b are from Z/8Z, and fractions a/b and (-a)/(-b) are equivalent, but=
a/b and (3a)/(3b) are different. If cells a/b, c/d and e/f have common ver=
tex, then a+p*c+q*e=3D0, b+p*d+q*f=3D0 for some p,q=3D1 or -1.
So we have 28 colors: 1/0,3/0,n/1,n/3,k/2,k/6,1/4,3/4, where n=3D0..7, k=
=3D1,3,5,7. If we unify colors a/b and 3a/3b, we'll get non-orientable 14-c=
olor puzzle.=20
And about your riddle - it is too simple for me. Is it OK to leave it to =
somebody else?

Good luck!
Andrey


--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> Interesting... so this means there should be an alternate, genus 5 puzzle
> based on {8,3}. And when I looked at genus 5 tilings at the tilings.org
> table , indeed it suggests
> there should be a 24-color version. It must have some tiling pattern whi=
ch
> does not fit into the simple rule I've used for the current puzzles. I'l=
l
> plan to investigate at some point, but it would be awesome if someone wan=
ted
> to see if they can figure out the pattern of 24 colors which fit together=
,
> then explain it to me :) I made some blank pictures of an {8,3} tiling,
> hopefully suitable for printing, to help out any who are interested in
> tackling the problem. They are
> here and
> here . Thanks fo=
r
> sharing the associated IRP Melinda - that arrangement of snub cubes reall=
y
> is beautiful.
>=20
> I'd love for MagicTile to handle the {3,7} and {3,8} triangle-faced puzzl=
es
> directly too, rather than just their duals, and have started investigatin=
g
> how these might be sliced up. Andrey's amazing observation of edge
> behavior on Alex's {4,4} puzzle made me wonder what else all the puzzles
> with non-simplex vertex figures have in store for us!
>=20
> By the way, anyone want to make a guess as to what this
> puzzleis?
> (hint: it is in conformal disguise)
>=20
> Cheers,
> Roice
>=20




From: "Andrey" <andreyastrelin@yahoo.com>
Date: Tue, 08 Feb 2011 10:56:34 -0000
Subject: Re: [MC4D] Interesting object




Small correction: We don't need k/6 set, because 1/2=3D7/6 and 3/2=3D5/6. I=
t gives 24 colors, not 28. And after the reduction (a/b=3D3a/3b) we get 1/2=
=3D3/6=3D5/2, so we have only 12 colors (and the puzzle is orientable).

Andrey

--- In 4D_Cubing@yahoogroups.com, "Andrey" wrote:
>
> Roice,
> Yes, their picture of the vertex shows that there are really 8 triagles=
meet at the point, but they call it (10,3) lattice... It looks like the su=
rface of the periodic subset of octahedral-tetrahedral lattice. May be you =
know how to describe this subset?
> My method of painting of (8,3) gives 28-color painting! Picture is here=
: http://astr73.narod.ru/pics/8x3_28.jpg Cells are indexed by frations a/b,=
where a,b are from Z/8Z, and fractions a/b and (-a)/(-b) are equivalent, b=
ut a/b and (3a)/(3b) are different. If cells a/b, c/d and e/f have common v=
ertex, then a+p*c+q*e=3D0, b+p*d+q*f=3D0 for some p,q=3D1 or -1.
> So we have 28 colors: 1/0,3/0,n/1,n/3,k/2,k/6,1/4,3/4, where n=3D0..7, =
k=3D1,3,5,7. If we unify colors a/b and 3a/3b, we'll get non-orientable 14-=
color puzzle.=20
> And about your riddle - it is too simple for me. Is it OK to leave it t=
o somebody else?
>=20
> Good luck!
> Andrey
>=20
>=20
> --- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
> >
> > Interesting... so this means there should be an alternate, genus 5 puzz=
le
> > based on {8,3}. And when I looked at genus 5 tilings at the tilings.or=
g
> > table , indeed it suggests
> > there should be a 24-color version. It must have some tiling pattern w=
hich
> > does not fit into the simple rule I've used for the current puzzles. I=
'll
> > plan to investigate at some point, but it would be awesome if someone w=
anted
> > to see if they can figure out the pattern of 24 colors which fit togeth=
er,
> > then explain it to me :) I made some blank pictures of an {8,3} tiling=
,
> > hopefully suitable for printing, to help out any who are interested in
> > tackling the problem. They are
> > here and
> > here . Thanks =
for
> > sharing the associated IRP Melinda - that arrangement of snub cubes rea=
lly
> > is beautiful.
> >=20
> > I'd love for MagicTile to handle the {3,7} and {3,8} triangle-faced puz=
zles
> > directly too, rather than just their duals, and have started investigat=
ing
> > how these might be sliced up. Andrey's amazing observation of edge
> > behavior on Alex's {4,4} puzzle made me wonder what else all the puzzle=
s
> > with non-simplex vertex figures have in store for us!
> >=20
> > By the way, anyone want to make a guess as to what this
> > puzzleis?
> > (hint: it is in conformal disguise)
> >=20
> > Cheers,
> > Roice
> >
>




From: Melinda Green <melinda@superliminal.com>
Date: Tue, 08 Feb 2011 03:13:49 -0800
Subject: Re: [MC4D] Interesting object



--------------030809050306020500020604
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: quoted-printable

Regarding the 24 coloring of the {8,3}, I don't think that the=20
hyperbolic view will help to visualize this very much because of the=20
extreme curvature, but I think that I can describe it fairly well. The=20
trick is to look at my {3,8}=20
screen shot and=20
hold the idea in your head that there is really only one red sphere and=20
one green one. The rest are just copies. So the repeat unit is this=20
two-sphere peanut shape. The octahedra are centered on each of the=20
vertices, with edge length 1/2, That is, pairs of octahedra meet at the=20
midpoints of each edge. Each octahedron consists of 4 green triangles=20
and 4 red ones folded into a sort of floppy butterfly shape. Since each=20
octahedron can be uniquely identified by the vertex at its center, the=20
coloring problem boils down to coloring each vertex of one faceted peanut.

Now notice that each vertex is part of a ring of 4 vertices surrounding=20
the squares where pairs of snub cubes meet. This says that all octagons=20
will participate in a exactly one ring of 4 octagons, stacked like 4=20
stop signs on the same pole. These will create straight parallel lines=20
across the hyperbolic plane. We can color these 1, 2, 3, 4, 1, 2, 3,=20
4,... Also, notice that to get from one ring to a copy of that ring, you=20
will have to first encounter another parallel ring which is on the=20
opposite side of a snub cube. Color those 5, 6, 7, 8, 5, 6, 7, 8,...=20
This pattern then repeats, alternating between parallel lines of the=20
first four colors with lines of the second four.

Since a cell of any given color now has two neighbor cells colored, that=20
accounts for 1/3 of all the cells. Another way to look at that is that=20
we've accounted for a stack of alternating red-green spheres in one of=20
the 3D coordinate axes. There are two more axes to go, accounting for=20
all 24 colors. In the hyperbolic plane, those lines neither parallel to=20
the first set, nor do they cross them, which I find *very* interesting!=20
I've numbered the coloring of one of Roice's nets here=20
sudwm97oHpdxPwKYDwVnBnbJMbonJoveIr0PrdjJg/Melinda/8-3_24.png>.=20
There are still some more large cells to fill in but I'm too tired to=20
try to fill in more. Hopefully this should give enough to show the pattern.

Regarding puzzles made from the {3,7} and {3,8} directly, that would=20
definitely be neat. It may even help to be doing this in the plane in=20
order to understand the general problem of non-simplex vertex figures.=20
What I'd *really* prefer would be to see those puzzles as painted onto=20
the surface of either repeating polyhedral models or onto curved minimal=20
surfaces. I can easily supply the coordinates and connectivity data for=20
the polyhedral models if anyone wants to try that. The animations could=20
be accomplished by animating textures mapped to those surfaces. That=20
would be very fascinating to see.

As for your mystery puzzle, it looks to me like a simple planar tiling=20
of hexagons inverted across a central circle. Did I get that right?

-Melinda

On 2/7/2011 9:02 PM, Roice Nelson wrote:
>
>
> Interesting... so this means there should be an alternate, genus 5=20
> puzzle based on {8,3}. And when I looked at genus 5 tilings at the=20
> tilings.org table ,=20
> indeed it suggests there should be a 24-color version. It must have=20
> some tiling pattern which does not fit into the simple rule I've used=20
> for the current puzzles. I'll plan to investigate at some point, but=20
> it would be awesome if someone wanted to see if they can figure out=20
> the pattern of 24 colors which fit together, then explain it to me :) =
=20
> I made some blank pictures of an {8,3} tiling, hopefully suitable for=20
> printing, to help out any who are interested in tackling the problem.=20
> They are here=20
> and here=20
> . Thanks for=20
> sharing the associated IRP Melinda - that arrangement of snub cubes=20
> really is beautiful.
> I'd love for MagicTile to handle the {3,7} and {3,8} triangle-faced=20
> puzzles directly too, rather than just their duals, and have started=20
> investigating how these might be sliced up. Andrey's amazing=20
> observation of edge behavior on Alex's {4,4} puzzle made me wonder=20
> what else all the puzzles with non-simplex vertex figures have in=20
> store for us!
> By the way, anyone want to make a guess as to what this puzzle=20
> is?=20=20
> (hint: it is in conformal disguise)
> Cheers,
> Roice
> P.S. I got to meet George at the Gathering for Gardner conference last=20
> March. His daughter Vi was there as well, and has been making some=20
> waves online of late. She has a unique site at www.vihart.com=20
> , which I'm sure many in this group would enjoy.
>
>
> On Sun, Feb 6, 2011 at 8:20 PM, Melinda Green=20
> > wrote:
>
>
>
> Yes, that's definitely an interesting object, and yes, it does
> relate to our particular interest. First, I think that George Hart
> is slightly obscuring what I feel is the more natural way of
> describing the polyhedron by having some edges crossing shared
> verticies as opposed to terminating there. To simplify this
> unusual construction, just subdivide each of those big triangles
> into four smaller ones and then the object is much more easily
> described. That version also appears to be missing from my
> collection of infinite regular polyhedra
> . George
> Hart helped me with these IRP's by copying an out-of-print book
> with a collection of many figures containing many that I didn't
> already know about. Even in its subdivided form, this polyhedron
> appears to be new to me and not the book.
>
> BTW, I know that George found my collection interesting because he
> once copied my VRML files for the {5,5} which I had painfully
> worked out on my own, and then he hosted it on his site without
> attribution, even carefully removing my name from the comments. At
> least he took it down when I confronted him. He's been a very nice
> and enthusiastic booster of highly symmetric geometry and a
> generally nice and brilliant guy.
>
> The way that his new surface relates to twisty puzzles is exactly
> the same way that Roice implemented the twisty version of the
> {7,3} (duel of the {3,7}
> ), also known
> as Klein's Quartic . Any
> of these sorts of finite hyperbolic polyhedra that live in
> infinitely repeating 3-spaces (and probably many more that don't)
> can be turned into similar twisty puzzles, especially ones in
> which 3 polygons meet at each vertex. All of the possibilities
> that I know of would be the duels of any of the polyhedra in the
> "Triangles" column of the table on my IRP page
> . George's
> gyrangle, once subdivided as I described above, can be seen as an
> infinite {8,3}, and it's {3,8} duel could be made into a puzzle.
> Another particularly beautiful {3,8} is this one
> which has an
> unusually high genus (five!). It is naturally modeled as a
> particular cubic packing of snub cubes
> meeting at their square
> faces, and with those faces removed. It was also the hardest of
> all the IRP for me to model as there does not seem to be a
> closed-form solution with which to compute the vertex coordinates.
> Don helped me out with a method of computing them with an
> iterative function to compute the coordinates to any required
> accuracy. I think that you will agree from the screen shot that
> the 3D form is particularly beautiful. I have no idea how
> difficult the resulting planar puzzle might be but I'd definitely
> love to see it implemented. I'm looking at you, Roice. :-)
>
> Thanks for reporting on this new object, David. It's definitely
> interesting and pertinent in several ways.
> -Melinda
>
>
> On 2/6/2011 4:48 PM, David Vanderschel wrote:
>> =EF=BB=BF
>> I just read the following article:
>> http://physicsworld.com/cws/article/indepth/44950
>> Hart's in-depth page on the construct is here:
>> http://www.georgehart.com/DC/index.html
>> Though I have not completely groked it yet, it struck me that
>> there might be yet another opportunity for a permutation puzzle
>> here; so I am curious to see what insights some of the folks on
>> the 4D_Cubing list might have. It would not surprise me if a
>> connection can be found with some objects which have been
>> discussed here.
>> Regards,
>> David V.
>
>
>
>=20

--------------030809050306020500020604
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable




">


Regarding the 24 coloring of the {8,3}, I don't think that the
hyperbolic view will help to visualize this very much because of the
extreme curvature, but I think that I can describe it fairly well.
The trick is to look at my href=3D"http://superliminal.com/geometry/infinite/3_8b.htm">{3,8}
screen shot and hold the idea in your head that there is really only
one red sphere and one green one. The rest are just copies. So the
repeat unit is this two-sphere peanut shape. The octahedra are
centered on each of the vertices, with edge length 1/2, That is,
pairs of octahedra meet at the midpoints of each edge. Each
octahedron consists of 4 green triangles and 4 red ones folded into
a sort of floppy butterfly shape. Since each octahedron can be
uniquely identified by the vertex at its center, the coloring
problem boils down to coloring each vertex of one faceted peanut.



Now notice that each vertex is part of a ring of 4 vertices
surrounding the squares where pairs of snub cubes meet. This says
that all octagons will participate in a exactly one ring of 4
octagons, stacked like 4 stop signs on the same pole. These will
create straight parallel lines across the hyperbolic plane. We can
color these 1, 2, 3, 4, 1, 2, 3, 4,... Also, notice that to get from
one ring to a copy of that ring, you will have to first encounter
another parallel ring which is on the opposite side of a snub cube.
Color those 5, 6, 7, 8, 5, 6, 7, 8,... This pattern then repeats,
alternating between parallel lines of the first four colors with
lines of the second four.



Since a cell of any given color now has two neighbor cells colored,
that accounts for 1/3 of all the cells. Another way to look at that
is that we've accounted for a stack of alternating red-green spheres
in one of the 3D coordinate axes. There are two more axes to go,
accounting for all 24 colors. In the hyperbolic plane, those lines
neither parallel to the first set, nor do they cross them, which I
find *very* interesting! I've numbered the coloring of one of
Roice's nets href=3D"http://f1.grp.yahoofs.com/v1/IBRRTcSJbUeUR6Bq46VVRa-3DmVBq8y0swff58=
QS0ULf2sudwm97oHpdxPwKYDwVnBnbJMbonJoveIr0PrdjJg/Melinda/8-3_24.png">herea>.
There are still some more large cells to fill in but I'm too tired
to try to fill in more. Hopefully this should give enough to show
the pattern.



Regarding puzzles made from the {3,7} and {3,8} directly, that would
definitely be neat. It may even help to be doing this in the plane
in order to understand the general problem of non-simplex vertex
figures. What I'd *really* prefer would be to see those puzzles as
painted onto the surface of either repeating polyhedral models or
onto curved minimal surfaces. I can easily supply the coordinates
and connectivity data for the polyhedral models if anyone wants to
try that. The animations could be accomplished by animating textures
mapped to those surfaces. That would be very fascinating to see.



As for your mystery puzzle, it looks to me like a simple planar
tiling of hexagons inverted across a central circle. Did I get that
right?



-Melinda



On 2/7/2011 9:02 PM, Roice Nelson wrote:
cite=3D"mid:AANLkTikNNd_ijau5QXgeY4172MUj338vLR=3DLij0fM=3DHD@mail.gm=
ail.com"
type=3D"cite">


Interesting... so this means there should be an alternate,
genus 5 puzzle based on {8,3}.=C2=A0 And when I looked at genus 5
tilings at the href=3D"http://tilings.org/pubs/tileclasstables.pdf"
target=3D"_blank">tilings.org table
, indeed=C2=A0it suggests
there should be a 24-color version. =C2=A0It must have some tiling
pattern which does not fit into the simple rule I've used for
the current puzzles.=C2=A0 I'll plan to investigate at some point,
but it=C2=A0would be awesome if someone wanted to see if they can
figure out the pattern of 24 colors which fit together, then
explain it to me :)=C2=A0 I made some blank pictures of an {8,3}
tiling, hopefully suitable for printing, to help out any who are
interested in tackling the problem. =C2=A0They are moz-do-not-send=3D"true"
href=3D"http://gravitation3d.com/magictile/8-3/8-3_tiling1.png">h=
ere=C2=A0and
href=3D"http://gravitation3d.com/magictile/8-3/8-3_tiling2.png">h=
ere
.=C2=A0
Thanks for sharing the associated IRP Melinda - that arrangement
of snub cubes really is beautiful.

=C2=A0

I'd love for MagicTile to handle the {3,7} and {3,8}
triangle-faced puzzles directly=C2=A0too, rather than just their
duals, and have started investigating how these might be sliced
up.=C2=A0 Andrey's amazing observation of edge behavior=C2=A0on Ale=
x's
{4,4} puzzle made me wonder what else=C2=A0all the puzzles with
non-simplex vertex figures have in store for us!

=C2=A0

By the way, anyone want to make a guess as to what moz-do-not-send=3D"true"
href=3D"http://www.gravitation3d.com/magictile/pics/what_am_i.png=
">this
puzzle is?=C2=A0 (hint: it is in conformal disguise)

=C2=A0

Cheers,

Roice

=C2=A0

P.S. I got to meet George at the Gathering for Gardner
conference last March.=C2=A0 His daughter Vi was there as well, and
has been making some waves online of late.=C2=A0 She has a unique
site at target=3D"_blank">www.vihart.com, which I'm sure=C2=A0many in
this group=C2=A0would enjoy.





=C2=A0

On Sun, Feb 6, 2011 at 8:20 PM, Melinda
Green < href=3D"mailto:melinda@superliminal.com" target=3D"_blank">meli=
nda@superliminal.com
>

wrote:

margin: 0px 0px 0px 0.8ex; padding-left: 1ex;"
class=3D"gmail_quote">




Yes, that's definitely an interesting object, and yes, it
does relate to our particular interest. First, I think that
George Hart is slightly obscuring what I feel is the more
natural way of describing the polyhedron by having some
edges crossing shared verticies as opposed to terminating
there. To simplify this unusual construction, just subdivide
each of those big triangles into four smaller ones and then
the object is much more easily described. That version also
appears to be missing from my collection of moz-do-not-send=3D"true"
href=3D"http://superliminal.com/geometry/infinite/infinite.ht=
m"
target=3D"_blank">infinite regular polyhedra. George
Hart helped me with these IRP's by copying an out-of-print
book with a collection of many figures containing many that
I didn't already know about. Even in its subdivided form,
this polyhedron appears to be new to me and not the book.



BTW, I know that George found my collection interesting
because he once copied my VRML files for the {5,5} which I
had painfully worked out on my own, and then he hosted it on
his site without attribution, even carefully removing my
name from the comments. At least he took it down when I
confronted him. He's been a very nice and enthusiastic
booster of highly symmetric geometry and a generally nice
and brilliant guy.



The way that his new surface relates to twisty puzzles is
exactly the same way that Roice implemented the twisty
version of the {7,3} (duel of the href=3D"http://superliminal.com/geometry/infinite/3_7a.htm"
target=3D"_blank">{3,7}
), also known as moz-do-not-send=3D"true"
href=3D"http://math.ucr.edu/home/baez/klein.html"
target=3D"_blank">Klein's Quartic. Any of these sorts of
finite hyperbolic polyhedra that live in infinitely
repeating 3-spaces (and probably many more that don't) can
be turned into similar twisty puzzles, especially ones in
which 3 polygons meet at each vertex. All of the
possibilities that I know of would be the duels of any of
the polyhedra in the "Triangles" column of the table on my moz-do-not-send=3D"true"
href=3D"http://superliminal.com/geometry/infinite/infinite.ht=
m"
target=3D"_blank">IRP page. George's gyrangle, once
subdivided as I described above, can be seen as an infinite
{8,3}, and it's {3,8} duel could be made into a puzzle.
Another particularly beautiful {3,8} is moz-do-not-send=3D"true"
href=3D"http://superliminal.com/geometry/infinite/3_8b.htm"
target=3D"_blank">this one which has an unusually high
genus (five!). It is naturally modeled as a particular cubic
packing of href=3D"http://en.wikipedia.org/wiki/Snub_cube"
target=3D"_blank">snub cubes
meeting at their square
faces, and with those faces removed. It was also the hardest
of all the IRP for me to model as there does not seem to be
a closed-form solution with which to compute the vertex
coordinates. Don helped me out with a method of computing
them with an iterative function to compute the coordinates
to any required accuracy. I think that you will agree from
the screen shot that the 3D form is particularly beautiful.
I have no idea how difficult the resulting planar puzzle
might be but I'd definitely love to see it implemented. I'm
looking at you, Roice. :-)



Thanks for reporting on this new object, David. It's
definitely interesting and pertinent in several ways.

-Melinda





On 2/6/2011 4:48 PM, David Vanderschel wrote:
=EF=BB=BF
I just read the following article:<=
/font>

moz-do-not-send=3D"true"
href=3D"http://physicsworld.com/cws/article/indepth=
/44950"
target=3D"_blank">http://physicsworld.com/cws/artic=
le/indepth/44950

=C2=A0

Hart's
in-depth page on the construct is here:

moz-do-not-send=3D"true"
href=3D"http://www.georgehart.com/DC/index.html"
target=3D"_blank">http://www.georgehart.com/DC/inde=
x.html

=C2=A0

Though I
have not completely groked it yet, it struck me
that there might be yet another opportunity for a
permutation puzzle here; so I am curious to see
what insights some of the folks on the 4D_Cubing
list might have.=C2=A0 It would not surprise me if a
connection can be found with some objects which
have been discussed here.

=C2=A0

Regards,ont>

=C2=A0 Davi=
d V.







=20=20=20=20=20=20





--------------030809050306020500020604--




From: Roice Nelson <roice3@gmail.com>
Date: Tue, 8 Feb 2011 11:05:12 -0600
Subject: Re: [MC4D] Interesting object



--00032555e47e05397d049bc85a23
Content-Type: text/plain; charset=ISO-8859-1

Wow, I pose a question before falling asleep and wake up to two great
answers. Thanks! I'm grateful for both. Andrey's modular arithmetic
approach seems like it can be very helpful with the code calculations that
will need to be done, and Melinda's explanation provides a great deal of
insight into the connections between faces.

With Melinda's description, I think I know why my existing coloring rule
could not generate this tiling. It is because the pattern arises from these
hyperbolic translations of 6
ultraparallellines
(with 4 faces on each line), whereas my tiling rule is based on
patterns that arise in a rotationally symmetric way. I need to study
further, but if I'm right, one can not do a 1/8th, face-centered rotation of
a 24 colored {8,3} - it does not appear to be a symmetry of the object.

A coincident aside related to these hyperbolic translations is that I found
out yesterday I missed a class of checkerboards on KQ which are due to
symmetries of this kind. You can have six translational 4-cycles of KQ,
much like what we are discussing here (KQ is a more symmetric object though,
having the rotational symmetries as well). There is no analogous
checkerboard of Megaminx, since there are no parallel lines in spherical
geometry. I'll likely make the actual checkerboard this weekend, and will
email if I do. It will indeed be the last possible type, because I found
this out reading part of this
paper(see
p32), which enumerates all the possible KQ symmetries.

And on the mystery puzzle, that's exactly it Melinda. The image turned up
unexpectedly when I was playing with some code this weekend, and I was so
pleased that it looked very similar to the stereo graphically projected
Megaminx. You can do the same inversion on KQ to get a similar looking
picture of that as well.

seeya,
Roice

P.S. I hope this email came across ok, as the formatting of my last
one seems to be all messed up on the yahoo group site. Gmail and Yahoo
groups appear to be a little incongruous lately.

--00032555e47e05397d049bc85a23
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

Wow, I pose a question before falling asleep=A0and wake up to two grea=
t answers.=A0 Thanks!=A0 I'm grateful for both.=A0 Andrey's modular=
arithmetic approach seems like it can be very helpful=A0with the code calc=
ulations that will need to be done, and Melinda's explanation provides =
a great deal of insight into the connections between faces.


=A0

With Melinda's description, I think I know why my existing colorin=
g rule could not generate this tiling.=A0 It is because the pattern arises =
from these hyperbolic translations of 6 wiki/Ultraparallel#Non-intersecting_lines" target=3D"_blank">ultraparallel<=
/a> lines (with 4 faces on each line), whereas my tiling rule is based on p=
atterns that arise in a rotationally symmetric way.=A0 I need to study furt=
her, but if I'm right, one can not do a 1/8th, face-centered rotation o=
f a 24 colored {8,3}=A0- it does not appear to be a symmetry of the object.=


=A0

A coincident=A0aside related to these hyperbolic translations is that=
=A0I found out yesterday I missed a class of checkerboards on KQ which are =
due to symmetries of this kind.=A0 You can have six translational=A04-cycle=
s of KQ, much like what we are discussing here (KQ is a more symmetric obje=
ct though, having the rotational symmetries as well).=A0 There is no analog=
ous checkerboard of Megaminx, since there are no parallel lines in spherica=
l geometry.=A0 I'll likely make the actual checkerboard this weekend, a=
nd will email if I do.=A0 It will indeed be the last possible type, because=
I found this out reading part of
Book35/files/karcher.pdf" target=3D"_blank">this paper (see p32), which=
enumerates all the possible KQ symmetries.


=A0

And on the mystery puzzle, that's exactly it Melinda.=A0 The image=
turned up unexpectedly when I was playing with some code this weekend, and=
I was so pleased that it looked very similar to the stereo graphically pro=
jected Megaminx.=A0 You can do the same inversion on KQ to get a similar lo=
oking picture of that as well.


=A0

seeya,

Roice

=A0

P.S.=A0 I hope this email came across ok, as the formatting of my last=
one=A0seems to be all messed up on the yahoo group site.=A0 Gmail and Yaho=
o groups appear to be a little incongruous lately.


--00032555e47e05397d049bc85a23--




From: Melinda Green <melinda@superliminal.com>
Date: Tue, 08 Feb 2011 17:14:18 -0800
Subject: Re: [MC4D] Interesting object



--------------000501070407060009060306
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit

Thank you Roice. We aim to please! :-)

I had a strong feeling that Andrey was working on your problem at the
same time that I was. I must admit that I didn't understand his answer
at all, so I'm just glad that I had something complementary to add to
the discussion.

I had never heard of the term "ultraparallel" before. Maybe it should be
obvious in hindsight but running into it this way was very surprising
and really quite magical for me. One thing that they seem to suggest is
that it should be possible to create surfaces of arbitrarily high genus
this way. They almost certainly won't have regular polygonal
counterparts but it seems like there's no topological reason why you
can't have more than three sets of ultraparallel lines repeating across
a hyperbolic surface of sufficient curvature. I'd love to see what a
genus-20 puzzle looks like in Magic Tile.

I don't understand what you mean about the symmetry of this {8,3} not
allowing 1/8th face twists. It looks to me like any uniform tiling with
a triangular vertex figure should make for a natural magic tile so I
must be missing something. Is there a simple diagram or something that
shows this?

Even though I don't understand the problem, I wonder if you might be
able to create one very fun puzzle here by allowing only 180 degree
twists? Instead of coloring each face a solid color like usual, imagine
painting half of each face red and the other half green to match my
screen shot. The puzzle is solved when each sphere is a solid color. If
this works then it might make for some beautiful patterns. Solved magic
tile puzzles sometimes look like like unorganized patches of solid
colors, but this sort of two-colored puzzle would really highlight those
ultraparallel lines and make for some really beautiful patterns.

-Melinda

On 2/8/2011 9:05 AM, Roice Nelson wrote:
>
>
> Wow, I pose a question before falling asleep and wake up to two great
> answers. Thanks! I'm grateful for both. Andrey's modular arithmetic
> approach seems like it can be very helpful with the code calculations
> that will need to be done, and Melinda's explanation provides a great
> deal of insight into the connections between faces.
> With Melinda's description, I think I know why my existing coloring
> rule could not generate this tiling. It is because the pattern arises
> from these hyperbolic translations of 6 ultraparallel
>
> lines (with 4 faces on each line), whereas my tiling rule is based on
> patterns that arise in a rotationally symmetric way. I need to study
> further, but if I'm right, one can not do a 1/8th, face-centered
> rotation of a 24 colored {8,3} - it does not appear to be a symmetry
> of the object.
> A coincident aside related to these hyperbolic translations is that I
> found out yesterday I missed a class of checkerboards on KQ which are
> due to symmetries of this kind. You can have six
> translational 4-cycles of KQ, much like what we are discussing here
> (KQ is a more symmetric object though, having the rotational
> symmetries as well). There is no analogous checkerboard of Megaminx,
> since there are no parallel lines in spherical geometry. I'll likely
> make the actual checkerboard this weekend, and will email if I do. It
> will indeed be the last possible type, because I found this out
> reading part of this paper
> (see p32),
> which enumerates all the possible KQ symmetries.
> And on the mystery puzzle, that's exactly it Melinda. The image
> turned up unexpectedly when I was playing with some code this weekend,
> and I was so pleased that it looked very similar to the stereo
> graphically projected Megaminx. You can do the same inversion on KQ
> to get a similar looking picture of that as well.
> seeya,
> Roice
> P.S. I hope this email came across ok, as the formatting of my last
> one seems to be all messed up on the yahoo group site. Gmail and
> Yahoo groups appear to be a little incongruous lately.
>
>
>

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http-equiv="Content-Type">


Thank you Roice. We aim to please!  :-)



I had a strong feeling that Andrey was working on your problem at
the same time that I was. I must admit that I didn't understand his
answer at all, so I'm just glad that I had something complementary
to add to the discussion.



I had never heard of the term "ultraparallel" before. Maybe it
should be obvious in hindsight but running into it this way was very
surprising and really quite magical for me. One thing that they seem
to suggest is that it should be possible to create surfaces of
arbitrarily high genus this way. They almost certainly won't have
regular polygonal counterparts but it seems like there's no
topological reason why you can't have more than three sets of
ultraparallel lines repeating across a hyperbolic surface of
sufficient curvature. I'd love to see what a genus-20 puzzle looks
like in Magic Tile.



I don't understand what you mean about the symmetry of this {8,3}
not allowing 1/8th face twists. It looks to me like any uniform
tiling with a triangular vertex figure should make for a natural
magic tile so I must be missing something. Is there a simple diagram
or something that shows this?



Even though I don't understand the problem, I wonder if you might be
able to create one very fun puzzle here by allowing only 180 degree
twists? Instead of coloring each face a solid color like usual,
imagine painting half of each face red and the other half green to
match my screen shot. The puzzle is solved when each sphere is a
solid color. If this works then it might make for some beautiful
patterns. Solved magic tile puzzles sometimes look like like
unorganized patches of solid colors, but this sort of two-colored
puzzle would really highlight those ultraparallel lines and make for
some really beautiful patterns.



-Melinda



On 2/8/2011 9:05 AM, Roice Nelson wrote:
cite="mid:AANLkTik0HFB6MNQODgkL0yWRyVZjY3v2zrM6B9JP3T6K@mail.gmail.com"
type="cite">


Wow, I pose a question before falling asleep and wake up to
two great answers.  Thanks!  I'm grateful for both.  Andrey's
modular arithmetic approach seems like it can be very
helpful with the code calculations that will need to be done,
and Melinda's explanation provides a great deal of insight into
the connections between faces.

 

With Melinda's description, I think I know why my existing
coloring rule could not generate this tiling.  It is because the
pattern arises from these hyperbolic translations of 6 moz-do-not-send="true"
href="http://en.wikipedia.org/wiki/Ultraparallel#Non-intersecting_lines"
target="_blank">ultraparallel lines (with 4 faces on each
line), whereas my tiling rule is based on patterns that arise in
a rotationally symmetric way.  I need to study further, but if
I'm right, one can not do a 1/8th, face-centered rotation of a
24 colored {8,3} - it does not appear to be a symmetry of the
object.

 

A coincident aside related to these hyperbolic translations
is that I found out yesterday I missed a class of checkerboards
on KQ which are due to symmetries of this kind.  You can have
six translational 4-cycles of KQ, much like what we are
discussing here (KQ is a more symmetric object though, having
the rotational symmetries as well).  There is no analogous
checkerboard of Megaminx, since there are no parallel lines in
spherical geometry.  I'll likely make the actual checkerboard
this weekend, and will email if I do.  It will indeed be the
last possible type, because I found this out reading part of moz-do-not-send="true"
href="http://library.msri.org/books/Book35/files/karcher.pdf"
target="_blank">this paper (see p32), which enumerates all
the possible KQ symmetries.

 

And on the mystery puzzle, that's exactly it Melinda.  The
image turned up unexpectedly when I was playing with some code
this weekend, and I was so pleased that it looked very similar
to the stereo graphically projected Megaminx.  You can do the
same inversion on KQ to get a similar looking picture of that as
well.

 

seeya,

Roice

 

P.S.  I hope this email came across ok, as the formatting of
my last one seems to be all messed up on the yahoo group site. 
Gmail and Yahoo groups appear to be a little incongruous lately.







--------------000501070407060009060306--




From: Roice Nelson <roice3@gmail.com>
Date: Wed, 9 Feb 2011 10:27:54 -0600
Subject: Re: [MC4D] Interesting object



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Hi Melinda,

I haven't fully grokked Andrey's approach yet either, but my hope is that i=
f
I can understand it, it could help with some of the features I hope can be
supported. For instance, if I could quickly calculate the coloring pattern
from a single starting tile, supporting hyperbolic rotations/translations
might become much easier.

I planned to email out some corrections after further thought yesterday, an=
d
your email beat me to it :) My hypothesis looks mistaken, and I think
you're right that you can rotate a 24-colored {8,3} surface an eighth
turn about a face. I was talking about rotating the whole object (not just
a twist of a face), and the thought was an attempt to understand the reason
why my face coloring algorithm couldn't generate this puzzle (but that
turned out to be something different, the details of which I won't bore you
all with). The simple way to show a lack of rotational symmetry would be t=
o
find copies (of the face being rotated about) that did not move into each
other during the rotation.

Also, I should clarify something else I said, that KQ is only "more
symmetric" in the sense that it is maximally symmetric for its genus, while
the {8,3} is not (see wikipedia's Hurwitz
surface).
KQ has less symmetries than the 24-colored {8,3}.

Onward to higher genus surfaces, I don't know if there is a limit or not,
but I did run across an interesting surface a while back which is analogous
to KQ in many ways and is genus 70. Here is a paper on that surface, which
I hope to study more in time, as I can only wonder at the moment how many
ultraparallel lines might compose the object. The relevant dual tilings ar=
e
{11,5} and {5,11}, and I do know an {11,5} MagicTile puzzle would have 60
faces (though I don't know how playable it would be!).

http://www.neverendingbooks.org/DATA/biplanesingerman.pdf

As for any obviousness of ultraparallel lines, I have to say that with all
of this, none of it ever feels obvious to me in hindsight. I could stare a=
t
ultraparallel lines for a while, feel like they make sense at times, but
still never feeling like I fully grasp it. And for that matter, I can stil=
l
do the same with a picture of a "simple" dodecahedron. I like this
paragraph from the 120-cell
article
:

Telling the story in contemporary language
> has the danger that certain connections become
> =93obvious=94, and it is hard to understand how our
> mathematical ancestors could have overlooked
> them. However, there is no turning back; we can-
> not stop seeing the connections we see now, so
> the best thing to do is describe them as clearly as
> possible and recognise that our ancestors lacked
> our advantages.


I like your new puzzle idea. Something a little like it is possible right
now by setting multiple faces to the same color. So via configuration you
can make a 3-colored KQ puzzle by coloring sets of 8 the same. But it is
still different than you're describing (since faces are restricted to havin=
g
a single color, and twisting is unrestricted). Your idea reminds me of som=
e
of the gelatinbrain puzzle extensions.

Anyway, thanks again for working out the coloring (and to Andrey for that a=
s
well, and to David for starting this thread)!

seeya,
Roice



On Tue, Feb 8, 2011 at 7:14 PM, Melinda Green wro=
te:

>
>
> Thank you Roice. We aim to please! :-)
>
> I had a strong feeling that Andrey was working on your problem at the sam=
e
> time that I was. I must admit that I didn't understand his answer at all,=
so
> I'm just glad that I had something complementary to add to the discussion=
.
>
> I had never heard of the term "ultraparallel" before. Maybe it should be
> obvious in hindsight but running into it this way was very surprising and
> really quite magical for me. One thing that they seem to suggest is that =
it
> should be possible to create surfaces of arbitrarily high genus this way.
> They almost certainly won't have regular polygonal counterparts but it se=
ems
> like there's no topological reason why you can't have more than three set=
s
> of ultraparallel lines repeating across a hyperbolic surface of sufficien=
t
> curvature. I'd love to see what a genus-20 puzzle looks like in Magic Til=
e.
>
> I don't understand what you mean about the symmetry of this {8,3} not
> allowing 1/8th face twists. It looks to me like any uniform tiling with a
> triangular vertex figure should make for a natural magic tile so I must b=
e
> missing something. Is there a simple diagram or something that shows this=
?
>
> Even though I don't understand the problem, I wonder if you might be able
> to create one very fun puzzle here by allowing only 180 degree twists?
> Instead of coloring each face a solid color like usual, imagine painting
> half of each face red and the other half green to match my screen shot. T=
he
> puzzle is solved when each sphere is a solid color. If this works then it
> might make for some beautiful patterns. Solved magic tile puzzles sometim=
es
> look like like unorganized patches of solid colors, but this sort of
> two-colored puzzle would really highlight those ultraparallel lines and m=
ake
> for some really beautiful patterns.
>
> -Melinda
>

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Hi Melinda,



I haven't fully grokked Andrey's approach yet either, but my h=
ope is that if I can understand it, it could help with some of the features=
I hope can be supported. =A0For instance, if I could quickly calculate the=
coloring pattern from a single starting tile, supporting hyperbolic rotati=
ons/translations might become much easier.=A0




I planned to email out some corrections after further thought yesterda=
y, and your email beat me to it :) =A0My hypothesis looks mistaken, and=A0I=
think you're right that=A0you can rotate a 24-colored {8,3} surface an=
eighth turn=A0about a face. =A0I was talking about rotating the whole obje=
ct (not just a twist of a face), and the thought was an attempt to understa=
nd the reason why my face coloring algorithm couldn't generate this puz=
zle (but that turned out to be something different, the details of which I =
won't bore you all with).=A0=A0The simple way to=A0show a lack of rotat=
ional symmetry would be=A0to find=A0copies (of the face being rotated about=
) that did not move=A0into each other during the rotation.




Also, I should clarify something else I said, that KQ=A0is only "=
more symmetric" in the sense that it is maximally symmetric for its ge=
nus, while the {8,3} is not (see wikipedia's=A0ipedia.org/wiki/Hurwitz_surface" target=3D"_blank">Hurwitz surface). =
=A0KQ has less symmetries than the 24-colored {8,3}.




Onward to higher genus surfaces, I don't know if there is a limit =
or not, but I did run across an interesting surface a while back which is a=
nalogous to KQ in many ways and is genus 70.=A0=A0Here is a paper on that s=
urface, which I hope to study more in time, as I can only wonder at the mom=
ent how many ultraparallel lines might compose the object. =A0The relevant =
dual tilings are {11,5} and {5,11}, and I do know an {11,5} MagicTile puzzl=
e would have 60 faces (though I don't know how playable it would be!).<=
/div>






As for any obviousness of ultraparallel lines, I have to say that with=
all of this, none of it ever feels obvious to me in hindsight. =A0I could =
stare at ultraparallel lines for a while, feel like=A0they make sense at ti=
mes, but still never feeling like I fully grasp it. =A0And for that matter,=
=A0I can still do the same with a picture of a "simple" dodecahed=
ron. =A0I like this paragraph from 0101/fea-stillwell.pdf" target=3D"_blank">the 120-cell article:




px 0px 0.8ex; PADDING-LEFT: 1ex" class=3D"gmail_quote">Telling the story in=
contemporary language=A0
has the danger that certain connections become=


=93obvious=94, and it is hard to understand how our
mathematical ancesto=
rs could have overlooked
them. However, there is no turning back; we can=
-
not stop seeing the connections we see now, so=A0
the best thing to=
do is describe them as clearly as

possible and recognise that our ancestors lacked
our advantages.uote>


I like your new puzzle idea. =A0Something a little like it is possible=
right now by setting multiple faces to the same color. =A0So via configura=
tion you can make a 3-colored KQ puzzle by coloring sets of 8 the same. =A0=
But it is still different than you're describing (since faces are restr=
icted to having a single color, and twisting is unrestricted). =A0Your idea=
reminds me of some of the gelatinbrain puzzle extensions.




Anyway, thanks again for working out the coloring (and to Andrey for t=
hat as well, and to David for starting this thread)!



seeya,

Roice



target=3D"_blank">


On Tue, Feb 8, 2011 at 7:14 PM, Melinda Green pan dir=3D"ltr"><blank">melinda@superliminal.com> wrote:

; PADDING-LEFT: 1ex" class=3D"gmail_quote">


Thank you Roice. We aim t=
o please!=A0 :-)

I had a strong feeling that Andrey was working on y=
our problem at the same time that I was. I must admit that I didn't und=
erstand his answer at all, so I'm just glad that I had something comple=
mentary to add to the discussion.


I had never heard of the term "ultraparallel" before. Maybe i=
t should be obvious in hindsight but running into it this way was very surp=
rising and really quite magical for me. One thing that they seem to suggest=
is that it should be possible to create surfaces of arbitrarily high genus=
this way. They almost certainly won't have regular polygonal counterpa=
rts but it seems like there's no topological reason why you can't h=
ave more than three sets of ultraparallel lines repeating across a hyperbol=
ic surface of sufficient curvature. I'd love to see what a genus-20 puz=
zle looks like in Magic Tile.


I don't understand what you mean about the symmetry of this {8,3} n=
ot allowing 1/8th face twists. It looks to me like any uniform tiling with =
a triangular vertex figure should make for a natural magic tile so I must b=
e missing something. Is there a simple diagram or something that shows this=
?


Even though I don't understand the problem, I wonder if you might b=
e able to create one very fun puzzle here by allowing only 180 degree twist=
s? Instead of coloring each face a solid color like usual, imagine painting=
half of each face red and the other half green to match my screen shot. Th=
e puzzle is solved when each sphere is a solid color. If this works then it=
might make for some beautiful patterns. Solved magic tile puzzles sometime=
s look like like unorganized patches of solid colors, but this sort of two-=
colored puzzle would really highlight those ultraparallel lines and make fo=
r some really beautiful patterns.


-Melinda


--000325555e76732a78049bdbf268--




From: Roice Nelson <roice3@gmail.com>
Date: Wed, 9 Feb 2011 19:40:01 -0600
Subject: Re: [MC4D] Interesting object



--0016e6da98b9f2d37e049be3a863
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On Tue, Feb 8, 2011 at 11:05 AM, Roice Nelson wrote:

> A coincident aside related to these hyperbolic translations is that I found
> out yesterday I missed a class of checkerboards on KQ which are due to
> symmetries of this kind. You can have six translational 4-cycles of KQ,
> much like what we are discussing here
>

Bummer, it turns out the class of hyperbolic translation KQ symmetries don't
lead to valid checkerboards. The symmetries are "fixed point free", and all
84 edges move in 4-cycles. That's 21 4-cycles, an odd number of odd
permutations, so no go (unless we cheat and pop the puzzle apart :D).

On the plus side, the translation checkerboards should work on the {8,3}
puzzles, both the existing 12-colored one and the 24-colored one which needs
to be implemented.

24-colored
There are 8*24/2 = 96 edges, so when they are all 4-cycled, we'll have 24
sets.

12-colored
There are 8*12/2 = 48 edges, and the translations are 3-cycles in this case,
16 sets of them.

The latter should be a fun exercise, and if anyone would like to try, here
is a sequence to 3-cycle edge pieces without affecting anything else, using
clicks on only two faces.

( R2 L' R' L R' L R ) ( L2' R L R' L R' L' )

Swapping the two sub-sequences in parenthesis is another useful variant, and
these are what I used to make the KQ checkerboards.

Best,
Roice

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On Tue, Feb 8, 2011 at 11:05 AM, Roice Nelson pan dir=3D"ltr"><roice3@gmail.com>> wrote:
0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
A coincident=A0aside related to these hyperbolic translations is that=
=A0I found out yesterday I missed a class of checkerboards on KQ which are =
due to symmetries of this kind.=A0 You can have six translational=A04-cycle=
s of KQ, much like what we are discussing here


Bummer, it turns out the class of hyperbol=
ic translation=A0KQ symmetries don't lead to valid checkerboards.=A0 Th=
e symmetries are=A0"fixed point free", and all 84 edges move in 4=
-cycles.=A0 That's 21 4-cycles, an odd number of odd permutations, so n=
o go=A0(unless we cheat and pop the puzzle apart :D).

=A0
On the plus side, the translation checkerboards should w=
ork on the {8,3} puzzles, both the existing 12-colored one and the 24-color=
ed one which needs to be implemented.=A0
=A0
24-colored=

There are 8*24/2 =3D 96 edges, so when they are all 4-cycled, we'l=
l have 24 sets.=A0
=A0
12-colored
There are 8=
*12/2 =3D 48 edges, and the translations are 3-cycles in this case, 16 sets=
of them.=A0

=A0
The latter=A0should be a fun exercise,=A0and if anyone w=
ould like to try, here is a sequence to 3-cycle edge pieces without affecti=
ng anything else, using clicks on only two faces.
=A0
=
=A0=A0 =A0 =A0 =A0( R2 L' R' L R' L R ) ( L2' R L R' L =
R' L' )


Swapping the two sub-sequences in parenthesis is anothe=
r useful variant, and these are what I used to make the KQ checkerboards.div>

Best,
Roice=A0


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