A very interesting question about this puzzle is "how many distinct solved =
states are there?"
Since there are five tetrahedra with five colors, if all the tetrahedra are=
numbered, there are 5!=3D120 ways to color them. But let's not number them=
, but use the common convention that two coloring schemes are identical if =
a global rotation can change one to another. Mirroring is not allowed becau=
se this shape is chiral and thus asymmetric with respect to mirroring. Sinc=
e the shape has 60 rotational symmetries, the number of distinct coloring s=
chemes is down to 2. One still need to show the existence of a sequence of =
valid moves to get from one coloring scheme to another. I just did that by =
solving the puzzle from one scheme to the other. So I've just confirmed tha=
t "there are two distinct states for Twisty Star".
The two states can be described in this way. If you look at the center of p=
uzzle in the default view, there are five petals just like a flower. From m=
agenta, going clockwise, the colors are (magenta, yellow, green, red, blue)=
. Because these five petals are corners of five different tetrahedra, the c=
oloring of a "flower" determines the coloring of the whole puzzle. If you l=
ook at the puzzles from other angle, you can find other 11 "flowers" with d=
ifferent color sequences. In fact, for any even permutation of (magenta, ye=
llow, green, red, blue), you can always find a flower and a starting color =
such that the clockwise color sequence is that permutation. But for any odd=
permutation, it's not there. All odd permutations are on the other solved =
state.=20
So from the original state, if you want to do a three cycle to three colors=
, like magenta -> yellow -> green -> magenta, you don't have to twist the p=
uzzle. Global rotation suffices. If you want to swap two colors, you need t=
o do a lot of twists.=20
--- In 4D_Cubing@yahoogroups.com, "schuma"
>
> Hi everyone,
>=20
> The compound of five tetrahedra is a geometric object that interests me f=
or a long time. As a freshman, I carefully drew it using compass and straig=
htedge on my notebook. I've always been intrigued by its relation to the do=
decahedron, the chirality, and its pretty shape. In short, it's my geometri=
c crush.=20
>=20
> So far I haven't seen a twisty puzzle based on this shape. Recently, insp=
ired by Leslie Le's Super Star, I decided to write a simulator. That's defi=
nitely a good motivation for me to learn Java applet programming. It's done=
and can be found here:
>=20
> http://people.bu.edu/nanma/TwistyStar/TwistyStar.html
>=20
> You may need to upgrade JRE to see it. I call it "Twisty Star" because it=
's a twisty puzzle, and also the shape looks twisted. Some screenshots:
>=20
> [http://twistypuzzles.com/forum/download/file.php?id=3D30776]
> [http://twistypuzzles.com/forum/download/file.php?id=3D30777]
>=20
> It's a compound of five tetrahedra intersecting with each other. The 20 v=
ertices coincide with the vertices of a dodecahedron. The five tetrahedra a=
re colored by five colors.=20
>=20
> It can be twisted around 20 vertices. Since the cuts are right above the =
faces of the tetrahedra, it can be regarded as "face-turning" as well as "v=
ertex-turning". In other words, the dual of this solid is its mirror. So it=
's almost self-dual. Therefore there's a one to one correspondence between =
the vertices and the faces. Mathematically it's related to the face-turning=
icosahedra.
>=20
> The vertex to twist is labeled by a small circle around it, and the movin=
g region is highlighted. Even with these assistances, it's not easy to see =
how it turns. Sometimes with only one twist away from the solved state, I j=
ust can't find the twist. I haven't solved it yet. For this color scheme, I=
guess it has multiple solved states, meaning that one can, for example, sw=
ap the red tetrahedron with the blue one.
>=20
> I think it's possible to build a physical version. I posted this simulato=
r on the Twistypuzzles forum [http://twistypuzzles.com/forum/viewtopic.php?=
f=3D1&p=3D281801] and request the builders over there to consider it. Maybe=
someone will make it someday.
>=20
> I'd like to thank Melinda, Roice, Jeremy and Brandon for their feedback.=
=20
>=20
> Please let me know if you see more glitches.
>=20
> Have fun solving it!
>=20
> -- schuma
>
Hello,
After I programmed this compound polyhedra, in a personal email, Roice Nels=
on raised the question about compound polytopes in 4D. We didn't know such =
compounds. After some research, we found a list of them in Coxeter's "Regul=
ar Polytopes" (actually both of us have copies of this book and didn't read=
it carefully enough nor have a good memory), and some related links as fol=
lows:
http://homepages.wmich.edu/~drichter/zomeindex.htm
http://userpages.monmouth.com/~chenrich/CompoundPolytope/NewCompound.html
http://www.bendwavy.org/klitzing/explain/compound.htm
The list includes, for example, the compound of five 24-cells, the one of f=
ifteen 16-cells, etc. I have to say I'm not able to picture five 24-cells i=
nscribed in a 600-cell. I wonder if any group members have more experience =
with them and have something to share.
I found the compound are natural candidates for twisty puzzles, because (1)=
in most illustrations, the components of the compound are colored differen=
tly, just like on a puzzle (2) the faces of a component always cut other co=
mponents, which provides natural "cuts".=20
I hope some day we'll have 4D puzzles based on 4D compound!
Nan
--- In 4D_Cubing@yahoogroups.com, "schuma"
>
> A very interesting question about this puzzle is "how many distinct solve=
d states are there?"
>=20
> Since there are five tetrahedra with five colors, if all the tetrahedra a=
re numbered, there are 5!=3D120 ways to color them. But let's not number th=
em, but use the common convention that two coloring schemes are identical i=
f a global rotation can change one to another. Mirroring is not allowed bec=
ause this shape is chiral and thus asymmetric with respect to mirroring. Si=
nce the shape has 60 rotational symmetries, the number of distinct coloring=
schemes is down to 2. One still need to show the existence of a sequence o=
f valid moves to get from one coloring scheme to another. I just did that b=
y solving the puzzle from one scheme to the other. So I've just confirmed t=
hat "there are two distinct states for Twisty Star".
>=20
> The two states can be described in this way. If you look at the center of=
puzzle in the default view, there are five petals just like a flower. From=
magenta, going clockwise, the colors are (magenta, yellow, green, red, blu=
e). Because these five petals are corners of five different tetrahedra, the=
coloring of a "flower" determines the coloring of the whole puzzle. If you=
look at the puzzles from other angle, you can find other 11 "flowers" with=
different color sequences. In fact, for any even permutation of (magenta, =
yellow, green, red, blue), you can always find a flower and a starting colo=
r such that the clockwise color sequence is that permutation. But for any o=
dd permutation, it's not there. All odd permutations are on the other solve=
d state.=20
>=20
> So from the original state, if you want to do a three cycle to three colo=
rs, like magenta -> yellow -> green -> magenta, you don't have to twist the=
puzzle. Global rotation suffices. If you want to swap two colors, you need=
to do a lot of twists.=20
>=20
>=20
>=20
> --- In 4D_Cubing@yahoogroups.com, "schuma"
> >
> > Hi everyone,
> >=20
> > The compound of five tetrahedra is a geometric object that interests me=
for a long time. As a freshman, I carefully drew it using compass and stra=
ightedge on my notebook. I've always been intrigued by its relation to the =
dodecahedron, the chirality, and its pretty shape. In short, it's my geomet=
ric crush.=20
> >=20
> > So far I haven't seen a twisty puzzle based on this shape. Recently, in=
spired by Leslie Le's Super Star, I decided to write a simulator. That's de=
finitely a good motivation for me to learn Java applet programming. It's do=
ne and can be found here:
> >=20
> > http://people.bu.edu/nanma/TwistyStar/TwistyStar.html
> >=20
> > You may need to upgrade JRE to see it. I call it "Twisty Star" because =
it's a twisty puzzle, and also the shape looks twisted. Some screenshots:
> >=20
> > [http://twistypuzzles.com/forum/download/file.php?id=3D30776]
> > [http://twistypuzzles.com/forum/download/file.php?id=3D30777]
> >=20
> > It's a compound of five tetrahedra intersecting with each other. The 20=
vertices coincide with the vertices of a dodecahedron. The five tetrahedra=
are colored by five colors.=20
> >=20
> > It can be twisted around 20 vertices. Since the cuts are right above th=
e faces of the tetrahedra, it can be regarded as "face-turning" as well as =
"vertex-turning". In other words, the dual of this solid is its mirror. So =
it's almost self-dual. Therefore there's a one to one correspondence betwee=
n the vertices and the faces. Mathematically it's related to the face-turni=
ng icosahedra.
> >=20
> > The vertex to twist is labeled by a small circle around it, and the mov=
ing region is highlighted. Even with these assistances, it's not easy to se=
e how it turns. Sometimes with only one twist away from the solved state, I=
just can't find the twist. I haven't solved it yet. For this color scheme,=
I guess it has multiple solved states, meaning that one can, for example, =
swap the red tetrahedron with the blue one.
> >=20
> > I think it's possible to build a physical version. I posted this simula=
tor on the Twistypuzzles forum [http://twistypuzzles.com/forum/viewtopic.ph=
p?f=3D1&p=3D281801] and request the builders over there to consider it. May=
be someone will make it someday.
> >=20
> > I'd like to thank Melinda, Roice, Jeremy and Brandon for their feedback=
.=20
> >=20
> > Please let me know if you see more glitches.
> >=20
> > Have fun solving it!
> >=20
> > -- schuma
> >
>
I think that the simplest and most understandable compound it 3 hypercubes =
inscribed in 24-cell. I can't say now what kind of 24-cell slicing it gives=
, but it should be not difficult to imagine... But to draw it in understabl=
e way may be a little tricky. Non-convex 4D shapes are nightmare when you t=
ry to work with them.
Andrey
--- In 4D_Cubing@yahoogroups.com, "schuma"
>
> Hello,
>=20
> After I programmed this compound polyhedra, in a personal email, Roice Ne=
lson raised the question about compound polytopes in 4D. We didn't know suc=
h compounds. After some research, we found a list of them in Coxeter's "Reg=
ular Polytopes" (actually both of us have copies of this book and didn't re=
ad it carefully enough nor have a good memory), and some related links as f=
ollows:
>=20
> http://homepages.wmich.edu/~drichter/zomeindex.htm
> http://userpages.monmouth.com/~chenrich/CompoundPolytope/NewCompound.html
> http://www.bendwavy.org/klitzing/explain/compound.htm
>=20
> The list includes, for example, the compound of five 24-cells, the one of=
fifteen 16-cells, etc. I have to say I'm not able to picture five 24-cells=
inscribed in a 600-cell. I wonder if any group members have more experienc=
e with them and have something to share.
>=20
> I found the compound are natural candidates for twisty puzzles, because (=
1) in most illustrations, the components of the compound are colored differ=
ently, just like on a puzzle (2) the faces of a component always cut other =
components, which provides natural "cuts".=20
>=20
> I hope some day we'll have 4D puzzles based on 4D compound!
>=20
> Nan
>=20
>=20
>=20
>=20
> --- In 4D_Cubing@yahoogroups.com, "schuma"
> >
> > A very interesting question about this puzzle is "how many distinct sol=
ved states are there?"
> >=20
> > Since there are five tetrahedra with five colors, if all the tetrahedra=
are numbered, there are 5!=3D120 ways to color them. But let's not number =
them, but use the common convention that two coloring schemes are identical=
if a global rotation can change one to another. Mirroring is not allowed b=
ecause this shape is chiral and thus asymmetric with respect to mirroring. =
Since the shape has 60 rotational symmetries, the number of distinct colori=
ng schemes is down to 2. One still need to show the existence of a sequence=
of valid moves to get from one coloring scheme to another. I just did that=
by solving the puzzle from one scheme to the other. So I've just confirmed=
that "there are two distinct states for Twisty Star".
> >=20
> > The two states can be described in this way. If you look at the center =
of puzzle in the default view, there are five petals just like a flower. Fr=
om magenta, going clockwise, the colors are (magenta, yellow, green, red, b=
lue). Because these five petals are corners of five different tetrahedra, t=
he coloring of a "flower" determines the coloring of the whole puzzle. If y=
ou look at the puzzles from other angle, you can find other 11 "flowers" wi=
th different color sequences. In fact, for any even permutation of (magenta=
, yellow, green, red, blue), you can always find a flower and a starting co=
lor such that the clockwise color sequence is that permutation. But for any=
odd permutation, it's not there. All odd permutations are on the other sol=
ved state.=20
> >=20
> > So from the original state, if you want to do a three cycle to three co=
lors, like magenta -> yellow -> green -> magenta, you don't have to twist t=
he puzzle. Global rotation suffices. If you want to swap two colors, you ne=
ed to do a lot of twists.=20
> >=20
> >=20
> >=20
> > --- In 4D_Cubing@yahoogroups.com, "schuma"
> > >
> > > Hi everyone,
> > >=20
> > > The compound of five tetrahedra is a geometric object that interests =
me for a long time. As a freshman, I carefully drew it using compass and st=
raightedge on my notebook. I've always been intrigued by its relation to th=
e dodecahedron, the chirality, and its pretty shape. In short, it's my geom=
etric crush.=20
> > >=20
> > > So far I haven't seen a twisty puzzle based on this shape. Recently, =
inspired by Leslie Le's Super Star, I decided to write a simulator. That's =
definitely a good motivation for me to learn Java applet programming. It's =
done and can be found here:
> > >=20
> > > http://people.bu.edu/nanma/TwistyStar/TwistyStar.html
> > >=20
> > > You may need to upgrade JRE to see it. I call it "Twisty Star" becaus=
e it's a twisty puzzle, and also the shape looks twisted. Some screenshots:
> > >=20
> > > [http://twistypuzzles.com/forum/download/file.php?id=3D30776]
> > > [http://twistypuzzles.com/forum/download/file.php?id=3D30777]
> > >=20
> > > It's a compound of five tetrahedra intersecting with each other. The =
20 vertices coincide with the vertices of a dodecahedron. The five tetrahed=
ra are colored by five colors.=20
> > >=20
> > > It can be twisted around 20 vertices. Since the cuts are right above =
the faces of the tetrahedra, it can be regarded as "face-turning" as well a=
s "vertex-turning". In other words, the dual of this solid is its mirror. S=
o it's almost self-dual. Therefore there's a one to one correspondence betw=
een the vertices and the faces. Mathematically it's related to the face-tur=
ning icosahedra.
> > >=20
> > > The vertex to twist is labeled by a small circle around it, and the m=
oving region is highlighted. Even with these assistances, it's not easy to =
see how it turns. Sometimes with only one twist away from the solved state,=
I just can't find the twist. I haven't solved it yet. For this color schem=
e, I guess it has multiple solved states, meaning that one can, for example=
, swap the red tetrahedron with the blue one.
> > >=20
> > > I think it's possible to build a physical version. I posted this simu=
lator on the Twistypuzzles forum [http://twistypuzzles.com/forum/viewtopic.=
php?f=3D1&p=3D281801] and request the builders over there to consider it. M=
aybe someone will make it someday.
> > >=20
> > > I'd like to thank Melinda, Roice, Jeremy and Brandon for their feedba=
ck.=20
> > >=20
> > > Please let me know if you see more glitches.
> > >=20
> > > Have fun solving it!
> > >=20
> > > -- schuma
> > >
> >
>
Yes, it's vertex-turn 24 cell with cutting spaces going through next vertic=
es:
Puzzle 24-cell_VT_1
Dim 4
NAxis 1
Faces 1,1,0,0
Group 0,-1,1,0/-1,1,1,1 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
Axis 1,0,0,0
Twists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2
Cuts 1 -1
FixedMask 2
But all parts visible in the 24-cell form are cut from it in the compound f=
orm.
--- In 4D_Cubing@yahoogroups.com, "Andrey"
>
> I think that the simplest and most understandable compound it 3 hypercube=
s inscribed in 24-cell. I can't say now what kind of 24-cell slicing it giv=
es, but it should be not difficult to imagine... But to draw it in understa=
ble way may be a little tricky. Non-convex 4D shapes are nightmare when you=
try to work with them.
>=20
> Andrey
>=20
>=20
> --- In 4D_Cubing@yahoogroups.com, "schuma"
> >
> > Hello,
> >=20
> > After I programmed this compound polyhedra, in a personal email, Roice =
Nelson raised the question about compound polytopes in 4D. We didn't know s=
uch compounds. After some research, we found a list of them in Coxeter's "R=
egular Polytopes" (actually both of us have copies of this book and didn't =
read it carefully enough nor have a good memory), and some related links as=
follows:
> >=20
> > http://homepages.wmich.edu/~drichter/zomeindex.htm
> > http://userpages.monmouth.com/~chenrich/CompoundPolytope/NewCompound.ht=
ml
> > http://www.bendwavy.org/klitzing/explain/compound.htm
> >=20
> > The list includes, for example, the compound of five 24-cells, the one =
of fifteen 16-cells, etc. I have to say I'm not able to picture five 24-cel=
ls inscribed in a 600-cell. I wonder if any group members have more experie=
nce with them and have something to share.
> >=20
> > I found the compound are natural candidates for twisty puzzles, because=
(1) in most illustrations, the components of the compound are colored diff=
erently, just like on a puzzle (2) the faces of a component always cut othe=
r components, which provides natural "cuts".=20
> >=20
> > I hope some day we'll have 4D puzzles based on 4D compound!
> >=20
> > Nan
> >=20
> >=20
> >=20
> >=20
> > --- In 4D_Cubing@yahoogroups.com, "schuma"
> > >
> > > A very interesting question about this puzzle is "how many distinct s=
olved states are there?"
> > >=20
> > > Since there are five tetrahedra with five colors, if all the tetrahed=
ra are numbered, there are 5!=3D120 ways to color them. But let's not numbe=
r them, but use the common convention that two coloring schemes are identic=
al if a global rotation can change one to another. Mirroring is not allowed=
because this shape is chiral and thus asymmetric with respect to mirroring=
. Since the shape has 60 rotational symmetries, the number of distinct colo=
ring schemes is down to 2. One still need to show the existence of a sequen=
ce of valid moves to get from one coloring scheme to another. I just did th=
at by solving the puzzle from one scheme to the other. So I've just confirm=
ed that "there are two distinct states for Twisty Star".
> > >=20
> > > The two states can be described in this way. If you look at the cente=
r of puzzle in the default view, there are five petals just like a flower. =
From magenta, going clockwise, the colors are (magenta, yellow, green, red,=
blue). Because these five petals are corners of five different tetrahedra,=
the coloring of a "flower" determines the coloring of the whole puzzle. If=
you look at the puzzles from other angle, you can find other 11 "flowers" =
with different color sequences. In fact, for any even permutation of (magen=
ta, yellow, green, red, blue), you can always find a flower and a starting =
color such that the clockwise color sequence is that permutation. But for a=
ny odd permutation, it's not there. All odd permutations are on the other s=
olved state.=20
> > >=20
> > > So from the original state, if you want to do a three cycle to three =
colors, like magenta -> yellow -> green -> magenta, you don't have to twist=
the puzzle. Global rotation suffices. If you want to swap two colors, you =
need to do a lot of twists.=20
> > >=20
> > >=20
> > >=20
> > > --- In 4D_Cubing@yahoogroups.com, "schuma"
> > > >
> > > > Hi everyone,
> > > >=20
> > > > The compound of five tetrahedra is a geometric object that interest=
s me for a long time. As a freshman, I carefully drew it using compass and =
straightedge on my notebook. I've always been intrigued by its relation to =
the dodecahedron, the chirality, and its pretty shape. In short, it's my ge=
ometric crush.=20
> > > >=20
> > > > So far I haven't seen a twisty puzzle based on this shape. Recently=
, inspired by Leslie Le's Super Star, I decided to write a simulator. That'=
s definitely a good motivation for me to learn Java applet programming. It'=
s done and can be found here:
> > > >=20
> > > > http://people.bu.edu/nanma/TwistyStar/TwistyStar.html
> > > >=20
> > > > You may need to upgrade JRE to see it. I call it "Twisty Star" beca=
use it's a twisty puzzle, and also the shape looks twisted. Some screenshot=
s:
> > > >=20
> > > > [http://twistypuzzles.com/forum/download/file.php?id=3D30776]
> > > > [http://twistypuzzles.com/forum/download/file.php?id=3D30777]
> > > >=20
> > > > It's a compound of five tetrahedra intersecting with each other. Th=
e 20 vertices coincide with the vertices of a dodecahedron. The five tetrah=
edra are colored by five colors.=20
> > > >=20
> > > > It can be twisted around 20 vertices. Since the cuts are right abov=
e the faces of the tetrahedra, it can be regarded as "face-turning" as well=
as "vertex-turning". In other words, the dual of this solid is its mirror.=
So it's almost self-dual. Therefore there's a one to one correspondence be=
tween the vertices and the faces. Mathematically it's related to the face-t=
urning icosahedra.
> > > >=20
> > > > The vertex to twist is labeled by a small circle around it, and the=
moving region is highlighted. Even with these assistances, it's not easy t=
o see how it turns. Sometimes with only one twist away from the solved stat=
e, I just can't find the twist. I haven't solved it yet. For this color sch=
eme, I guess it has multiple solved states, meaning that one can, for examp=
le, swap the red tetrahedron with the blue one.
> > > >=20
> > > > I think it's possible to build a physical version. I posted this si=
mulator on the Twistypuzzles forum [http://twistypuzzles.com/forum/viewtopi=
c.php?f=3D1&p=3D281801] and request the builders over there to consider it.=
Maybe someone will make it someday.
> > > >=20
> > > > I'd like to thank Melinda, Roice, Jeremy and Brandon for their feed=
back.=20
> > > >=20
> > > > Please let me know if you see more glitches.
> > > >=20
> > > > Have fun solving it!
> > > >=20
> > > > -- schuma
> > > >
> > >
> >
>