Thread: "hemi-puzzles!"

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Hi all,

I added some hemi-puzzles, all ones we haven't seen before. The
hemi-dodecahedron and hemi-cube are not new, but I made them vertex turning
this time. There are also hemi-octahedron and hemi-icosahedron puzzles
now. All of these have the topology of the projective plane.

I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not
symmetrical (like the {8,3} 10-Color and some of the other hyperbolic
puzzles). It turns out to have 8 faces, 10 edges, and 4 vertices, so the Euler
Characteristic shows it
has the topology of a sphere. I'll try to write a little more about this
8C puzzle soon.

You can download the latest by clicking
here
.

Happy Holidays,
Roice

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Hi all,

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Here's a little more on the {3,5} 8-Color. These puzzles with asymmetrical
colorings are strange, but they arise naturally from the math that
identifies cells with each other, so I wanted to understand things a little
better.

To do that, I made a graph of the 4 vertices, 10 edges, and 8 faces all
"rolled up". By that I mean each of these features is only shown once,
rather than shown multiple times (with hand waving that "this face is
identified with that one", as the MagicTile presentation requires). An
image of my graph with default MagicTile colors is
here,
and some observations about it are:

- The cyan and purple cells are
henagons (polygons
with one vertex and one edge). The face twisting of these two scrambles
nothing.
- The blue and orange cells are degenerate triangles. They have three
sides, but only two vertices. I initially thought they were
digons,
but they are more like a digon with a henagon subtracted out. I don't know
if there is a special name for this polygon.
- The red, yellow, white, and green cells are proper triangles. (By the
way, if you want to trace out the green and white triangles in the graph,
note that the edge that goes off the top of the screen is the same edge
that comes up from the bottom.)
- Since there are different cell types, this puzzle represents a *
non-regular* spherical polyhedron. It was cool to realize this could be
done with MagicTile's abstraction :)
- Two of the vertices have 4 colors surrounding them, and two have 5
colors surrounding them. Even so, the repeated color on a 4C vertex piece
comes from different parts of a triangle, so the behavior is still 5C-like.
- Since I was able to make this planar
graphrepresentation of the
object, it was easier to see how it has the topology
of the sphere.

I haven't tried to make sequences to solve it yet, but will. If anyone
solves this puzzle, I'd love to hear about your experience with it!

Roice


On Fri, Dec 23, 2011 at 12:59 PM, Roice Nelson wrote:

> Hi all,
>
> I added some hemi-puzzles, all ones we haven't seen before. The
> hemi-dodecahedron and hemi-cube are not new, but I made them vertex turning
> this time. There are also hemi-octahedron and hemi-icosahedron puzzles
> now. All of these have the topology of the projective plane.
>
> I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not
> symmetrical (like the {8,3} 10-Color and some of the other hyperbolic
> puzzles). It turns out to have 8 faces, 10 edges, and 4 vertices, so the Euler
> Characteristic shows
> it has the topology of a sphere. I'll try to write a little more about
> this 8C puzzle soon.
>
> You can download the latest by clicking here
> .
>
> Happy Holidays,
> Roice
>

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Here's a little more on the {3,5} 8-Color. =A0These puzzles with asymme=
trical colorings are strange, but they arise naturally from the math that i=
dentifies cells with each other, so I wanted to understand things a little =
better.

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What a beautiful graph!
I have to fool around with this strange little puzzle.

-Melinda

On 12/23/2011 10:47 PM, Roice Nelson wrote:
>
>
> Here's a little more on the {3,5} 8-Color. These puzzles with
> asymmetrical colorings are strange, but they arise naturally from the
> math that identifies cells with each other, so I wanted to understand
> things a little better.
>
> To do that, I made a graph of the 4 vertices, 10 edges, and 8 faces
> all "rolled up". By that I mean each of these features is only shown
> once, rather than shown multiple times (with hand waving that "this
> face is identified with that one", as the MagicTile presentation
> requires). An image of my graph with default MagicTile colors is here
> ,
> and some observations about it are:
>
> * The cyan and purple cells are henagons
> (polygons with one vertex
> and one edge). The face twisting of these two scrambles nothing.
> * The blue and orange cells are degenerate triangles. They have
> three sides, but only two vertices. I initially thought they were
> digons , but they are more
> like a digon with a henagon subtracted out. I don't know if there
> is a special name for this polygon.
> * The red, yellow, white, and green cells are proper triangles. (By
> the way, if you want to trace out the green and white triangles in
> the graph, note that the edge that goes off the top of the screen
> is the same edge that comes up from the bottom.)
> * Since there are different cell types, this puzzle represents a
> /non-regular/ spherical polyhedron. It was cool to realize this
> could be done with MagicTile's abstraction :)
> * Two of the vertices have 4 colors surrounding them, and two have 5
> colors surrounding them. Even so, the repeated color on a 4C
> vertex piece comes from different parts of a triangle, so the
> behavior is still 5C-like.
> * Since I was able to make this planar graph
> representation of the
> object, it was easier to see how it has the topology of the sphere.
>
> I haven't tried to make sequences to solve it yet, but will. If
> anyone solves this puzzle, I'd love to hear about your experience with it!
>
> Roice
>
>
> On Fri, Dec 23, 2011 at 12:59 PM, Roice Nelson > > wrote:
>
> Hi all,
>
> I added some hemi-puzzles, all ones we haven't seen before. The
> hemi-dodecahedron and hemi-cube are not new, but I made them
> vertex turning this time. There are also hemi-octahedron and
> hemi-icosahedron puzzles now. All of these have the topology of
> the projective plane.
>
> I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not
> symmetrical (like the {8,3} 10-Color and some of the other
> hyperbolic puzzles). It turns out to have 8 faces, 10 edges, and
> 4 vertices, so the Euler Characteristic
> shows it has
> the topology of a sphere. I'll try to write a little more about
> this 8C puzzle soon.
>
> You can download the latest by clicking here
> .
>
> Happy Holidays,
> Roice
>
>
>
>
>

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What a beautiful graph!

I have to fool around with this strange little puzzle.



-Melinda



On 12/23/2011 10:47 PM, Roice Nelson wrote:
cite="mid:CAEMuGXoNRvb55234EiaZD=1jaZz9chd=Lw=pWy6W4C0cx_nUwQ@mail.gmail.com"
type="cite">


Here's a little more on the {3,5} 8-Color.  These puzzles with
asymmetrical colorings are strange, but they arise naturally from
the math that identifies cells with each other, so I wanted to
understand things a little better.

Awesome, this colorful analysis of a colorful puzzle. I see I have to
return to MagicTile (after MPUlt and FlatRubik).

--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> Here's a little more on the {3,5} 8-Color. These puzzles with
asymmetrical
> colorings are strange, but they arise naturally from the math that
> identifies cells with each other, so I wanted to understand things a
little
> better.
>
> To do that, I made a graph of the 4 vertices, 10 edges, and 8 faces
all
> "rolled up". By that I mean each of these features is only shown once,
> rather than shown multiple times (with hand waving that "this face is
> identified with that one", as the MagicTile presentation requires). An
> image of my graph with default MagicTile colors is
>
here/1602345595/view>,
> and some observations about it are:
>
> - The cyan and purple cells are
> henagons (polygons
> with one vertex and one edge). The face twisting of these two
scrambles
> nothing.
> - The blue and orange cells are degenerate triangles. They have three
> sides, but only two vertices. I initially thought they were
> digons,
> but they are more like a digon with a henagon subtracted out. I don't
know
> if there is a special name for this polygon.
> - The red, yellow, white, and green cells are proper triangles. (By
the
> way, if you want to trace out the green and white triangles in the
graph,
> note that the edge that goes off the top of the screen is the same
edge
> that comes up from the bottom.)
> - Since there are different cell types, this puzzle represents a *
> non-regular* spherical polyhedron. It was cool to realize this could
be
> done with MagicTile's abstraction :)
> - Two of the vertices have 4 colors surrounding them, and two have 5
> colors surrounding them. Even so, the repeated color on a 4C vertex
piece
> comes from different parts of a triangle, so the behavior is still
5C-like.
> - Since I was able to make this planar
> graphrepresentation of the
> object, it was easier to see how it has the topology
> of the sphere.
>
> I haven't tried to make sequences to solve it yet, but will. If anyone
> solves this puzzle, I'd love to hear about your experience with it!
>
> Roice
>
>
> On Fri, Dec 23, 2011 at 12:59 PM, Roice Nelson roice3@... wrote:
>
> > Hi all,
> >
> > I added some hemi-puzzles, all ones we haven't seen before. The
> > hemi-dodecahedron and hemi-cube are not new, but I made them vertex
turning
> > this time. There are also hemi-octahedron and hemi-icosahedron
puzzles
> > now. All of these have the topology of the projective plane.
> >
> > I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not
> > symmetrical (like the {8,3} 10-Color and some of the other
hyperbolic
> > puzzles). It turns out to have 8 faces, 10 edges, and 4 vertices, so
the Euler
> > Characteristic
shows
> > it has the topology of a sphere. I'll try to write a little more
about
> > this 8C puzzle soon.
> >
> > You can download the latest by clicking
hereew.zip>
> > .
> >
> > Happy Holidays,
> > Roice
> >
>

Roice,

Thank you for such a special Xmas gift. After seeing your description of th=
e topology, I solved this puzzle. The solution is quite traditional, solvin=
g the pieces type by type. For each type I solved the "special" polygons fi=
rst to make sure nothing weird happens. For the pieces related to special p=
olygons, I had to carefully select the moves so that they don't intersect i=
n an unexpected way. Aside from this there's no more difficulty.

But I have to say your analysis is very very helpful and illuminating. With=
out the planar graph I won't understand how it works.

Maybe I'll apply your method to analyze and try the other asymmetric puzzle=
s in the future. Before today I don't really know how to deal with them.

The other hemi-polyhedral puzzles are pretty straightforward though.=20

Merry Christmas and happy new year to everyone!

Nan


--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> Here's a little more on the {3,5} 8-Color. These puzzles with asymmetric=
al
> colorings are strange, but they arise naturally from the math that
> identifies cells with each other, so I wanted to understand things a litt=
le
> better.
>=20
> To do that, I made a graph of the 4 vertices, 10 edges, and 8 faces all
> "rolled up". By that I mean each of these features is only shown once,
> rather than shown multiple times (with hand waving that "this face is
> identified with that one", as the MagicTile presentation requires). An
> image of my graph with default MagicTile colors is
> here1602345595/view>,
> and some observations about it are:
>=20
> - The cyan and purple cells are
> henagons (polygons
> with one vertex and one edge). The face twisting of these two scramble=
s
> nothing.
> - The blue and orange cells are degenerate triangles. They have three
> sides, but only two vertices. I initially thought they were
> digons,
> but they are more like a digon with a henagon subtracted out. I don't=
know
> if there is a special name for this polygon.
> - The red, yellow, white, and green cells are proper triangles. (By t=
he
> way, if you want to trace out the green and white triangles in the gra=
ph,
> note that the edge that goes off the top of the screen is the same edg=
e
> that comes up from the bottom.)
> - Since there are different cell types, this puzzle represents a *
> non-regular* spherical polyhedron. It was cool to realize this could =
be
> done with MagicTile's abstraction :)
> - Two of the vertices have 4 colors surrounding them, and two have 5
> colors surrounding them. Even so, the repeated color on a 4C vertex p=
iece
> comes from different parts of a triangle, so the behavior is still 5C-=
like.
> - Since I was able to make this planar
> graphrepresentation of the
> object, it was easier to see how it has the topology
> of the sphere.
>=20
> I haven't tried to make sequences to solve it yet, but will. If anyone
> solves this puzzle, I'd love to hear about your experience with it!
>=20
> Roice
>=20
>=20
> On Fri, Dec 23, 2011 at 12:59 PM, Roice Nelson wrote:
>=20
> > Hi all,
> >
> > I added some hemi-puzzles, all ones we haven't seen before. The
> > hemi-dodecahedron and hemi-cube are not new, but I made them vertex tur=
ning
> > this time. There are also hemi-octahedron and hemi-icosahedron puzzles
> > now. All of these have the topology of the projective plane.
> >
> > I also stumbled upon a {3,5} 8-Color puzzle. The coloring is not
> > symmetrical (like the {8,3} 10-Color and some of the other hyperbolic
> > puzzles). It turns out to have 8 faces, 10 edges, and 4 vertices, so t=
he Euler
> > Characteristic show=
s
> > it has the topology of a sphere. I'll try to write a little more about
> > this 8C puzzle soon.
> >
> > You can download the latest by clicking hereom/magictile/downloads/MagicTile_v2_Preview.zip>
> > .
> >
> > Happy Holidays,
> > Roice
> >
>