Thread: "Regular abstract polytopes based on {5,3,4} and {4,3,5}"

Hello,

The regular abstract polytopes based on hyperbolic tessellations {5,3,4} an=
d {4,3,5} have been mentioned by Andrey several times here. Recently I read=
more about them and found Gruenbaum talked about a polytope formed by 32 h=
emidodecahedra, which was related to {5,3,4}. It should be this one:

http://www.abstract-polytopes.com/atlas/1920/240995/5.html

According to this page, it has 32 cells, each of which is a hemidodecahedro=
n. It has 96 faces, 120 edges and 40 vertices. The vertex figure is an octa=
hedron (note: not hemi-octahedron). Compared with the 11-cell and the 57-ce=
ll, this 32-cell received little attention.

It has a dual, which is based on {4,3,5}:

http://www.abstract-polytopes.com/atlas/1920/240995/2.html

The 40 faces are cubes (not hemi-cubes). The vertex figure is a hemi-icosah=
edron.=20

The vertex coordinates of {4,3,5} and {5,3,4} have been computed analytical=
ly and can be found in this paper [Garner, Coordinates for Vertices of Regu=
lar Honeycombs in Hyperbolic Space, www.jstor.org/stable/2415373, Table 1].=
This is of course a good news for implementation. If any one wants to see =
the paper but has no access to Jstor please email me. The coordinates are i=
n a Minkowskian space. I need to learn more hyperbolic geometry to understa=
nd the model.=20

According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISING F=
ROM HONEYCOMBS, http://www.sciencedirect.com/science/article/pii/0012365X84=
900323], there are more abstract polytopes based on {5,3,4} and {4,3,5}. Bu=
t they cannot be found in [http://www.abstract-polytopes.com/atlas/] becaus=
e this atlas contains information of "small" polytopes with up to 2000 symm=
etries. Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-ce=
ll have 1920 symmetries, which is just below the boundary. Something like 1=
20-cell, and 57-cell etc are not there because they are too large. But thes=
e two things can keep me excited for a while.=20

Nan

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Hi Nan,

This is wonderful information, and puzzle versions definitely sound
realizable. I do not have access to Jstor, but would love to see a copy of
the paper.

As far as the coordinates being in Minkowski space, that means they are in
the Hyperboloid Model . I
have written code to go between this model and some other models
(Poincare/Klein), and I'm happy to share if you think it could help in your
adventures on this topic. It is code for 2D geometry, but should be
adaptable to the 3D case.

Roice


On Fri, Jun 8, 2012 at 12:48 AM, schuma wrote:

> Hello,
>
> The regular abstract polytopes based on hyperbolic tessellations {5,3,4}
> and {4,3,5} have been mentioned by Andrey several times here. Recently I
> read more about them and found Gruenbaum talked about a polytope formed by
> 32 hemidodecahedra, which was related to {5,3,4}. It should be this one:
>
> http://www.abstract-polytopes.com/atlas/1920/240995/5.html
>
> According to this page, it has 32 cells, each of which is a
> hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The vertex
> figure is an octahedron (note: not hemi-octahedron). Compared with the
> 11-cell and the 57-cell, this 32-cell received little attention.
>
> It has a dual, which is based on {4,3,5}:
>
> http://www.abstract-polytopes.com/atlas/1920/240995/2.html
>
> The 40 faces are cubes (not hemi-cubes). The vertex figure is a
> hemi-icosahedron.
>
> The vertex coordinates of {4,3,5} and {5,3,4} have been computed
> analytically and can be found in this paper [Garner, Coordinates for
> Vertices of Regular Honeycombs in Hyperbolic Space,
> www.jstor.org/stable/2415373, Table 1]. This is of course a good news for
> implementation. If any one wants to see the paper but has no access to
> Jstor please email me. The coordinates are in a Minkowskian space. I need
> to learn more hyperbolic geometry to understand the model.
>
> According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISING
> FROM HONEYCOMBS,
> http://www.sciencedirect.com/science/article/pii/0012365X84900323], there
> are more abstract polytopes based on {5,3,4} and {4,3,5}. But they cannot
> be found in [http://www.abstract-polytopes.com/atlas/] because this atlas
> contains information of "small" polytopes with up to 2000 symmetries.
> Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell have
> 1920 symmetries, which is just below the boundary. Something like 120-cell,
> and 57-cell etc are not there because they are too large. But these two
> things can keep me excited for a while.
>
> Nan
>
>
>

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Hi Nan,

Hi all,

After learning a little bit of hyperbolic geometry, I made an applet to vis=
ualize the {5,3,4} honeycomb:

http://people.bu.edu/nanma/InsideH3/H3.html

This visualization is similar to Andrey's MHT633. Imagine that you have a s=
paceship flying in a {5,3,4} honeycomb in a hyperbolic space. Then this is =
what you will see (suppose light travels along geodesics). Dragging is to r=
otate the spaceship. Shift+up/down dragging is to drive the spaceship forwa=
rd and backward. Try navigating in this space!

Because it's just for my own education, I haven't implement the automatic e=
xtension of the tessellation when you move "outside" of the several initial=
cells. So we can drive the spaceship away from these cells and look back f=
rom outside.

In case anyone is curious, internally I'm using a hyperboloid model. Among =
the several models, I found this one intuitive for me.

Nan

--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> Hi Nan,
>=20
> This is wonderful information, and puzzle versions definitely sound
> realizable. I do not have access to Jstor, but would love to see a copy =
of
> the paper.
>=20
> As far as the coordinates being in Minkowski space, that means they are i=
n
> the Hyperboloid Model . =
I
> have written code to go between this model and some other models
> (Poincare/Klein), and I'm happy to share if you think it could help in yo=
ur
> adventures on this topic. It is code for 2D geometry, but should be
> adaptable to the 3D case.
>=20
> Roice
>=20
>=20
> On Fri, Jun 8, 2012 at 12:48 AM, schuma wrote:
>=20
> > Hello,
> >
> > The regular abstract polytopes based on hyperbolic tessellations {5,3,4=
}
> > and {4,3,5} have been mentioned by Andrey several times here. Recently =
I
> > read more about them and found Gruenbaum talked about a polytope formed=
by
> > 32 hemidodecahedra, which was related to {5,3,4}. It should be this one=
:
> >
> > http://www.abstract-polytopes.com/atlas/1920/240995/5.html
> >
> > According to this page, it has 32 cells, each of which is a
> > hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The verte=
x
> > figure is an octahedron (note: not hemi-octahedron). Compared with the
> > 11-cell and the 57-cell, this 32-cell received little attention.
> >
> > It has a dual, which is based on {4,3,5}:
> >
> > http://www.abstract-polytopes.com/atlas/1920/240995/2.html
> >
> > The 40 faces are cubes (not hemi-cubes). The vertex figure is a
> > hemi-icosahedron.
> >
> > The vertex coordinates of {4,3,5} and {5,3,4} have been computed
> > analytically and can be found in this paper [Garner, Coordinates for
> > Vertices of Regular Honeycombs in Hyperbolic Space,
> > www.jstor.org/stable/2415373, Table 1]. This is of course a good news f=
or
> > implementation. If any one wants to see the paper but has no access to
> > Jstor please email me. The coordinates are in a Minkowskian space. I ne=
ed
> > to learn more hyperbolic geometry to understand the model.
> >
> > According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISI=
NG
> > FROM HONEYCOMBS,
> > http://www.sciencedirect.com/science/article/pii/0012365X84900323], the=
re
> > are more abstract polytopes based on {5,3,4} and {4,3,5}. But they cann=
ot
> > be found in [http://www.abstract-polytopes.com/atlas/] because this atl=
as
> > contains information of "small" polytopes with up to 2000 symmetries.
> > Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell h=
ave
> > 1920 symmetries, which is just below the boundary. Something like 120-c=
ell,
> > and 57-cell etc are not there because they are too large. But these two
> > things can keep me excited for a while.
> >
> > Nan
> >
> >
> >
>

Hey, that's a really neat visualization, Nan!

It appears to be perfectly understandable given your description. It
just looks like a simple wireframe structure very much like a bunch of
soap bubbles in a foam. The distortion simply appears to be a wide-angle
view rather than a hyperbolic space. Perhaps it will be more confusing
when tiled much further but I think you have chosen a very good
projection and UI.

-Melinda

On 6/10/2012 3:13 PM, schuma wrote:
> Hi all,
>
> After learning a little bit of hyperbolic geometry, I made an applet to visualize the {5,3,4} honeycomb:
>
> http://people.bu.edu/nanma/InsideH3/H3.html
>
> This visualization is similar to Andrey's MHT633. Imagine that you have a spaceship flying in a {5,3,4} honeycomb in a hyperbolic space. Then this is what you will see (suppose light travels along geodesics). Dragging is to rotate the spaceship. Shift+up/down dragging is to drive the spaceship forward and backward. Try navigating in this space!
>
> Because it's just for my own education, I haven't implement the automatic extension of the tessellation when you move "outside" of the several initial cells. So we can drive the spaceship away from these cells and look back from outside.
>
> In case anyone is curious, internally I'm using a hyperboloid model. Among the several models, I found this one intuitive for me.
>
> Nan
>
> --- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>> Hi Nan,
>>
>> This is wonderful information, and puzzle versions definitely sound
>> realizable. I do not have access to Jstor, but would love to see a copy of
>> the paper.
>>
>> As far as the coordinates being in Minkowski space, that means they are in
>> the Hyperboloid Model. I
>> have written code to go between this model and some other models
>> (Poincare/Klein), and I'm happy to share if you think it could help in your
>> adventures on this topic. It is code for 2D geometry, but should be
>> adaptable to the 3D case.
>>
>> Roice
>>
>>
>> On Fri, Jun 8, 2012 at 12:48 AM, schuma wrote:
>>
>>> Hello,
>>>
>>> The regular abstract polytopes based on hyperbolic tessellations {5,3,4}
>>> and {4,3,5} have been mentioned by Andrey several times here. Recently I
>>> read more about them and found Gruenbaum talked about a polytope formed by
>>> 32 hemidodecahedra, which was related to {5,3,4}. It should be this one:
>>>
>>> http://www.abstract-polytopes.com/atlas/1920/240995/5.html
>>>
>>> According to this page, it has 32 cells, each of which is a
>>> hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The vertex
>>> figure is an octahedron (note: not hemi-octahedron). Compared with the
>>> 11-cell and the 57-cell, this 32-cell received little attention.
>>>
>>> It has a dual, which is based on {4,3,5}:
>>>
>>> http://www.abstract-polytopes.com/atlas/1920/240995/2.html
>>>
>>> The 40 faces are cubes (not hemi-cubes). The vertex figure is a
>>> hemi-icosahedron.
>>>
>>> The vertex coordinates of {4,3,5} and {5,3,4} have been computed
>>> analytically and can be found in this paper [Garner, Coordinates for
>>> Vertices of Regular Honeycombs in Hyperbolic Space,
>>> www.jstor.org/stable/2415373, Table 1]. This is of course a good news for
>>> implementation. If any one wants to see the paper but has no access to
>>> Jstor please email me. The coordinates are in a Minkowskian space. I need
>>> to learn more hyperbolic geometry to understand the model.
>>>
>>> According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISING
>>> FROM HONEYCOMBS,
>>> http://www.sciencedirect.com/science/article/pii/0012365X84900323], there
>>> are more abstract polytopes based on {5,3,4} and {4,3,5}. But they cannot
>>> be found in [http://www.abstract-polytopes.com/atlas/] because this atlas
>>> contains information of "small" polytopes with up to 2000 symmetries.
>>> Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell have
>>> 1920 symmetries, which is just below the boundary. Something like 120-cell,
>>> and 57-cell etc are not there because they are too large. But these two
>>> things can keep me excited for a while.
>>>
>>> Nan
>>>
>>>
>>>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

Thanks!

I'm computing the coordinates of {5,3,5} now. In Coxeter's [The Beauty of G=
eometry: Twelve Essays (Chapter 10, Regular Honeycombs in Hyperbolic Space)=
]
there's a summary table for some parameters for the tessellations. I don't =
have access to this book online (google book has it, but the scan was not c=
lear enough, I can't even read some numbers). So I went to the library and =
took a few photos of the table.

http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/Coxeter1.JPG=20
http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/Coxeter2.JPG=20
http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/Coxeter3.JPG=20

I'm sharing these numbers because they're useful and hard to find online.=20

As explained in Coxeter1.JPG, 2*phi is the edge length of the honeycomb. Th=
e expressions for phi turns out to be pretty useful when calculating the co=
ordinates of vertices. It's not very hard to derive but it's convenient if =
we have them. In Garner (1966, a paper I mentioned earlier in this thread),=
he cited [Coxeter, 1954] for the edge length. Chapter 10 of the "Twelve Es=
says" is nothing but a reprint of the 1954 paper. So Garner was actually ci=
ting this exact table.=20

Nan

--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>
> Hey, that's a really neat visualization, Nan!
>=20
> It appears to be perfectly understandable given your description. It=20
> just looks like a simple wireframe structure very much like a bunch of=20
> soap bubbles in a foam. The distortion simply appears to be a wide-angle=
=20
> view rather than a hyperbolic space. Perhaps it will be more confusing=20
> when tiled much further but I think you have chosen a very good=20
> projection and UI.
>=20
> -Melinda
>=20
> On 6/10/2012 3:13 PM, schuma wrote:
> > Hi all,
> >
> > After learning a little bit of hyperbolic geometry, I made an applet to=
visualize the {5,3,4} honeycomb:
> >
> > http://people.bu.edu/nanma/InsideH3/H3.html
> >
> > This visualization is similar to Andrey's MHT633. Imagine that you have=
a spaceship flying in a {5,3,4} honeycomb in a hyperbolic space. Then this=
is what you will see (suppose light travels along geodesics). Dragging is =
to rotate the spaceship. Shift+up/down dragging is to drive the spaceship f=
orward and backward. Try navigating in this space!
> >
> > Because it's just for my own education, I haven't implement the automat=
ic extension of the tessellation when you move "outside" of the several ini=
tial cells. So we can drive the spaceship away from these cells and look ba=
ck from outside.
> >
> > In case anyone is curious, internally I'm using a hyperboloid model. Am=
ong the several models, I found this one intuitive for me.
> >
> > Nan
> >
> > --- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
> >> Hi Nan,
> >>
> >> This is wonderful information, and puzzle versions definitely sound
> >> realizable. I do not have access to Jstor, but would love to see a co=
py of
> >> the paper.
> >>
> >> As far as the coordinates being in Minkowski space, that means they ar=
e in
> >> the Hyperboloid Model.=
I
> >> have written code to go between this model and some other models
> >> (Poincare/Klein), and I'm happy to share if you think it could help in=
your
> >> adventures on this topic. It is code for 2D geometry, but should be
> >> adaptable to the 3D case.
> >>
> >> Roice
> >>
> >>
> >> On Fri, Jun 8, 2012 at 12:48 AM, schuma wrote:
> >>
> >>> Hello,
> >>>
> >>> The regular abstract polytopes based on hyperbolic tessellations {5,3=
,4}
> >>> and {4,3,5} have been mentioned by Andrey several times here. Recentl=
y I
> >>> read more about them and found Gruenbaum talked about a polytope form=
ed by
> >>> 32 hemidodecahedra, which was related to {5,3,4}. It should be this o=
ne:
> >>>
> >>> http://www.abstract-polytopes.com/atlas/1920/240995/5.html
> >>>
> >>> According to this page, it has 32 cells, each of which is a
> >>> hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The ver=
tex
> >>> figure is an octahedron (note: not hemi-octahedron). Compared with th=
e
> >>> 11-cell and the 57-cell, this 32-cell received little attention.
> >>>
> >>> It has a dual, which is based on {4,3,5}:
> >>>
> >>> http://www.abstract-polytopes.com/atlas/1920/240995/2.html
> >>>
> >>> The 40 faces are cubes (not hemi-cubes). The vertex figure is a
> >>> hemi-icosahedron.
> >>>
> >>> The vertex coordinates of {4,3,5} and {5,3,4} have been computed
> >>> analytically and can be found in this paper [Garner, Coordinates for
> >>> Vertices of Regular Honeycombs in Hyperbolic Space,
> >>> www.jstor.org/stable/2415373, Table 1]. This is of course a good news=
for
> >>> implementation. If any one wants to see the paper but has no access t=
o
> >>> Jstor please email me. The coordinates are in a Minkowskian space. I =
need
> >>> to learn more hyperbolic geometry to understand the model.
> >>>
> >>> According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARI=
SING
> >>> FROM HONEYCOMBS,
> >>> http://www.sciencedirect.com/science/article/pii/0012365X84900323], t=
here
> >>> are more abstract polytopes based on {5,3,4} and {4,3,5}. But they ca=
nnot
> >>> be found in [http://www.abstract-polytopes.com/atlas/] because this a=
tlas
> >>> contains information of "small" polytopes with up to 2000 symmetries.
> >>> Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell=
have
> >>> 1920 symmetries, which is just below the boundary. Something like 120=
-cell,
> >>> and 57-cell etc are not there because they are too large. But these t=
wo
> >>> things can keep me excited for a while.
> >>>
> >>> Nan
> >>>
> >>>
> >>>
> >
> >
> > ------------------------------------
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
>

Hi,

{4,3,5} and {5,3,5} are online now, at the same address:

http://people.bu.edu/nanma/InsideH3/H3.html

These two honeycombs with the icosahedral vertex figure seems less intuitiv=
e than {5,3,4}.

Nan

> --- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
> >
> > Hey, that's a really neat visualization, Nan!
> >=20
> > It appears to be perfectly understandable given your description. It=20
> > just looks like a simple wireframe structure very much like a bunch of=
=20
> > soap bubbles in a foam. The distortion simply appears to be a wide-angl=
e=20
> > view rather than a hyperbolic space. Perhaps it will be more confusing=
=20
> > when tiled much further but I think you have chosen a very good=20
> > projection and UI.
> >=20
> > -Melinda
> >=20
> > On 6/10/2012 3:13 PM, schuma wrote:
> > > Hi all,
> > >
> > > After learning a little bit of hyperbolic geometry, I made an applet =
to visualize the {5,3,4} honeycomb:
> > >
> > > http://people.bu.edu/nanma/InsideH3/H3.html
> > >
> > > This visualization is similar to Andrey's MHT633. Imagine that you ha=
ve a spaceship flying in a {5,3,4} honeycomb in a hyperbolic space. Then th=
is is what you will see (suppose light travels along geodesics). Dragging i=
s to rotate the spaceship. Shift+up/down dragging is to drive the spaceship=
forward and backward. Try navigating in this space!
> > >
> > > Because it's just for my own education, I haven't implement the autom=
atic extension of the tessellation when you move "outside" of the several i=
nitial cells. So we can drive the spaceship away from these cells and look =
back from outside.
> > >
> > > In case anyone is curious, internally I'm using a hyperboloid model. =
Among the several models, I found this one intuitive for me.
> > >
> > > Nan
> > >
> > > --- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
> > >> Hi Nan,
> > >>
> > >> This is wonderful information, and puzzle versions definitely sound
> > >> realizable. I do not have access to Jstor, but would love to see a =
copy of
> > >> the paper.
> > >>
> > >> As far as the coordinates being in Minkowski space, that means they =
are in
> > >> the Hyperboloid Model>. I
> > >> have written code to go between this model and some other models
> > >> (Poincare/Klein), and I'm happy to share if you think it could help =
in your
> > >> adventures on this topic. It is code for 2D geometry, but should be
> > >> adaptable to the 3D case.
> > >>
> > >> Roice
> > >>
> > >>
> > >> On Fri, Jun 8, 2012 at 12:48 AM, schuma wrote:
> > >>
> > >>> Hello,
> > >>>
> > >>> The regular abstract polytopes based on hyperbolic tessellations {5=
,3,4}
> > >>> and {4,3,5} have been mentioned by Andrey several times here. Recen=
tly I
> > >>> read more about them and found Gruenbaum talked about a polytope fo=
rmed by
> > >>> 32 hemidodecahedra, which was related to {5,3,4}. It should be this=
one:
> > >>>
> > >>> http://www.abstract-polytopes.com/atlas/1920/240995/5.html
> > >>>
> > >>> According to this page, it has 32 cells, each of which is a
> > >>> hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The v=
ertex
> > >>> figure is an octahedron (note: not hemi-octahedron). Compared with =
the
> > >>> 11-cell and the 57-cell, this 32-cell received little attention.
> > >>>
> > >>> It has a dual, which is based on {4,3,5}:
> > >>>
> > >>> http://www.abstract-polytopes.com/atlas/1920/240995/2.html
> > >>>
> > >>> The 40 faces are cubes (not hemi-cubes). The vertex figure is a
> > >>> hemi-icosahedron.
> > >>>
> > >>> The vertex coordinates of {4,3,5} and {5,3,4} have been computed
> > >>> analytically and can be found in this paper [Garner, Coordinates fo=
r
> > >>> Vertices of Regular Honeycombs in Hyperbolic Space,
> > >>> www.jstor.org/stable/2415373, Table 1]. This is of course a good ne=
ws for
> > >>> implementation. If any one wants to see the paper but has no access=
to
> > >>> Jstor please email me. The coordinates are in a Minkowskian space. =
I need
> > >>> to learn more hyperbolic geometry to understand the model.
> > >>>
> > >>> According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA A=
RISING
> > >>> FROM HONEYCOMBS,
> > >>> http://www.sciencedirect.com/science/article/pii/0012365X84900323],=
there
> > >>> are more abstract polytopes based on {5,3,4} and {4,3,5}. But they =
cannot
> > >>> be found in [http://www.abstract-polytopes.com/atlas/] because this=
atlas
> > >>> contains information of "small" polytopes with up to 2000 symmetrie=
s.
> > >>> Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-ce=
ll have
> > >>> 1920 symmetries, which is just below the boundary. Something like 1=
20-cell,
> > >>> and 57-cell etc are not there because they are too large. But these=
two
> > >>> things can keep me excited for a while.
> > >>>
> > >>> Nan
> > >>>
> > >>>
> > >>>
> > >
> > >
> > > ------------------------------------
> > >
> > > Yahoo! Groups Links
> > >
> > >
> > >
> > >
> >
>

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Really nice Nan. It's instructive to be able to switch between these three
honeycombs. It's a great applet, which I'll definitely share with
others as I have the chance!

Of these three, I think I'd most like to see a puzzle based on the {4,3,5}
- perhaps with topological rather than purely geometric slicing, done in a
way to make it feel MC4D-like.

There's only one suggestion I have. Some of the thick lines can be covered
by thin lines behind them, so it seems there is a z-ordering problem. In
OpenGL, the depth buffer does all the work for you, but I don't imagine the
drawing framework you are working with has something like this. It can be
a difficult problem, and you may decide it's not worth the effort to
address, but I thought I'd mention it.

Best,
Roice

On Tue, Jun 12, 2012 at 1:04 PM, schuma wrote:

> Hi,
>
> {4,3,5} and {5,3,5} are online now, at the same address:
>
> http://people.bu.edu/nanma/InsideH3/H3.html
>
> These two honeycombs with the icosahedral vertex figure seems less
> intuitive than {5,3,4}.
>
> Nan
>
>

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