Animal enumerations on regular tilings in Spherical, Euclidean, and Hyperbolic 2-dimensional spaces

Jonathan Post jvospost3 at
Mon Mar 5 17:49:12 CET 2007

Dear Tomás Oliveira e Silva,

Thank you for verifying my
A119611  Number of free polyominoes in {4,5}tesselation of the hyperbolic plane.

Your sequence might then be described as Number of fixed polyominoes in
{4,5}tesselation of the hyperbolic plane, enantiamorphs counted twice.

1, 4, 18, 88, 439, 2232, 11522, 60256, 318627, 1700996

Yours is itself a new sequence not yet in OEIS.  Your sequence and
mine could usefully crossreference each other, as well as the older
and better known Euclidean sequences. As you extend your table of
animals in hyperbolic tessellations, you will get many more new
sequences. There are a number of people eager to see more such work by

For additional crossreferences to OEIS, see my comments below my copy
of your email to me.


Jonathan Vos Post

On 3/5/07, jonathan post <jvospost2 at> wrote:
> --- Tomás Oliveira e Silva <tos at> wrote:
> > Date: Mon, 05 Mar 2007 08:55:29 +0000
> > From: Tomás Oliveira e Silva <tos at>
> > To: jonathan post <jvospost2 at>
> > Subject: Re: Animal enumerations on regular tilings
> > in Spherical, Euclidean,
> >  and Hyperbolic 2-dimensional spaces
> >
> > Dear Jonathan,
> >
> > Your figures appear to be correct. I can count very
> > easily the total number
> > of configurations for the {4,5} case:
> >
> >   1: 1
> >   2: 4
> >   3: 18
> >   4: 88
> >   5: 439
> >   6: 2232
> >   7: 11522
> >   8: 60256
> >   9: 318627
> > 10: 1700996
> >
> > Of course, the same "animal" may appear more than
> > once in this enumeration;
> > it depends on the symmetry class it belongs to. I'm
> > in the middle of other
> > projects right now, but I will take a look at the
> > problem of identifying
> > identical "animals" sometime in the future.
> >
> > By the way, I think that allowing mirror image
> > polyominoes to be the same
> > polyomino is conceptually wrong; mirror imaging
> > requires another dimension.
> > What about 3-dimensional "animals": are mirror
> > images the same or are they
> > different?
> >
> > Best regards,
> > --
> >
> --------------------------------------------------------------------------
> >     Tomás Oliveira e Silva    tos at
> >
> >
> --------------------------------------------------------------------------
> > "God's Final Message to His Creation: We apologise
> > for the inconvenience."
> >                        So Long, and Thanks for all
> > the Fish, Douglas Adams
> >
> >

For a nice colorful animation of a hyperbolic
pentagonal tiling, and the generating functions, and
the formulae, see:

which illustrates:

A054888  Layer counting sequence for hyperbolic
tessellation by regular pentagons of angle Pi/2
Comment: "The layer sequence is the sequence of the
cardinalities of the layers accumulating around a
(finite-sided) polygon of the tessellation under
successive side-reflections. "

For other related OEIS seuences, starting with mine:

A119611  Number of free polyominoes in {4,5}
tesselation of the hyperbolic plane

1, 1, 1, 2, 5, 15, 56

 OFFSET  0,4

 COMMENT  Each tesselation in the hyperbolic plane is
represented by a Schlafli symbol of the form {p,q},
which means that q regular p-gons surround each
vertex. There exists a hyperbolic tesselation {p,q}
for every p,q such that (p-2)*(q-2) > 4. I have, in a
paper not referenced here explicitly, described
polyiamonds and enumerated in the Klein curve,
topologically derived from the {3,7} or dually the
{7,3} hyperbolic tesselation, as a side-effect of
defining polyheptagons in the Klein quartic.

 LINKS  Don Hatch, Hyperbolic Planar Tesselations:

Eric W. Weisstein, Polyomino.

 EXAMPLE  For n = 0,1,2,3 the polyominoes in Euclidean
Golombics A000105 are essentially same as in the {4,5}
tesselation of the hyperbolic plane, with redfinition
of "straight line" and angular deficiency at a vertex.
For n = 4, the square tetromino does not exist. In its
place is the cut-square, a pentagonal pentomino with
one cell removed; see n = 5. For n = 5, we have
modified versions of 11 of the 12 Euclidean
pentominoes, but not the P-pentamino, as that has the
square tetromino as a subpolyomino, with one adjacent
monomino. In place of the P we have 4 unique
hyperbolic pentominoes. First, the aforementioned
pentagonal pentomino, with 5-fold symmetry, embedded
in the space where 5 right angles define a full
rotation. Next, the cut-square tetromino can have an
adjacent monomino in 4 nonisomorphic positions. 12 - 1
+ 4 = 15 hyperbolic pentominoes. For n = 6 we lose the
8 Euclidean hexominoes that have the square tetromino
as a subpolyomino. In their place, we have the
pentagonal pentomino with an adjacent monomino; the
cut-square with an adjacent domino in 12 nonisomorphic
positions; and the cut-square with two separate
adjacent monominoes in 16 nonisomorphic positions. 35
- 8 + 29 = 56 hyperbolic hexominoes.

 CROSSREFS  Cf. A000105.

 KEYWORD  nonn,uned

 AUTHOR  Jonathan Vos Post (jvospost2(AT),
Jun 04 2006

A000105  Number of polyominoes (or square animals)
with n cells

This cites and links to:

Tomas Oliveira e Silva, Animal enumerations on regular
tilings in Spherical, Euclidean, and Hyperbolic
2-dimensional spaces

January 4, 2007


A019503  Minimal cardinality of triangulation of
n-cube using n-simplices whose vertices are vertices
of the n-cube .

Formula: "5522 <= a(8) <= 11944 [Haiman, Ziegler]. For
large n, a method due to Smith, using volume estimates
in hyperbolic geometry, yields the best lower bounds
on a(n) so far. - Jonathan Vos Post
(jvospost2(AT), Jul 13 2005"

A001352  a(n) = 4a(n-1) - a(n-2), n >= 3.

Comment: "Also the coordination sequence of a {4,6}
tiling of the hyperbolic plane, where there are 6
squares (with vertex angles Pi/3) around every vertex.
- toen (tca110(AT), May 16 2005"

A001354  Coordination sequence for hyperbolic
tesselation 3^7 (from triangle group (2,3,7)).

A054887  Layer counting sequence for hyperbolic
tessellation by cuspidal triangles of angles

A054889  Layer counting sequence for hyperbolic
tessellation by regular pentagons of angle 2*Pi/5.

A054890  Layer counting sequence for hyperbolic
tessellation by regular heptagons of angle Pi/3.

A096231  Number of n-th generation triangles in the
tiling of the hyperbolic plane by triangles with
angles {pi/2, pi/3, 0}.

A076765  Partial sums of Chebyshev sequence
Comment: "In the tiling {5,3,4} of 3-dimensional
hyperbolic space, the number of regular dodecahedra
with right angles of the n generation which are
contained in an eighth of space (intersection of three
pairwise perpendicular hyperplanes which are supported
by the faces of a dodecahedron at a vertex)."

A112407  Decimal expansion of a semiprime analogue of
a Ramanujan formula.
KEYWORD  cons,nonn,uned

 AUTHOR  Jonathan Vos Post (jvospost2(AT),
Dec 21 2005

A119602  Number of nonisomorphic polytetrahedra with n
identical regular tetrahedra connected face-to-face
and/or edge-to-edge (chiral shapes counted twice).

 1, 1, 2, 7, 39 (list; graph; listen)

 OFFSET  0,3

 COMMENT  Polytetrahedra (abbreviated polytets or
n-tets) are a 3-dimensional generalization of

Polytetrahedra are composed of identically-sized
regular tetrahedra in Euclidean 3-space. More
tetrahedra can be placed in proximity in Hyperbolic
space, but that is beyond the scope of this paper.
Herein we make use of the dihedral angle of the
regular tetrahedron (the angle between two faces) as
being arccos(1/3) radians, or (180/pi) arccos(1/3) ~
70.5287794 degrees. For n running from 0 through 4, we
enumerate different classes of polytetrahedra, where
tetrahedra are variously connected in a "floppy"
edge-to-edge (e2e) manner, 2, 3, 4, or 5 sharing an
edge; or in a rigid face-to-face (f2f) manner; or in a
combination of the two ("semifloppy"). Note that a
"floppy" polytetrahedron with only e2e connections can
actually be mechanically rigid.

 REFERENCES  Andrew I. Campbell, Valerie J. Anderson,
Jeroen S. van Duijneveldt, and Paul Bartlett,
"Dynamical Arrest in Attractive Colloids: The Effect
of Long-Range Repulsion", Phys. Rev. Lett. 94, 208301

Jonathan Vos Post, Polytetrahedra, preprint, Draft
4.0, approx. 6750 words, 15 pages, available as Word
file by email upon request.

J. F. Sadoc, "Boerdijk-Coxeter helix and biological
helices", Eur. Phys. J. B 12, 309-318.

 LINKS  Eric W. Weisstein et al., Tetrahedron.

Wikipedia, Polyiamond.

 EXAMPLE  a(0) = 1 because there is only one kind of
n-tet with zero tetrahedra, namely the null set.

a(1) = 1 because there is only one kind of n-tet with
one tetrahedron, namely the regular tetrahedron

a(2) = 2 because there are two ways to establish
vertex-to-vertex connections between two regular
tetrahedra, neglecting a pair of tetrahedra which only
touch at a single vertex (beyond the scope of this
paper). First, we may join the two tetrahedra
face-to-face (f2f) to get the Triangular Dipyramid.
The triangular (or trigonal) dipyramid is one of the 8
convex deltahedra, and Johnson solid J12. It has 5
vertices (2 tips and a girdle of three around the
joined triangular face), 9 edges, and degree sequence
(3, 3, 3, 3, 3, 3). This is a rigid hexahedron, with 6
vertices, 11 edges, and 8 faces. It is one of the 7
convex hexahedra. Second, we may join the two
tetrahedra edge-to-edge (f2f) to get the "floppy
2-tet." It is floppy because there is no constraint on
the angle that the two tetrahedra may make about the
"hinge" between them, until reaching the dihedral
angle arccosine (1/3) ~ 70.53 degrees, upon which it
has folded into a triangular dipyramid. The floppy
2-tet has 6 vertices, 11 edges, and 8 faces.

a(3) = 7 because the unique rigid 3-tet is called a
"boat." The boat is a concave irregular octahedron
which, since all faces are identical equilateral
triangles, is a deltahedron. It has 6 vertices: two
tips (prow and stern), the two extrema of the concave
hinge, and the two extrema of the convex "keel." It
has 12 edges: 3 each adjacent to the stern and bow,
the unique concave edge, 4 connecting the concave edge
to the keel, and one keel edge. We also have 2 purely
floppy 3-tets and 2 semifloppy 3-tets, as described

(a) triangular dipyramid with e2e tet along one of the
3 triangular girdle edges (semifloppy, as it has 1 f2f
and 1 e2e connection);

(b) triangular dipyramid with e2e tet along one of the
6 edges adjacent to a tip (semifloppy, as it has 1 f2f
and 1 e2e connection);

(c) floppy 2-tet with 3rd tet added e2e so that the
three tets' centroids are coplanar, and can form a
straight line when the hinges are both at zero

(d) floppy 2-tet with 3rd tet added e2e so that, when
both hinges are both at zero degrees, the three tets'
centroids are coplanar, but the lines connecting one
pair of the tets is perpendicular to the line
connecting the other tets' centroids. Comparing (c)
with (d), we may look at the shadows of the edges on a
plane, i.e. the projection. A single tetrahedron may
be oriented so that its projection onto a plane is a
square with crossing diagonals. Similarly, the
projection of (c) onto a plane parallel to the plane
of the 3 centroids is three squares end-to-end, i.e. a
straight triomino, with each square containing crossed
diagonals. The projection of (d) onto a plane parallel
to the plane of the 3 centroids is three squares in an
L shape, i.e. a L-triomino, with each square
containing crossed diagonals. In the hydrocarbon
world, we analogize these two to n-propane and
iso-propane. We can generalize this to an infinite
class of purely floppy polytetrahedra whose planar

are polyominoes. The unique 1-tet projects to a
monomino; the unique floppy 2-tet projects to a
domino. Since there are five free tetrominoes there
are at least five fully floppy 4-tets (a sixth from
the square teromiono with one e2e connection broken).
Since there are 12 free pentominoes there are at least
12 fully floppy 5-tets. Since there are 35 free
hexominoes there are at least 35 fully floppy 5-tets.
And so on, where the published enumerations of
polyominoes immediately define a partial set of
polytetrahedra: 108 floppy 7-tets, 369 floppy 8-tets
(which do not include the floppy 8-tet ring which has
no 2-D equivalent), and so forth. As Andrew Carmichael
Post pointed out, if we bend (c) above into a ring,
and close the ring with an additional e2e connection,
we have a mechanically rigid "floppy" 3-tet with a
partially (3/4) surrounded tetrahedral hole. We can
describe this also as a central tetrahedron with 3
external tetrahedra joined f2f with 3 of the 4 faces

and then the central tetrahedron removed. We have the

3-tet, in which 3 tets share an edge. This is possible
because 3 x 70.53 degrees = 211.59 degrees, which is
sufficiently smaller than 360 degrees that the
floppiness allows for 148.41 degrees to be distributed
between the tets. The planar projection of this onto a
plane parallel to the plane of the 3 tets's centroids,
when bent to equiangular, is three identical triangles
meeting at a vertex, akin to the radiation warning

 CROSSREFS  Cf. A000577, A000105.

Sequence in context: A000366 A106211 A014058
this_sequence A121752 A054133 A032118

Adjacent sequences: A119599 A119600 A119601
this_sequence A119603 A119604 A119605

 KEYWORD  nonn,uned,obsc

 AUTHOR  Jonathan Vos Post (jvospost2(AT),
Jun 02 2006

 EXTENSIONS  This entry should be shortened and much
of this material put into a text file to be attached
to this entry. - njas, Jun 05 2006

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