Tilings of the hyperbolic space and their visualization
Vladimir Bulatov
Corvallis, Oregon, USA
Joint MAA/AMS meeting, New Orleans, January 7, 2011

Abstract
Visual representation of tiling of 3D hyperbolic space attracted very little attention compare to tilings of hyperbolic
plane, which were popularized by M.C.Escher circle limit woodcuts. Although there is a lot of activity on theoreticalside of the problem starting from work of H.Poincare on Kleinian groups and continuing with breakthrough ofW.Thurston in the development of low dimensional topology and G.Perelman's proof of Poincare conjecture.
The book "Indra's Pearl" have popularized visualization of 2D limit set of Kleinian groups, which is located at theinfinity of hyperbolic space. In this talk we present our attempts to build and visualize actual 3D tilings. We study
tilings with symmetry group generated by reflections in the faces of Coxeter polyhedron, which also is thefundamental polyhedron of the group.
Online address of the talk:bulatov.org/math/1101/

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Outline
Introduction. Who, what and when
2D tiling with Coxeter polygons
3D tiling with Coxeter polyhedra
Hyperbolic polyhedra existence and construction
Building isometry group from generators
Visualization of the group and it's subgroups
Tiling zoo
Implementation in metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Introduction
These are some of the people involved in the development of the hyperbolic geometry and classification of symmetrygroups of hyperbolic plane and space:
N.Lobachevski, Janos Bolyai, F.Klein, H.Poincare, D.Coxeter, J.Milnor, J.Hubbard, W.Thurston, B.Mandelbrot,
G.Perelman

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Visualization of 2D hyperbolic tiling

V.Bulatov. Tilings of the hyperbolic space and their visualization

The first high quality drawing of 2D
hyperbolic tiling apperas 1890 in book by
Felix Klein and Robert Fricke.
This is drawing of *237 tessellation from
F.Klein and R.Fricke "Vorlesungen überdie Theorie der Elliptischen
Modulfunctionen," Vol. 1, Leipzig(1890)

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Visualization of 2D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

M.C.Escher Circle Limit III (1958)
He was inspired by drawing of hyperbolic
tiling in paper of H.S.M.Coxeter.

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Visualization of 2D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Many hyperbolic patterns by D.Dunham

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Visualization of 2D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

C.Goodman-Strauss's graphics work.

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Visualization of 2D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

C.Kaplan's Islamic pattern.

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Visualization of 2D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Martin von Gagern "Hyperbolization of
Euclidean Ornaments"

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Limit set of Kleinian groups

V.Bulatov. Tilings of the hyperbolic space and their visualization

3D tilings were visualized byrendering it's 2D limit set on theinfinity of the hyperbolic space.
One of the first know drawings oflimit set from F.Klein and R.Fricke"Vorlesungen über die Theorie derAutomorphen Functionen"
Leipzig(1897)
This hand drafted image was thebest available to mathematicians
until 1970s, when B.Mandelbrot
started to make computer
renderings of Kleinian groups

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Limit set of Kleinian groups

V.Bulatov. Tilings of the hyperbolic space and their visualization

B.Mandelbrot's computer
rendering of Kleinian group(from Un ensemble-limite parMichele Audin et Arnaud
Cheritat (2009))

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Limit set of Kleinian groups

V.Bulatov. Tilings of the hyperbolic space and their visualization

Interactive applet of the kleiniangroup visualization by Arnaud
Cheritat. From Un ensemble-limite.

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Limit set of Kleinian groups

V.Bulatov. Tilings of the hyperbolic space and their visualization"Indra Pearl" by D.Mumford,
C.Series and D.Wright (2004)
popularized visualization of limit setsof Kleinian groups

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Limit set of Kleinian groups

V.Bulatov. Tilings of the hyperbolic space and their visualization

Kleinian group images by Jos Leys.
His very artistic images rendered as
3D images represent essentially2-dimensional limit set of Kleiniangroup.

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Tiling with Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

Coxeter polygon:
All angles are submultiples of
i =

mi

.

Here all four angles of euclidean
polygon are equal2

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Tiling with Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

The tiling is generated by reflectionsin the sides of the Coxeter polygon.
Here we have original tile andsecond generation of tiles.

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Tiling with Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

First, second and third generation oftiles.
Condition on angles

i =miguaranties, that tiles fit without gaps.
2mi tiles meet at each vertex.

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Tiling with Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

Complete tiling - regular rectangular
grid.

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Coxeter Polygons in the Hyperbolic Plane

Hyperbolic Coxeter pentagon with all
shown in the Poincare disk
2model of hyperbolic plane.
i =The shape of the right angled
pentagon has 2 independent
parameters (two arbitrary sides
lengths).
Lengths of any two sides can bechosen independently. The length of
three remaining sides can be foundfrom hyperbolic trigonometry
identities.

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Coxeter Polygons in the Hyperbolic Plane

V.Bulatov. Tilings of the hyperbolic space and their visualization

Base tile is hyperbolically reflected in
the sides of the Coxeter polygon.

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Coxeter Polygons in the Hyperbolic PlaneFirst, second and third generationsof tiles.
guaranties, that 2mi tiles
miaround each vertex fit without gaps
and overlaps.
i =V.Bulatov. Tilings of the hyperbolic space and their visualization

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Coxeter Polygons in the Hyperbolic Plane

V.Bulatov. Tilings of the hyperbolic space and their visualization

Four generations of tiles.

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Coxeter tiles in the Hyperbolic Plane

V.Bulatov. Tilings of the hyperbolic space and their visualization

Complete tiling in the Poincare discmodel of hyperbolic plane.

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Coxeter tiles in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

To make another view of the sametiling we can conformally stretch the
Poincare disk into an infinite band.

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Coxeter tiles in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

To make another view of the sametiling we can conformally stretch the
Poincare disk into an infinite band.

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Coxeter tiles in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling in the band model of thehyperbolic plane.
The hyperbolic metric alonghorizontal axis of the band iseuclidean. As a result the tiling has
euclidean translation symmetry inhorizontal direction.
The tiling has translation symmetry
along each of the geodesics shown,but only horizontal translation is
obvious to our euclidean eyes.

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Motion in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Appropriate hyperbolic isometry cansend any 2 selected points toofthe band model.

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Motion in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Appropriate hyperbolic isometry cansend any 2 selected points toofthe band model.

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Motion in the band model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Appropriate hyperbolic isometry cansend any 2 selected points toofthe band model.

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Existence of Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic Coxeter N-gon exist for
any N

5 and angles

i =

mi

mi

2

Hyperbolic triangle exist if
1
1
1
+
+
<1
m1 m2 m3
Hyperbolic quadrangle exist if
1
1
1
1
+
+
+
< 2.
m1 m2 m3 m4
The space of shapes of Coxeter
N-gon has dimension N 3.

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Existence of Coxeter Polygons

V.Bulatov. Tilings of the hyperbolic space and their visualization

Example.
Tiling of hyperbolic plane with
Coxeter hexagons with all angles
equal .
3

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Coxeter Polyhedron

Dihedral angles of Coxeter
polyhedron are submultiples of :
i =

mi

.

Reflections in sides of Coxeter
polyhedron satisfy conditions ofPoincare Polyhedron Theorem.
Therefore the reflected copies ofsuch polyhedron fill the space
without gaps and overlaps.
Example - euclidean rectangularparallelepiped.

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Coxeter Polyhedra Tiling

V.Bulatov. Tilings of the hyperbolic space and their visualization

First and second generation of tilingof the euclidean space byrectangular parallelepipeds.
We can easy imagine the rest of thetiling - infinite regular grid.

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Coxeter polyhedra in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Coxeter polyhedra in hyperbolic
space.
Do they exist?

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Coxeter polyhedra in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

From Andreev's Theorem(1967) oncompact polyhedra with non-optusedihedral angles in hyperbolic space
follows:
There exist unique compact Coxeter
polyhedron in hyperbolic space iff:
1) Each vertex has 3 adjacent faces.
2) Tripple of dihedral angles at eachvertex is from the set, , , , , , , , ,
2 3 3
2 3 4
2 3 5
, , , n 2,
2 2 n3) Dihedral angles for each prismatic
3-cycle satisfy 1 + 2 + 3 <4) Dihedral angles for each prismatic
4-cycle satisfy 1 + 2 + 3 + 4 < 2

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Coxeter polyhedra in H3

Example of polyhedron satisfying
Andreev's theorem.
Edges with dihedral angles=m

are marked with m if m > 2
Unmarked edges have dihedral
angles .
2
The polyhedron is combinatoriallyequivalent to a cube with one
truncated vertex.

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Example. Truncated cube in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Visualization of the actual geometricrealization of truncated cube in thehyperbolic space shown in the
Poincare ball model.

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Example. Right angled dodecahedron

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled hyperbolic
dodecahedron
All dihedral angles are equal2
All faces are regular right angled
hyperbolic pentagons

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Non-compact Coxeter polyhedra in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

From Andreev's Theorem(1967) on
polyhedra of finite volume with
non-optuse dihedral angles inhyperbolic space follows:
There exist unique Coxeter
polyhedron in H 3 if
1) the additional triplets of dihedral
angles are allowed:, , , , , , , ,2 4 4 3 3 3 2 3 6
Vertices with such angles arelocated at the infinity of hyperbolic
space (ideal vertices).
2) Ideal vertices with 4 adjacentfaces and all four dihedral angles
equal

2

are permitted.

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Example. Ideal hyperbolic tetrahedron

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tetrahedron with all dihedral angles
3. All 4 vertices are ideal.

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Example. Ideal hyperbolic octahedron

V.Bulatov. Tilings of the hyperbolic space and their visualization

Octahedron with all dihedral angles
2. All 6 vertices are ideal.

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Coxeter polyhedra in E3 vs H3

Coxeter polyhedra in Euclidean space

Coxeter polyhedra in Hyperbolic space

Only 7 non-equivalent types of compact polyhedra:

V.Bulatov. Tilings of the hyperbolic space and their visualization

Infinitely many combinatorially non-equivalent
compact polyhedra.
Each combinatorial type allows wide selection ofdifferent dihedral angles satisfying Andreev's
theorem.

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Coxeter polyhedra in E3 vs H3 (continued)

Coxeter polyhedra in Euclidean space

Coxeter polyhedra in Hyperbolic space

Shape may have continuous parameters, forexample height of a prism.

Shape of polyhedron of finite volume is fixedby the choice of it's dihedral angles.

Few families of infinite polyhedra

V.Bulatov. Tilings of the hyperbolic space and their visualization

Infinite number of infinite families of infinite
polyhedra of finite and infinite volume.

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Example: 32 Coxeter tetrahedra in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

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How to construct polyhedra in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization"Hand" calculation for simpler polyhedra:
Construct small sphere around each vertex.
Intersections of adjacent faces with thespheres form spherical triangles with angles
equal to dihedral angles.
Using spherical sines laws find sides ofspherical triangles from angles. These sides
are flat angles of polyhedron's faces.
From known flat angles of triangular faces findtheir edges using hyperbolic sines laws.
From known angles and some edges findremaining edges using hyperbolic
trigonometry.
General method: Roeder R. (2007) Constructing
Hyperbolic Polyhedra Using Newton's Method.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
12 generators of the tiling arereflections in each of 12
dodecahedron's faces

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
First iteration of reflections.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
2nd iteration.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
3rd iteration.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
4th iteration.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Regular right angled dodecahedron.
10-th iteration.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Just kidding!
It is a sphere, which is good
approximation to 10 10 dodecahedral
tiles.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different approach - rendering onlyedges
(image by Claudio Rocchini fromwikipedia.org).
4 generations of tiles are shown. Not
much is possible to see.

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Visualization of tiling in H3

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another approach - looking at thetiling from inside of the space (W.
Thurston, J. Weeks). This allows to
see clearly the local structure of thetiling.
Image by Tom Ruen fromwikipedia.org generated by Jeff
Weeks interactive software.
Similar image from "Knot plot" video
was used on cover of dozens ofmath books.

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Visualization of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic space tessellation as it looks frominside the space. Frame from "Not Knot" video(1994)

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Looking from inside we can see only
nearest neighbours.
Looking from outside we can only
see the outer boundary.
Let's try to look outside, but see not
the whole tiling, but only it's part.
We select only subset of all
generators. The subset will generate
subgroup of the whole group.

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

First iteration of subgroup
generation.

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Second iteration of subgroup
generation.

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Third (and last) iteration of subgroup
generation.

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Move center of the shape into thecenter of the Poincare ball. Shapehas obvious cubic symmetry.

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Visualization of tiling by subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another view of the same finite
subtiling. We have 8 dodecahedra
around center

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Visualization of infinite subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

We can select generators, which
generate infinite subgroup.

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Visualization of infinite subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

First iteration.

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Visualization of infinite subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Second iteration.

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Visualization of infinite subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Third iteration.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Few more iterations.
All tiles are lying in one hyperbolic
plane.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Rotation of the tiling around thecenter of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Rotation of the tiling around thecenter of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Rotation of the tiling around thecenter of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Rotation of the tiling around thecenter of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic translation of plane into
the center of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic translation of plane into
the center of the ball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

More rotation about the center of theball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

More rotation about the center of theball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

More rotation about the center of theball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

More rotation about the center of theball.

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Infinite Fuchsian subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

More rotation about the center of theball.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Stretching the Poincare ball into
infinite cylinder.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Cylinder model is straighforward
axially symmetric 3D generalizationof the conformal band model of thehyperbolic plane.
Cylinder model of hyperbolic space
is not conformal, it has some limited
angular distorsions. However, it hassuch a nice property as euclidean
metrics along cylinder's axis. It isespecially usefull when we want tovisualize some specific hyperbolic
geodesic which we aling in that casewith the cylinder's axis. All theplanes orthoginal to that geodesic
are represented by circular disks
orthogonal to the axis.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
If cylinder's axis is aligned with theaxis of a hyperbolic transformation of
the group, tiling will have euclideantranslational symmetry along thecylinder's axis. If it is aligned with theaxis of a loxodromic transformation,
tiling will spiral around the cylinder's
axis.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
This group has no loxodromictransformatons.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
Transformation with longer period isaligned with the cylinder's axis.

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Tiling in Cylinder Model

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
No period is visible.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Different subset of generators
generates tiling, which is not flat
anymore.
The limit set of the tiling is continous
fractal curve. Axis of cylinder isaligned with the axis of an hyperbolictransformation of the subgroup.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation of the sametiling.
Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.
Tiling is periodic along cylinder'sboundary.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation of the sametiling.
Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.
Tiling is periodic along cylinder'sboundary.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation of the sametiling.
Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.
Tiling is periodic along cylinder'sboundary.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Axis of cylinder is aligned with theaxis of a loxodromic transform.
Tiling is spiraling around thecylinder's axis.

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Quasifuchsian Subgroup

V.Bulatov. Tilings of the hyperbolic space and their visualization

Axis of cylinder is aligned with theaxis of different loxodromictransform.
Tiling is spiraling around thecylinder's axis at different speed.

94 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Quasifuchsian tiling with Lambert cubes
show in the Poincare ball model. Only 4
out of 6 generators are used.

95 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Quasifuchsian tiling with 5 and 9 sided
prisms.

96 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Quasifuchsian tiling with 5 and 18 sided
prisms.

97 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with right angled dodecahedra.
Limit set is point set.

98 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with rhombic triacontahedra in thecylinder model.

99 of 138Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with rhombic triacontahedra

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with truncated tetrahedra. Axis
of cylinder model is vertical and it isaligned with common perpendicular
to two selected planes. These
planes become flat in the cylindermodel and are at the top and bottomof the piece.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with different truncatedtetrahedra in cylinder model. Axis ofcylinder is aligned with axis of some
hyperbolic transformation of thegroup, which makes tiling periodic.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with truncated Lambert cubes.
The limit set is complex fractal.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another tiling with truncated cubes.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with two generator free group.
Shown in the cylinder model.
Generators have parabollic commutator,
which is responsible for cusps.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

The same tiling in shown in the Poincareball model.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

The same tiling in shown in the cylindermodel.
Axis of cylinder is aligned with axis ofsome hyperbolic transformation. It makes
tiling periodic.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with two generator free groupshown in the cylinder model.
Generators have almost parabolic
commutator.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling with group with parabolic
commutator shown in the cylinder model.
Fixed points of two parabolictransformations are located at thecylinder's axis at infinity.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Some quasifuchsian group with 5
generators shown in the cylinder model.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Wild quasifuchsian group.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Wilder quasifuchsian group.

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Samples of 3D hyperbolic tilings

Tiling with truncated cubes in thecylinder model.
Axis of the cylinder model is allignedwith common perpendicular to 2
planes. This makes both theseplanes flat.
Placing repeating pattern on thesurface reveals the *455 tiling
structure on the hyperplanes, whichcorresponds to , , triangle formed455
at the truncated vertex.

V.Bulatov. Tilings of the hyperbolic space and their visualization

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation the same tilingwith cubes in the cylinder model.
The axis of cylinder model isperpendicular to one selected plane,
which make this plane at the bottom
flat.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation the same tiling.
*455 pattern is repeated at every
exposed plane.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation the same tiling.
To find this orientation we selected aloxodromic transformation with small
rotational and large translational
components. Next we selected 2
hyperbolic planes with poles nearthe 2 fixed points of thetransformation and have alligned
common perpendicular to theseplanes with the axis of the cylindermodel.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation of the sametiling.
One plane at the bottom isperpendicular to the cylinder axis.
Another plane is almost parallel thethe axis, making long tong at thetop.

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Samples of 3D hyperbolic tilings

V.Bulatov. Tilings of the hyperbolic space and their visualization

Another orientation of the same tilingwith cubes in the cylinder model.
One of the hyperplanes is very close
to the cylinder's axis and is stretchedalmost to the vertical band.
On the top and at the bottom of thetiling there are flat hyperplanes,
which have their common
perpendicular aligned with thecylinder's axis.

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Samples of tiling at infinity

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling at the boundary of hyperbolic
space shown in stereographic projection
to 2D.

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Samples of tiling at infinity

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling at the boundary of hyperbolic
space shown in stereographic projection
to 2D.

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Samples of tiling at infinity

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling at the boundary of hyperbolic
space shown in stereographic projection
to 2D.

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Samples of tiling at infinity

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling at the boundary of hyperbolic
space shown in stereographic projection
to 2D.

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Samples of tiling at infinity

V.Bulatov. Tilings of the hyperbolic space and their visualization

Tiling at the boundary of hyperbolic
space shown in stereographic projection
to 2D.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Metal sculpture. 20 Dodecahedra.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Metal sculpture. First iteration of Weber-Seifert
dodecahedral tiling.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Metal sculpture. 20 hyperbolic cubes.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Metal sculpture. Tiling by Fuchsian group.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic bracelet.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic bracelet.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic bracelet.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic bracelet.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic pendant.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic pendant.

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Implementation in Metal

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic pendant.

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Hyperbolic Sculpture

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic sculpture. Color 3D print.

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Hyperbolic Sculpture

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic sculpture. Color 3D print.

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Hyperbolic Sculpture

V.Bulatov. Tilings of the hyperbolic space and their visualization

Hyperbolic sculpture. Color 3D print.

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Hyperbolic Sculpture

V.Bulatov. Tilings of the hyperbolic space and their visualization

Online address of the talk:bulatov.org/math/1101/

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