-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
1/138
Tilings of the hyperbolic space and their visualization
Vladimir BulatovCorvallis, 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 hyperbolicplane, 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 studytilings 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/
.Bulatov. Tilings of the hyperbolic space and their visualization 1 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
2/138
Outline
Introduction. Who, what and when
2D tiling with Coxeter polygons3D 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 2 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
3/138
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, J anos 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 3 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
4/138
Visualization of 2D hyperbolic t iling
The first high quality drawing of 2Dhyperbolic tiling apperas 1890 in book byFelix Klein and Robert Fricke.This is drawing of *237 tessellation fromF.Klein and R.Fricke "Vorlesungen berdie Theorie der ElliptischenModulfunctionen," Vol. 1, Leipzig(1890)
V.Bulatov. Tilings of the hyperbolic space and their visualization 4 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
5/138
Visualization of 2D hyperbolic t ilings
M.C.Escher Circle Limit III (1958)He was inspired by drawing of hyperbolictiling in paper of H.S.M.Coxeter.
V.Bulatov. Tilings of the hyperbolic space and their visualization 5 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
6/138
Visualization of 2D hyperbolic t ilings
Many hyperbolic patterns by D.Dunham
V.Bulatov. Tilings of the hyperbolic space and their visualization 6 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
7/138
Visualization of 2D hyperbolic t ilings
C.Goodman-Strauss's graphics work.
.Bulatov. Tilings of the hyperbolic space and their visualization 7 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
8/138
Visualization of 2D hyperbolic t ilings
C.Kaplan's Islamic pattern.
.Bulatov. Tilings of the hyperbolic space and their visualization 8 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
9/138
Visualization of 2D hyperbolic t ilings
Martin von Gagern "Hyperbolization ofEuclidean Ornaments"
V.Bulatov. Tilings of the hyperbolic space and their visualization 9 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
10/138
Limit set of Kleinian groups
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 ber die Theorie derAutomorphen Functionen"Leipzig(1897)
This hand drafted image was thebest available to mathematiciansuntil 1970s, when B.Mandelbrotstarted to make computerrenderings of Kleinian groups
V.Bulatov. Tilings of the hyperbolic space and their visualization 10 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
11/138
Limit set of Kleinian groups
B.Mandelbrot's computerrendering of Kleinian group(fromUn ensemble-limite parMichele Audin et ArnaudCheritat (2009))
.Bulatov. Tilings of the hyperbolic space and their visualization 11 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
12/138
Limit set of Kleinian groups
Interactive applet of the kleiniangroup visualization by ArnaudCheritat. From Un ensemble-limite.
.Bulatov. Tilings of the hyperbolic space and their visualization 12 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
13/138
Limit set of Kleinian groups
"Indra Pearl" by D.Mumford,C.Series and D.Wright (2004)popularized visualization of limit setsof Kleinian groups
V.Bulatov. Tilings of the hyperbolic space and their visualization 13 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
14/138
Limit set of Kleinian groups
Kleinian group images byJ os Leys.
His very artistic images rendered as3D images represent essentially2-dimensional limit set of Kleiniangroup.
.Bulatov. Tilings of the hyperbolic space and their visualization 14 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
15/138
Tiling with Coxeter Polygons
Coxeter polygon:
All angles are submultiples of
i =
mi
.
Here all four angles of euclidean
polygon are equal
2
V.Bulatov. Tilings of the hyperbolic space and their visualization 15 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
16/138
Tiling with Coxeter Polygons
The tiling is generated by reflectionsin the sides of the Coxeter polygon.
Here we have original tile andsecond generation of tiles.
.Bulatov. Tilings of the hyperbolic space and their visualization 16 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
17/138
Tiling with Coxeter Polygons
First, second and third generation oftiles.
Condition on angles i =
mi
guaranties, that tiles fit without gaps.2mi tiles meet at each vertex.
.Bulatov. Tilings of the hyperbolic space and their visualization 17 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
18/138
Tiling with Coxeter Polygons
Complete tiling - regular rectangulargrid.
V.Bulatov. Tilings of the hyperbolic space and their visualization 18 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
19/138
Coxeter Polygons in the Hyperbolic Plane
Hyperbolic Coxeter pentagon with all
i =
2shown in the Poincare disk
model of hyperbolic plane.
The shape of the right angledpentagon has 2 independentparameters (two arbitrary sideslengths).
Lengths of any two sides can bechosen independently. The length ofthree remaining sides can be foundfrom hyperbolic trigonometryidentities.
V.Bulatov. Tilings of the hyperbolic space and their visualization 19 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
20/138
Coxeter Polygons in the Hyperbolic Plane
Base tile is hyperbolically reflected inthe sides of the Coxeter polygon.
.Bulatov. Tilings of the hyperbolic space and their visualization 20 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
21/138
Coxeter Polygons in the Hyperbolic Plane
First, second and third generationsof tiles.
i =
mi
guaranties, that 2mi tiles
around each vertex fit without gapsand overlaps.
V.Bulatov. Tilings of the hyperbolic space and their visualization 21 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
22/138
Coxeter Polygons in the Hyperbolic Plane
Four generations of tiles.
V.Bulatov. Tilings of the hyperbolic space and their visualization 22 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
23/138
Coxeter ti les in the Hyperbolic Plane
Complete tiling in the Poincare discmodel of hyperbolic plane.
V.Bulatov. Tilings of the hyperbolic space and their visualization 23 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
24/138
Coxeter ti les in the band model
To make another view of the sametiling we can conformally stretch thePoincare disk into an infinite band.
V.Bulatov. Tilings of the hyperbolic space and their visualization 24 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
25/138
Coxeter ti les in the band model
To make another view of the sametiling we can conformally stretch thePoincare disk into an infinite band.
V.Bulatov. Tilings of the hyperbolic space and their visualization 25 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
26/138
Coxeter ti les in the band model
Tiling in the band model of thehyperbolic plane.
The hyperbolic metric alonghorizontal axis of the band iseuclidean. As a result the tiling haseuclidean translation symmetry inhorizontal direction.
The tiling has translation symmetryalong each of the geodesics shown,but only horizontal translation is
obvious to our euclidean eyes.
V.Bulatov. Tilings of the hyperbolic space and their visualization 26 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
27/138
Motion in the band model
Appropriate hyperbolic isometry cansend any 2 selected points to of
the band model.
.Bulatov. Tilings of the hyperbolic space and their visualization 27 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
28/138
Motion in the band model
Appropriate hyperbolic isometry cansend any 2 selected points to of
the band model.
.Bulatov. Tilings of the hyperbolic space and their visualization 28 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
29/138
Motion in the band model
Appropriate hyperbolic isometry cansend any 2 selected points to of
the band model.
.Bulatov. Tilings of the hyperbolic space and their visualization 29 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
30/138
Existence of Coxeter Polygons
Hyperbolic Coxeter N-gon exist for
anyN 5 and angles i =
mi
mi 2
Hyperbolic triangle exist if1
m1
+1
m2
+1
m3
< 1
Hyperbolic quadrangle exist if1
m1
+1
m2
+1
m3
+1
m4
< 2.
The space of shapes of Coxeter
N-gon has dimension N 3.
.Bulatov. Tilings of the hyperbolic space and their visualization 30 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
31/138
Existence of Coxeter Polygons
Example.
Tiling of hyperbolic plane withCoxeter hexagons with all angles
equal
3.
V.Bulatov. Tilings of the hyperbolic space and their visualization 31 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
32/138
Coxeter Polyhedron
Dihedral angles ofCoxeter
polyhedron are submultiples of:
i =
mi
.
Reflections in sides of Coxeterpolyhedron satisfy conditions of
Poincare Polyhedron Theorem.
Therefore the reflected copies ofsuch polyhedron fill the spacewithout gaps and overlaps.
Example - euclidean rectangularparallelepiped.
V.Bulatov. Tilings of the hyperbolic space and their visualization 32 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
33/138
Coxeter Polyhedra Tiling
First and second generation of tilingof the euclidean space byrectangular parallelepipeds.
We can easy imagine the rest of thetiling - infinite regular grid.
.Bulatov. Tilings of the hyperbolic space and their visualization 33 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
34/138
Coxeter polyhedra in H3
Coxeter polyhedra in hyperbolicspace.
Do they exist?
.Bulatov. Tilings of the hyperbolic space and their visualization 34 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
35/138
Coxeter polyhedra in H3
From Andreev's Theorem(1967) oncompact polyhedra with non-optusedihedral angles in hyperbolic spacefollows:
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
,
2,
2,
n, n 2,
3) Dihedral angles for each prismatic3-cycle satisfy 1 + 2 + 3 <
4) Dihedral angles for each prismatic4-cycle satisfy 1 + 2 + 3 + 4 < 2
.Bulatov. Tilings of the hyperbolic space and their visualization 35 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
36/138
Coxeter polyhedra in H3
Example of polyhedron satisfyingAndreev's theorem.
Edges with dihedral angles =
m
are marked with m ifm > 2Unmarked edges have dihedral
angles
2.
The polyhedron is combinatoriallyequivalent to a cube with one
truncated vertex.
V.Bulatov. Tilings of the hyperbolic space and their visualization 36 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
37/138
Example. Truncated cube in H3
Visualization of the actual geometricrealization of truncated cube in thehyperbolic space shown in thePoincare ball model.
V.Bulatov. Tilings of the hyperbolic space and their visualization 37 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
38/138
Example. Right angled dodecahedron
Regular right angled hyperbolicdodecahedron
All dihedral angles are equal
2
All faces are regular right angledhyperbolic pentagons
V.Bulatov. Tilings of the hyperbolic space and their visualization 38 of 138
3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
39/138
Non-compact Coxeter polyhedra in H3
From Andreev's Theorem(1967) onpolyhedra of finite volume withnon-optuse dihedral angles inhyperbolic space follows:
There exist unique Coxeter
polyhedron in H3
if1) the additional triplets of dihedralangles are allowed:
2,
4,
4,
3,
3,
3
,
2,
3,
6
Vertices with such angles arelocated at the infinity of hyperbolicspace (ideal vertices).
2) Ideal vertices with 4 adjacentfaces and all four dihedral angles
equal
2are permitted.
V.Bulatov. Tilings of the hyperbolic space and their visualization 39 of 138
E l Id l h b li t t h d
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
40/138
Example. Ideal hyperbolic tetrahedron
Tetrahedron with all dihedral angles
3. All 4 vertices are ideal.
V.Bulatov. Tilings of the hyperbolic space and their visualization 40 of 138
E l Id l h b li t h d
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
41/138
Example. Ideal hyperbolic octahedron
Octahedron with all dihedral angles
2. All 6 vertices are ideal.
V.Bulatov. Tilings of the hyperbolic space and their visualization 41 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
42/138
3 3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
43/138
Coxeter polyhedra in E3 vs H3 (continued)
Coxeter polyhedra in Euclidean space Coxeter polyhedra in Hyperbolic space
Shape may have continuous parameters, for
example height of a prism.
Shape of polyhedron of finite volume is fixed
by the choice of it's dihedral angles.
Few families of infinite polyhedra
Infinite number of infinite families of infinite
polyhedra of finite and infinite volume.
V.Bulatov. Tilings of the hyperbolic space and their visualization 43 of 138
3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
44/138
Example: 32 Coxeter tetrahedra in H3
V.Bulatov. Tilings of the hyperbolic space and their visualization 44 of 138
3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
45/138
How to construct polyhedra in H3
"Hand" calculation for simpler polyhedra:
Construct small sphere around each vertex.
Intersections of adjacent faces with the
spheres form spherical triangles with angles
equal to dihedral angles.
Using spherical sines laws find sides of
spherical triangles from angles. These sides
are flat angles of polyhedron's faces.
From known flat angles of triangular faces find
their edges using hyperbolic sines laws.
From known angles and some edges find
remaining edges using hyperbolic
trigonometry.
General method: Roeder R. (2007) ConstructingHyperbolic Polyhedra Using Newton's Method.
V.Bulatov. Tilings of the hyperbolic space and their visualization 45 of 138
3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
46/138
Visualization of t iling in H3
Regular right angled dodecahedron.12 generators of the tiling arereflections in each of 12dodecahedron's faces
V.Bulatov. Tilings of the hyperbolic space and their visualization 46 of 138
3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
47/138
Visualization of t iling in H3
Regular right angled dodecahedron.First iteration of reflections.
V.Bulatov. Tilings of the hyperbolic space and their visualization 47 of 138
Vi li ti f t ili i H3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
48/138
Visualization of t iling in H3
Regular right angled dodecahedron.2nd iteration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 48 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
49/138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
50/138
Vi li ti f t ili i H3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
51/138
Visualization of t iling in H3
Regular right angled dodecahedron.10-th iteration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 51 of 138
Vi li ti f t ili i H3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
52/138
Visualization of t iling in H3
J ust kidding!
It is a sphere, which is good
approximation to 1010 dodecahedral
tiles.
V.Bulatov. Tilings of the hyperbolic space and their visualization 52 of 138
Vis ali ation of tiling in H3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
53/138
Visualization of t iling in H3
Different approach - rendering onlyedges(image by Claudio Rocchini fromwikipedia.org).4 generations of tiles are shown. Notmuch is possible to see.
V.Bulatov. Tilings of the hyperbolic space and their visualization 53 of 138
Visualization of tiling in H3
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
54/138
Visualization of t iling in H3
Another approach - looking at thetiling from inside of the space (W.Thurston, J . Weeks). This allows tosee clearly the local structure of thetiling.
Image by Tom Ruen fromwikipedia.org generated byJ effWeeks interactive software.
Similar image from "Knot plot" videowas used on cover of dozens of
math books.
V.Bulatov. Tilings of the hyperbolic space and their visualization 54 of 138
Visualization of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
55/138
Visualization of 3D hyperbolic t ilings
Hyperbolic space tessellation as it looks frominside the space. Frame from "Not Knot" video(1994)
V.Bulatov. Tilings of the hyperbolic space and their visualization 55 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
56/138
Visualization of tiling by subgroup
Looking from inside we can see onlynearest neighbours.
Looking from outside we can onlysee the outer boundary.
Let's try to look outside, but see notthe whole tiling, but only it's part.
We select only subset of allgenerators. The subset will generatesubgroup of the whole group.
.Bulatov. Tilings of the hyperbolic space and their visualization 56 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
57/138
Visualization of tiling by subgroup
First iteration of subgroupgeneration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 57 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
58/138
Visualization of tiling by subgroup
Second iteration of subgroupgeneration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 58 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
59/138
Visualization of tiling by subgroup
Third (and last) iteration of subgroupgeneration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 59 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
60/138
g y g p
Move center of the shape into thecenter of the Poincare ball. Shapehas obvious cubic symmetry.
V.Bulatov. Tilings of the hyperbolic space and their visualization 60 of 138
Visualization of tiling by subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
61/138
g y g p
Another view of the same finitesubtiling. We have 8 dodecahedraaround center
V.Bulatov. Tilings of the hyperbolic space and their visualization 61 of 138
Visualization of infinite subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
62/138
g p
We can select generators, whichgenerate infinite subgroup.
.Bulatov. Tilings of the hyperbolic space and their visualization 62 of 138
Visualization of infinite subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
63/138
g p
First iteration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 63 of 138
Visualization of infinite subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
64/138
g p
Second iteration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 64 of 138
Visualization of infinite subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
65/138
g p
Third iteration.
V.Bulatov. Tilings of the hyperbolic space and their visualization 65 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
66/138
Few more iterations.
All tiles are lying in one hyperbolicplane.
V.Bulatov. Tilings of the hyperbolic space and their visualization 66 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
67/138
Rotation of the tiling around thecenter of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 67 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
68/138
Rotation of the tiling around thecenter of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 68 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
69/138
Rotation of the tiling around thecenter of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 69 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
70/138
Rotation of the tiling around thecenter of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 70 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
71/138
Hyperbolic translation of plane intothe center of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 71 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
72/138
Hyperbolic translation of plane intothe center of the ball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 72 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
73/138
More rotation about the center of theball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 73 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
74/138
More rotation about the center of theball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 74 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
75/138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
76/138
More rotation about the center of theball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 76 of 138
Infinite Fuchsian subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
77/138
More rotation about the center of theball.
V.Bulatov. Tilings of the hyperbolic space and their visualization 77 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
78/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 78 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
79/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 79 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
80/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 80 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
81/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 81 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
82/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 82 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
83/138
Stretching the Poincare ball intoinfinite cylinder.
V.Bulatov. Tilings of the hyperbolic space and their visualization 83 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
84/138
Cylinder model is straighforwardaxially symmetric 3D generalizationof the conformal band model of thehyperbolic plane.
Cylinder model of hyperbolic space
is not conformal, it has some limitedangular distorsions. However, it hassuch a nice property as euclideanmetrics along cylinder's axis. It isespecially usefull when we want tovisualize some specific hyperbolicgeodesic which we aling in that casewith the cylinder's axis. All the
planes orthoginal to that geodesicare represented by circular disksorthogonal to the axis.
V.Bulatov. Tilings of the hyperbolic space and their visualization 84 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
85/138
Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
If cylinder's axis is aligned with theaxis of a hyperbolic transformation ofthe 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'saxis.
V.Bulatov. Tilings of the hyperbolic space and their visualization 85 of 138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
86/138
Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
This group has no loxodromictransformatons.
V.Bulatov. Tilings of the hyperbolic space and their visualization 86 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
87/138
Tiling in Cylinder Model
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
88/138
Different orientation of the sametiling inside of cylinder model ofhyperbolic space.
No period is visible.
V.Bulatov. Tilings of the hyperbolic space and their visualization 88 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
89/138
Different subset of generators
generates tiling, which is not flatanymore.The limit set of the tiling is continousfractal curve. Axis of cylinder isaligned with the axis of an hyperbolictransformation of the subgroup.
V.Bulatov. Tilings of the hyperbolic space and their visualization 89 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
90/138
Another orientation of the same
tiling.Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.Tiling is periodic along cylinder'sboundary.
V.Bulatov. Tilings of the hyperbolic space and their visualization 90 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
91/138
Another orientation of the same
tiling.Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.Tiling is periodic along cylinder'sboundary.
V.Bulatov. Tilings of the hyperbolic space and their visualization 91 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
92/138
Another orientation of the same
tiling.Axis of cylinder is aligned with theaxis of an hyperbolic transformationof the subgroup.Tiling is periodic along cylinder'sboundary.
V.Bulatov. Tilings of the hyperbolic space and their visualization 92 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
93/138
Axis of cylinder is aligned with theaxis of a loxodromic transform.Tiling is spiraling around thecylinder's axis.
V.Bulatov. Tilings of the hyperbolic space and their visualization 93 of 138
Quasifuchsian Subgroup
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
94/138
Axis of cylinder is aligned with theaxis of different loxodromictransform.Tiling is spiraling around thecylinder's axis at different speed.
V.Bulatov. Tilings of the hyperbolic space and their visualization 94 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
95/138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
96/138
Quasifuchsian tiling with 5 and 9 sidedprisms.
V.Bulatov. Tilings of the hyperbolic space and their visualization 96 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
97/138
Quasifuchsian tiling with 5 and 18 sidedprisms.
V.Bulatov. Tilings of the hyperbolic space and their visualization 97 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
98/138
Tiling with right angled dodecahedra.Limit set is point set.
V.Bulatov. Tilings of the hyperbolic space and their visualization 98 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
99/138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
100/138
Tiling with rhombic triacontahedra
V.Bulatov. Tilings of the hyperbolic space and their visualization 100 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
101/138
Tiling with truncated tetrahedra. Axis
of cylinder model is vertical and it isaligned with common perpendicularto two selected planes. Theseplanes become flat in the cylindermodel and are at the top and bottomof the piece.
V.Bulatov. Tilings of the hyperbolic space and their visualization 101 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
102/138
Tiling with different truncatedtetrahedra in cylinder model. Axis ofcylinder is aligned with axis of somehyperbolic transformation of thegroup, which makes tiling periodic.
V.Bulatov. Tilings of the hyperbolic space and their visualization 102 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
103/138
Tiling with truncated Lambert cubes.The limit set is complex fractal.
V.Bulatov. Tilings of the hyperbolic space and their visualization 103 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
104/138
Another tiling with truncated cubes.
V.Bulatov. Tilings of the hyperbolic space and their visualization 104 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
105/138
Tiling with two generator free group.Shown in the cylinder model.Generators have parabollic commutator,which is responsible for cusps.
V.Bulatov. Tilings of the hyperbolic space and their visualization 105 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
106/138
The same tiling in shown in the Poincareball model.
V.Bulatov. Tilings of the hyperbolic space and their visualization 106 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
107/138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
108/138
Tiling with two generator free groupshown in the cylinder model.Generators have almost paraboliccommutator.
V.Bulatov. Tilings of the hyperbolic space and their visualization 108 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
109/138
Tiling with group with paraboliccommutator shown in the cylinder model.Fixed points of two parabolictransformations are located at thecylinder's axis at infinity.
V.Bulatov. Tilings of the hyperbolic space and their visualization 109 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
110/138
Some quasifuchsian group with 5generators shown in the cylinder model.
V.Bulatov. Tilings of the hyperbolic space and their visualization 110 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
111/138
Wild quasifuchsian group.
V.Bulatov. Tilings of the hyperbolic space and their visualization 111 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
112/138
Wilder quasifuchsian group.
V.Bulatov. Tilings of the hyperbolic space and their visualization 112 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
113/138
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 tilingstructure on the hyperplanes, which
corresponds to
4,
5,
5triangle formed
at the truncated vertex.
V.Bulatov. Tilings of the hyperbolic space and their visualization 113 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
114/138
Another orientation the same tiling
with cubes in the cylinder model.
The axis of cylinder model isperpendicular to one selected plane,which make this plane at the bottomflat.
V.Bulatov. Tilings of the hyperbolic space and their visualization 114 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
115/138
Another orientation the same tiling.*455 pattern is repeated at everyexposed plane.
V.Bulatov. Tilings of the hyperbolic space and their visualization 115 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
116/138
Another orientation the same tiling.
To find this orientation we selected aloxodromic transformation with small
rotational and large translationalcomponents. Next we selected 2hyperbolic planes with poles nearthe 2 fixed points of thetransformation and have allignedcommon perpendicular to theseplanes with the axis of the cylindermodel.
V.Bulatov. Tilings of the hyperbolic space and their visualization 116 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
117/138
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.
V.Bulatov. Tilings of the hyperbolic space and their visualization 117 of 138
Samples of 3D hyperbolic t ilings
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
118/138
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 commonperpendicular aligned with thecylinder's axis.
V.Bulatov. Tilings of the hyperbolic space and their visualization 118 of 138
Samples of tiling at infinity
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
119/138
Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.
V.Bulatov. Tilings of the hyperbolic space and their visualization 119 of 138
Samples of tiling at infinity
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
120/138
Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.
V.Bulatov. Tilings of the hyperbolic space and their visualization 120 of 138
Samples of tiling at infinity
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
121/138
Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.
V.Bulatov. Tilings of the hyperbolic space and their visualization 121 of 138
Samples of tiling at infinity
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
122/138
Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.
V.Bulatov. Tilings of the hyperbolic space and their visualization 122 of 138
Samples of tiling at infinity
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
123/138
Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.
V.Bulatov. Tilings of the hyperbolic space and their visualization 123 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
124/138
Metal sculpture. 20 Dodecahedra.
V.Bulatov. Tilings of the hyperbolic space and their visualization 124 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
125/138
Metal sculpture. First iteration of Weber-Seifert
dodecahedral tiling.
V.Bulatov. Tilings of the hyperbolic space and their visualization 125 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
126/138
Metal sculpture. 20 hyperbolic cubes.
V.Bulatov. Tilings of the hyperbolic space and their visualization 126 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
127/138
Metal sculpture. Tiling by Fuchsian group.
V.Bulatov. Tilings of the hyperbolic space and their visualization 127 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
128/138
Hyperbolic bracelet.
V.Bulatov. Tilings of the hyperbolic space and their visualization 128 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
129/138
Hyperbolic bracelet.
V.Bulatov. Tilings of the hyperbolic space and their visualization 129 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
130/138
Hyperbolic bracelet.
V.Bulatov. Tilings of the hyperbolic space and their visualization 130 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
131/138
Hyperbolic bracelet.
V.Bulatov. Tilings of the hyperbolic space and their visualization 131 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
132/138
Hyperbolic pendant.
V.Bulatov. Tilings of the hyperbolic space and their visualization 132 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
133/138
Hyperbolic pendant.
V.Bulatov. Tilings of the hyperbolic space and their visualization 133 of 138
Implementation in Metal
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
134/138
Hyperbolic pendant.
V.Bulatov. Tilings of the hyperbolic space and their visualization 134 of 138
Hyperbolic Sculpture
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
135/138
Hyperbolic sculpture. Color 3D print.
V.Bulatov. Tilings of the hyperbolic space and their visualization 135 of 138
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
136/138
Hyperbolic Sculpture
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
137/138
Hyperbolic sculpture. Color 3D print.
V.Bulatov. Tilings of the hyperbolic space and their visualization 137 of 138
Hyperbolic Sculpture
-
7/30/2019 Bulatov v. (2011) Tilings of the Hyperbolic Space and Their Visualization
138/138
Online address of the talk:bulatov.org/math/1101/