Tilings of the hyperbolic space and their visualizationbulatov.org/math/1101/Bulatov V. (2011) Tilings of the hyperbolic... · Tilings of the hyperbolic space and their visualization

Tilings of the hyperbolic space and their visualization

Vladimir BulatovCorvallis, Oregon, USAJoint MAA/AMS meeting, New Orleans, January 7, 2011

AbstractVisual 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/

V.Bulatov. Tilings of the hyperbolic space and their visualization 1 of 138

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


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


Visualization of 2D hyperbolic tiling

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 überdie Theorie der ElliptischenModulfunctionen," Vol. 1, Leipzig(1890)


Visualization of 2D hyperbolic tilings

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



Many hyperbolic patterns by D.Dunham



C.Goodman-Strauss's graphics work.



C.Kaplan's Islamic pattern.



Martin von Gagern "Hyperbolization ofEuclidean Ornaments"


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 über 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



B.Mandelbrot's computerrendering of Kleinian group(from Un ensemble-limite parMichele Audin et ArnaudCheritat (2009))



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



"Indra Pearl" by D.Mumford,C.Series and D.Wright (2004)popularized visualization of limit setsof Kleinian groups



Kleinian group images by Jos Leys.

His very artistic images rendered as3D images represent essentially2-dimensional limit set of Kleiniangroup.


Tiling with Coxeter Polygons

Coxeter polygon:

All angles are submultiples of

i =mi

.

Here all four angles of euclidean

polygon are equal 2



The tiling is generated by reflectionsin the sides of the Coxeter polygon.

Here we have original tile andsecond generation of tiles.



First, second and third generation oftiles.

Condition on angles i =mi

guaranties, that tiles fit without gaps.2mi tiles meet at each vertex.



Complete tiling - regular rectangulargrid.


Coxeter Polygons in the Hyperbolic Plane

Hyperbolic Coxeter pentagon with all

i =2 shown 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.



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



First, second and third generationsof tiles.

i =mi

guaranties, that 2mi tiles

around each vertex fit without gapsand overlaps.



Four generations of tiles.


Coxeter tiles in the Hyperbolic Plane

Complete tiling in the Poincare discmodel of hyperbolic plane.


Coxeter tiles in the band model

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



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



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 isobvious to our euclidean eyes.


Motion in the band model

Appropriate hyperbolic isometry cansend any 2 selected points to ofthe band model.








Existence of Coxeter Polygons

Hyperbolic Coxeter N-gon exist for

any N 5 and angles i =mi

mi 2

Hyperbolic triangle exist if1

m1+

1m2

+1

m3< 1

Hyperbolic quadrangle exist if1

m1+

1m2

+1

m3+

1m4

< 2.

The space of shapes of CoxeterN-gon has dimension N 3.


Existence of Coxeter Polygons

Example.

Tiling of hyperbolic plane withCoxeter hexagons with all angles

equal 3.


Coxeter Polyhedron

Dihedral angles of Coxeterpolyhedron are submultiples of :

i =mi

.

Reflections in sides of Coxeterpolyhedron satisfy conditions ofPoincare Polyhedron Theorem.

Therefore the reflected copies ofsuch polyhedron fill the spacewithout gaps and overlaps.

Example - euclidean rectangularparallelepiped.


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.


Coxeter polyhedra in H3

Coxeter polyhedra in hyperbolicspace.

Do they exist?



From Andreev's Theorem(1967) oncompact polyhedra with non-optusedihedral angles in hyperbolic spacefollows:

There exist unique compact Coxeterpolyhedron 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



Example of polyhedron satisfyingAndreev's theorem.

Edges with dihedral angles =m

are marked with m if m > 2Unmarked edges have dihedral

angles 2.

The polyhedron is combinatoriallyequivalent to a cube with onetruncated vertex.


Example. Truncated cube in H3

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


Example. Right angled dodecahedron

Regular right angled hyperbolicdodecahedron

All dihedral angles are equal 2

All faces are regular right angledhyperbolic pentagons


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 Coxeterpolyhedron in H 3 if1) the additional triplets of dihedralangles are allowed:

2,

4,

4,

3,

3,

3,

2,

3,

6Vertices with such angles arelocated at the infinity of hyperbolicspace (ideal vertices).2) Ideal vertices with 4 adjacentfaces and all four dihedral angles

equal 2 are permitted.


Example. Ideal hyperbolic tetrahedron

Tetrahedron with all dihedral angles

3. All 4 vertices are ideal.


Example. Ideal hyperbolic octahedron

Octahedron with all dihedral angles

2. All 6 vertices are ideal.


Coxeter polyhedra in E3 vs H3

Coxeter polyhedra in Euclidean space Coxeter polyhedra in Hyperbolic space

Only 7 non-equivalent types of compact polyhedra:

Infinitely many combinatorially non-equivalentcompact polyhedra.

Each combinatorial type allows wide selection ofdifferent dihedral angles satisfying Andreev'stheorem.


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

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


Example: 32 Coxeter tetrahedra in H3


How to construct polyhedra in H3

"Hand" calculation for simpler polyhedra:

Construct small sphere around each vertex.Intersections of adjacent faces with thespheres form spherical triangles with anglesequal to dihedral angles.

Using spherical sines laws find sides ofspherical triangles from angles. These sidesare 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 hyperbolictrigonometry.

General method: Roeder R. (2007) ConstructingHyperbolic Polyhedra Using Newton's Method.


Visualization of tiling in H3

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



Regular right angled dodecahedron.First iteration of reflections.



Regular right angled dodecahedron.2nd iteration.



Regular right angled dodecahedron.3rd iteration.



Regular right angled dodecahedron.4th iteration.



Regular right angled dodecahedron.10-th iteration.



Just kidding!

It is a sphere, which is goodapproximation to 1010 dodecahedraltiles.



Different approach - rendering onlyedges(image by Claudio Rocchini fromwikipedia.org).4 generations of tiles are shown. Notmuch is possible to see.



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 by JeffWeeks interactive software.

Similar image from "Knot plot" videowas used on cover of dozens ofmath books.



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


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.



First iteration of subgroupgeneration.



Second iteration of subgroupgeneration.



Third (and last) iteration of subgroupgeneration.



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



Another view of the same finitesubtiling. We have 8 dodecahedraaround center


Visualization of infinite subgroup

We can select generators, whichgenerate infinite subgroup.



First iteration.



Second iteration.



Third iteration.


Infinite Fuchsian subgroup

Few more iterations.

All tiles are lying in one hyperbolicplane.



Rotation of the tiling around thecenter of the ball.












Hyperbolic translation of plane intothe center of the ball.



Hyperbolic translation of plane intothe center of the ball.



More rotation about the center of theball.














Tiling in Cylinder Model

Stretching the Poincare ball intoinfinite cylinder.


















Cylinder model is straighforwardaxially symmetric 3D generalizationof the conformal band model of thehyperbolic plane.

Cylinder model of hyperbolic spaceis 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 theplanes orthoginal to that geodesicare represented by circular disksorthogonal to the axis.



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.




This group has no loxodromictransformatons.




Transformation with longer period isaligned with the cylinder's axis.




No period is visible.


Quasifuchsian Subgroup

Different subset of generatorsgenerates 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.



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









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



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


Samples of 3D hyperbolic tilings

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



Quasifuchsian tiling with 5 and 9 sidedprisms.



Quasifuchsian tiling with 5 and 18 sidedprisms.



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



Tiling with rhombic triacontahedra in thecylinder model.



Tiling with rhombic triacontahedra



Tiling with truncated tetrahedra. Axisof 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.



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



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



Another tiling with truncated cubes.



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



The same tiling in shown in the Poincareball model.



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



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



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



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



Wild quasifuchsian group.



Wilder quasifuchsian group.



Tiling with truncated cubes in thecylinder model.Axis of the cylinder model is allignedwith common perpendicular to 2planes. This makes both theseplanes flat.

Placing repeating pattern on thesurface reveals the *455 tilingstructure on the hyperplanes, which

corresponds to 4

,5

,5

triangle formed

at the truncated vertex.



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 bottomflat.



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



Another orientation the same tiling.

To find this orientation we selected aloxodromic transformation with smallrotational 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.



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.



Another orientation of the same tilingwith cubes in the cylinder model.

One of the hyperplanes is very closeto 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.


Samples of tiling at infinity

Tiling at the boundary of hyperbolicspace shown in stereographic projectionto 2D.














Implementation in Metal

Metal sculpture. 20 Dodecahedra.



Metal sculpture. First iteration of Weber-Seifertdodecahedral tiling.



Metal sculpture. 20 hyperbolic cubes.



Metal sculpture. Tiling by Fuchsian group.



Hyperbolic bracelet.












Hyperbolic pendant.



Hyperbolic pendant.



Hyperbolic pendant.


Hyperbolic Sculpture

Hyperbolic sculpture. Color 3D print.









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


Documents

Tilings of the hyperbolic space and their visualizationbulatov.org/math/1101/Bulatov V. (2011) Tilings of the hyperbolic... · Tilings of the hyperbolic space and their visualization