tiling the hyperbolic plane - long version

Gaussian curvatureElliptic geometry

Hyperbolic geometryTiling

M.C. Escher’s work

Tiling the hyperbolic plane - longversion

Daniel Czegel

Eotvos Lorand University, Budapest

ICPS, HeidelbergAugust 14, 2014

Daniel Czegel Tiling the hyperbolic plane - long version

Definition of Gaussian curvature

1 find a normal vector N

2 rotate the normal planecontaining N

3 intersection of the surfaceand the normal plane: aplane curve, curvature:κ = 1

4 Gaussian curvature of thesurface:

K (r) = κmin(r) κmax(r)

5 sign!

Theorema Egregium

Does K (r) change if we wrap, bend, twist (i.e. change theembedding in the 3 dim space)?

Theorema Egregium (Great Theorem): No!

Gaussian curvature is an intrinsic measure of the surface!

In a 2 dimensional Universe, K fully determines howspacetime curves: GR, Einstein eqs.:

Rµν −R

2gµν + Λgµν =

c4Tµν

In 2 dim:Rµν = f (K , gµν), R = 2K

Theorema Egregium

Rµν −R

2gµν + Λgµν =

c4Tµν

Theorema Egregium

Rµν −R

2gµν + Λgµν =

c4Tµν

Theorema Egregium

Rµν −R

2gµν + Λgµν =

c4Tµν

Theorema Egregium

Rµν −R

2gµν + Λgµν =

c4Tµν

Cylinder, Oloid

⇒ a surface can be unfolded without distortion ⇔ K ≡ 0

cylinder:

κmax =1

R, κmin =

∞= 0 ⇒ K ≡ 0

nontrivial example: oloid: convex hull of two perpendicularcircles

every point of its surface touches the floor during rolling!

Cylinder, Oloid

cylinder:

κmax =1

R, κmin =

∞= 0 ⇒ K ≡ 0

Cylinder, Oloid

cylinder:

κmax =1

R, κmin =

∞= 0 ⇒ K ≡ 0

Cylinder, Oloid

cylinder:

κmax =1

R, κmin =

∞= 0 ⇒ K ≡ 0

Spherical triangles

sphere: K =?

K ≡ 1R2 > 0, a model of elliptic geometry;

”straight lines”=geodesics: great circles

What is the sum of angles Σ in a triangle?

Spherical triangles

sphere: K =?

Spherical triangles

sphere: K =?

Spherical triangles

sphere: K =?

Spherical triangles

i) Σ =3

2π, A =

2R2 ii) Σ = 3π, A = 2πR2

Generally?

A = (Σ− π)R2 ⇒ KA = Σ− π

The sum of angles Σ is size-independent: only if K = 0(euclidean geometry)

elliptic geometry:

KA > 0 ⇒ Σ > π

Spherical triangles

i) Σ =3

2π, A =

2R2 ii) Σ = 3π, A = 2πR2

Generally?

A = (Σ− π)R2 ⇒ KA = Σ− π

elliptic geometry:

KA > 0 ⇒ Σ > π

Spherical triangles

i) Σ =3

2π, A =

2R2 ii) Σ = 3π, A = 2πR2

Generally?

A = (Σ− π)R2 ⇒ KA = Σ− π

elliptic geometry:

KA > 0 ⇒ Σ > π

Spherical triangles

i) Σ =3

2π, A =

2R2 ii) Σ = 3π, A = 2πR2

Generally?

A = (Σ− π)R2 ⇒ KA = Σ− π

elliptic geometry:

KA > 0 ⇒ Σ > π

Stereographic projection

sphere: K 6= 0 ⇒ cannot be unfolded:there is no sphere → plane map that preserves both distanceand angle!

We have to choose; e.g.: preserves angle (conformal), butdoes not preserve distance: stereographic projection

unit sphere, equator ⊂ plane, project from the north pole

sphere: K 6= 0 ⇒ cannot be unfolded:there is no sphere → plane map that preserves both distanceand angle!We have to choose; e.g.: preserves angle (conformal), butdoes not preserve distance: stereographic projection

unit sphere, equator ⊂ plane, project from the north pole

sphere: K 6= 0 ⇒ cannot be unfolded:there is no sphere → plane map that preserves both distanceand angle!We have to choose; e.g.: preserves angle (conformal), butdoes not preserve distance: stereographic projection

unit sphere, equator ⊂ plane, project from the north poleDaniel Czegel Tiling the hyperbolic plane - long version

Properties of stereographic projection

northern (southern) hemisphere 7→ outside (inside) of the unitcircle (north pole 7→ ∞)

circle 7→ circle

special case: geodetic (”straight line”)=great circle 7→ circle

circle 7→ circle

Metric induced by stereographic projection

The spherical geometry can be modeled on a plane

Do not forget: this model does not preserve distance, instead,the metric:

ds2 = (dx2 + dy2)

(1 + x2 + y2)2

)metric tensor:

g = λ(x , y)

(1 00 1

)⇒ isotropic scaling ⇒ conformal (angle-preserving)

distance on the sphere, if (0, 0)→∞ on the plane?∫ ∞0

∫ ∞0

1 + x2dx = π

ds2 = (dx2 + dy2)

(1 + x2 + y2)2

metric tensor:

g = λ(x , y)

(1 00 1

∫ ∞0

1 + x2dx = π

ds2 = (dx2 + dy2)

(1 + x2 + y2)2

)metric tensor:

g = λ(x , y)

(1 00 1

∫ ∞0

1 + x2dx = π

ds2 = (dx2 + dy2)

(1 + x2 + y2)2

)metric tensor:

g = λ(x , y)

(1 00 1

distance on the sphere, if (0, 0)→∞ on the plane?

∫ ∞0

1 + x2dx = π

ds2 = (dx2 + dy2)

(1 + x2 + y2)2

)metric tensor:

g = λ(x , y)

(1 00 1

∫ ∞0

1 + x2dx = π

The Earth

The image of the Earth under stereographic projection from thesouth pole:

Hyperbolic geometry

Is K < 0 (negative curvature) possible at a point?

Saddle point: κmin < 0, κmax > 0

Hyperbolic plane: saddle points everywhere! (e.g. K ≡ −1)Can you imagine it? Is it possible to embed it into a 3 dimeuclidean space?

Hyperbolic geometry

Is K < 0 (negative curvature) possible at a point?Saddle point: κmin < 0, κmax > 0

Hyperbolic geometry

Hyperbolic plane: saddle points everywhere! (e.g. K ≡ −1)

Can you imagine it? Is it possible to embed it into a 3 dimeuclidean space?

Hyperbolic geometry

Hyperbolic paper

Cut equilateral triangles

At every vertex, glue 7 (instead of 6) triangles to each other!

What happens, if 5 triangles are glued at a vertex?

Icosahedron!

Hyperbolic paper

Icosahedron!

Hyperbolic paper

Icosahedron!

Hyperbolic paper

Icosahedron!

Hyperbolic paper

Icosahedron!

Hyperbolic triangles

Sum of angles Σ in a triangle?

elliptic & euclidean case:

KA = Σ− π

good news: it is also valid for K < 0!

A > 0 ⇒ KA < 0 ⇒ Σ < π

special case: K = −1

A = π − Σ ≤ π

Despite the hyperbolic plane is infinite, no triangle can havelarger area than π!!

KA = Σ− π

A > 0 ⇒ KA < 0 ⇒ Σ < π

A = π − Σ ≤ π

KA = Σ− π

A > 0 ⇒ KA < 0 ⇒ Σ < π

A = π − Σ ≤ π

KA = Σ− π

A > 0 ⇒ KA < 0 ⇒ Σ < π

A = π − Σ ≤ π

KA = Σ− π

A > 0 ⇒ KA < 0 ⇒ Σ < π

A = π − Σ ≤ π

KA = Σ− π

A > 0 ⇒ KA < 0 ⇒ Σ < π

A = π − Σ ≤ π

Poincare model of the hyperbolic plane

K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane

Poincare model: angle preserving, but not distance preserving(like the stereographic projection)

infinite hyperbolic plane → unit disc

infinity 7→ edge of the unit disk

geodesics (”straight lines”) 7→ circles that meet the edge ofthe unit disc at 90◦

parallel lines 7→ not intersecting such circles

Ideal triangles

Triangles with largest area?

A = π − Σ

largest: if Σ = 0 ( ⇔ α = β = γ = 0). How does it look like?

Figure 1 : Ideal triangles having all verices at infinity. Note thatthese triangles are congruent (having the same area A = π)!

Ideal triangles

A = π − Σ

Ideal triangles

A = π − Σ

Ideal triangles

A = π − Σ

Hyperbolic metric

Metric?

ds2 = (dx2 + dy2)

(1−(x2 + y2))2

)(compare with stereographic projection!)

again: isotropic scaling ⇒ angle-preserving

Near the edge (x2 + y2 ≈ 1): ds2 >> dx2 + dy2

center → edge in the Poincare model: ∞ distance in thehyperbolic plane!∫ 1

1− x2dx =∞

Hyperbolic metric

Metric?

ds2 = (dx2 + dy2)

(1−(x2 + y2))2

1− x2dx =∞

Hyperbolic metric

Metric?

ds2 = (dx2 + dy2)

(1−(x2 + y2))2

1− x2dx =∞

Hyperbolic metric

Metric?

ds2 = (dx2 + dy2)

(1−(x2 + y2))2

1− x2dx =∞

Hyperbolic metric

Metric?

ds2 = (dx2 + dy2)

(1−(x2 + y2))2

1− x2dx =∞

Figure 2 : Hyperbolic man takes a walk to infinity

Schlafli-symbol

Regular tiling: tiling by i) congruent regular poligons (n-gons),ii) at every vertex, m poligons meet

Euclidean plane?

Schlafli-symbol: {n,m}Any relationship between these three Schlafli-symbols?

2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling

Schlafli-symbol

Euclidean plane?

Schlafli-symbol

Euclidean plane?

Schlafli-symbol

Euclidean plane?

Schlafli-symbol: {n,m}

Any relationship between these three Schlafli-symbols?

Schlafli-symbol

Euclidean plane?

Schlafli-symbol

Euclidean plane?

Tiling on the sphere

Regular tilings on the sphere? (assume n,m ≥ 3, i.e.nondegenrate cases)

n = 3, m = 3 ?

A blown tetrahedron!

n = 3, m = 3 ?

{4, 3} ?

A cube:

{3, 4}, {3, 5}, {5, 3} ?

Octahedron, icosahedron, dodecahedron.

{4, 3} ?

A cube:

{3, 4}, {3, 5}, {5, 3} ?

{4, 3} ?

A cube:

{3, 4}, {3, 5}, {5, 3} ?

{4, 3} ?

A cube:

{3, 4}, {3, 5}, {5, 3} ?

Any more?

No. Why?

only for these five!

Figure 3 : The five Platonic solids

Any more?

No. Why?

Any more?

No. Why?

Hyperbolic tiling

Hyperbolic tiling?

If 1n + 1

m < 12

How many such tilings?Infinite!