tiling the hyperbolic plane
TRANSCRIPT
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling the hyperbolic plane
Daniel Czegel
Eotvos Lorand University, Budapest
ICPS, HeidelbergAugust 14, 2014
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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
R
4 Gaussian curvature of thesurface:
K (r) = κmin(r) κmax(r)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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
R
4 Gaussian curvature of thesurface:
K (r) = κmin(r) κmax(r)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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
R
4 Gaussian curvature of thesurface:
K (r) = κmin(r) κmax(r)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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
R
4 Gaussian curvature of thesurface:
K (r) = κmin(r) κmax(r)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Gaussian curvature of the unit sphere?
unit sphere: K ≡ 1 (elliptic geometry)
plane: K ≡ 0 (euclidean geometry)
K ≡ −1: saddle points everywhere! (hyperbolic geometry)
How to imagine the hyperbolic plane? Is it possible to embedit into the 3 dim euclidean space?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Gaussian curvature of the unit sphere?
unit sphere: K ≡ 1 (elliptic geometry)
plane: K ≡ 0 (euclidean geometry)
K ≡ −1: saddle points everywhere! (hyperbolic geometry)
How to imagine the hyperbolic plane? Is it possible to embedit into the 3 dim euclidean space?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Gaussian curvature of the unit sphere?
unit sphere: K ≡ 1 (elliptic geometry)
plane: K ≡ 0 (euclidean geometry)
K ≡ −1: saddle points everywhere! (hyperbolic geometry)
How to imagine the hyperbolic plane? Is it possible to embedit into the 3 dim euclidean space?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Gaussian curvature of the unit sphere?
unit sphere: K ≡ 1 (elliptic geometry)
plane: K ≡ 0 (euclidean geometry)
K ≡ −1: saddle points everywhere! (hyperbolic geometry)
How to imagine the hyperbolic plane? Is it possible to embedit into the 3 dim euclidean space?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Gaussian curvature of the unit sphere?
unit sphere: K ≡ 1 (elliptic geometry)
plane: K ≡ 0 (euclidean geometry)
K ≡ −1: saddle points everywhere! (hyperbolic geometry)
How to imagine the hyperbolic plane? Is it possible to embedit into the 3 dim euclidean space?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane
Poincare model of the hyperbolic plane: angle preserving, butnot 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◦
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane
Poincare model of the hyperbolic plane: angle preserving, butnot 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◦
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane
Poincare model of the hyperbolic plane: angle preserving, butnot 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◦
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane
Poincare model of the hyperbolic plane: angle preserving, butnot 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◦
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
K 6= 0 ⇒ There is no distance-preserving ANDangle-preserving map to the euclidean plane
Poincare model of the hyperbolic plane: angle preserving, butnot 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◦
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Poincare model of the hyperbolic plane
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Figure 1 : Hyperbolic man takes a walk to infinity
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
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?
1
n+
1
m=
1
2There is no other such n,m ∈ N ⇔ no other regulareuclidean tiling
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Regular tilings on the sphere?
n = 3, m = 3 ?
A blown tetrahedron!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Regular tilings on the sphere?
n = 3, m = 3 ?
A blown tetrahedron!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Regular tilings on the sphere?
n = 3, m = 3 ?
A blown tetrahedron!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Regular tilings on the sphere?
n = 3, m = 3 ?
A blown tetrahedron!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
{4, 3} ?
A cube:
{3, 4}, {3, 5}, {5, 3} ?
Octahedron, icosahedron, dodecahedron.
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
{4, 3} ?
A cube:
{3, 4}, {3, 5}, {5, 3} ?
Octahedron, icosahedron, dodecahedron.
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
{4, 3} ?
A cube:
{3, 4}, {3, 5}, {5, 3} ?
Octahedron, icosahedron, dodecahedron.
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
{4, 3} ?
A cube:
{3, 4}, {3, 5}, {5, 3} ?
Octahedron, icosahedron, dodecahedron.
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Any more?
No. Why?
1
n+
1
m>
1
2
only for these five!
Figure 2 : The five Platonic solids
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Any more?
No. Why?
1
n+
1
m>
1
2
only for these five!
Figure 2 : The five Platonic solids
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Tiling on the sphere
Any more?
No. Why?
1
n+
1
m>
1
2
only for these five!
Figure 2 : The five Platonic solids
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Hyperbolic tiling?
If 1n + 1
m < 12
How many such tilings?Infinite!
Figure 3 : {3, 7} Figure 4 : {7, 3}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Hyperbolic tiling?If 1
n + 1m < 1
2
How many such tilings?Infinite!
Figure 3 : {3, 7} Figure 4 : {7, 3}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Hyperbolic tiling?If 1
n + 1m < 1
2How many such tilings?
Infinite!
Figure 3 : {3, 7} Figure 4 : {7, 3}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Hyperbolic tiling?If 1
n + 1m < 1
2How many such tilings?Infinite!
Figure 3 : {3, 7} Figure 4 : {7, 3}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Hyperbolic tiling?If 1
n + 1m < 1
2How many such tilings?Infinite!
Figure 3 : {3, 7} Figure 4 : {7, 3}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
n or m can even be infinite!
Figure 5 : {3,∞} Figure 6 : {∞, 3},”aperiogon”
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
n or m can even be infinite!
Figure 5 : {3,∞} Figure 6 : {∞, 3},”aperiogon”
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Hyperbolic tiling
Or both!
Figure 7 : {∞,∞}
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Classification of regular tilings
Figure 8 : Classification of regular tilings
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Figure 9 : M.C. Escher
Figure 10 : H.S.M.Coxeter
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Figure 11 : Escher’s Circle Limit I. (1958)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Figure 12 : Circle Limit I.: nonregular tiling of the hyperbolic plane(m = 4, 6, angles: 60◦ − 90◦ − 60◦ − 90◦)
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Any regular tiling?
Figure 13 : Escher’s Circle Limit III. (1959).
Schlafli symbol?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Any regular tiling?
Figure 13 : Escher’s Circle Limit III. (1959).
Schlafli symbol?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Any regular tiling?
Figure 13 : Escher’s Circle Limit III. (1959).
Schlafli symbol?
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
Figure 14 : Schlafli symbol of Circle Limit III.: {8, 3}!
Daniel Czegel Tiling the hyperbolic plane
The three geometriesPoincare model
TilingM.C. Escher’s work
References
Weeks, J. R. (2001). The shape of space. CRC press.
http://aleph0.clarku.edu/
~djoyce/poincare/poincare.html
http://en.wikipedia.org/wiki/
Uniform_tilings_in_hyperbolic_plane
http://euler.slu.edu/escher/index.php/
Math_and_the_Art_of_M._C._Escher
http://www.reed.edu/reed_magazine/march2010/
features/capturing_infinity/3.html
Thank You!
Daniel Czegel Tiling the hyperbolic plane