Tiling the hyperbolic plane - long version
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The euclidean plane can be only tiled regularly by triangles, squares or hexagons. But what about other geometries with constant curvature? How to tile the sphere or the hyperbolic plane by regular polygons? After a short introduction to the Poincaré model of the hyperbolic plane, its regular tilings are discussed with illustrative examples from the work of M.C. Escher.
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Tiling the hyperbolic plane - long version
- 1. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling the hyperbolic plane - long version Daniel Czegel Eotvos Lorand University, Budapest ICPS, Heidelberg August 14, 2014 Daniel Czegel Tiling the hyperbolic plane - long version
- 2. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Daniel Czegel Tiling the hyperbolic plane - long version
- 3. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Definition of Gaussian curvature 1
- 4. nd a normal vector N Daniel Czegel Tiling the hyperbolic plane - long version
- 5. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Definition of Gaussian curvature 1
- 6. nd a normal vector N 2 rotate the normal plane containing N Daniel Czegel Tiling the hyperbolic plane - long version
- 7. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Definition of Gaussian curvature 1
- 8. nd a normal vector N 2 rotate the normal plane containing N 3 intersection of the surface and the normal plane: a plane curve, curvature: = 1R Daniel Czegel Tiling the hyperbolic plane - long version
- 9. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Definition of Gaussian curvature 1
- 10. nd a normal vector N 2 rotate the normal plane containing N 3 intersection of the surface and the normal plane: a plane curve, curvature: = 1R 4 Gaussian curvature of the surface: K(r) = min(r) max (r) Daniel Czegel Tiling the hyperbolic plane - long version
- 11. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Definition of Gaussian curvature 1
- 12. nd a normal vector N 2 rotate the normal plane containing N 3 intersection of the surface and the normal plane: a plane curve, curvature: = 1R 4 Gaussian curvature of the surface: K(r) = min(r) max (r) 5 sign! Daniel Czegel Tiling the hyperbolic plane - long version
- 13. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Theorema Egregium Does K(r) change if we wrap, bend, twist (i.e. change the embedding in the 3 dim space)? aniel Czegel Tiling the hyperbolic plane - long version D
- 14. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Theorema Egregium Does K(r) change if we wrap, bend, twist (i.e. change the embedding in the 3 dim space)? Theorema Egregium (Great Theorem): No! aniel Czegel Tiling the hyperbolic plane - long version D
- 15. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Theorema Egregium Does K(r) change if we wrap, bend, twist (i.e. change the embedding in the 3 dim space)? Theorema Egregium (Great Theorem): No! Gaussian curvature is an intrinsic measure of the surface! aniel Czegel Tiling the hyperbolic plane - long version D
- 16. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Theorema Egregium Does K(r) change if we wrap, bend, twist (i.e. change the embedding 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 how spacetime curves: GR, Einstein eqs.: R R 2 g + g = 8G c4 T aniel Czegel Tiling the hyperbolic plane - long version D
- 17. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Theorema Egregium Does K(r) change if we wrap, bend, twist (i.e. change the embedding 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 how spacetime curves: GR, Einstein eqs.: R R 2 g + g = 8G c4 T In 2 dim: R = f (K; g); R = 2K Daniel Czegel Tiling the hyperbolic plane - long version
- 18. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Cylinder, Oloid ) a surface can be unfolded without distortion , K 0 aniel Czegel Tiling the hyperbolic plane - long version D
- 19. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Cylinder, Oloid ) a surface can be unfolded without distortion , K 0 cylinder: max = 1 R ; min = 1 1 = 0 ) K 0 aniel Czegel Tiling the hyperbolic plane - long version D
- 20. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Cylinder, Oloid ) a surface can be unfolded without distortion , K 0 cylinder: max = 1 R ; min = 1 1 = 0 ) K 0 nontrivial example: oloid: convex hull of two perpendicular circles aniel Czegel Tiling the hyperbolic plane - long version D
- 21. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Cylinder, Oloid ) a surface can be unfolded without distortion , K 0 cylinder: max = 1 R ; min = 1 1 = 0 ) K 0 nontrivial example: oloid: convex hull of two perpendicular circles every point of its surface touches the oor during rolling! Daniel Czegel Tiling the hyperbolic plane - long version
- 22. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles sphere: K =? aniel Czegel Tiling the hyperbolic plane - long version D
- 23. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles sphere: K =? K 1 R2 0, a model of elliptic geometry; straight lines=geodesics: great circles aniel Czegel Tiling the hyperbolic plane - long version D
- 24. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles sphere: K =? K 1 R2 0, a model of elliptic geometry; straight lines=geodesics: great circles What is the sum of angles in a triangle? Daniel Czegel Tiling the hyperbolic plane - long version
- 25. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles sphere: K =? K 1 R2 0, a model of elliptic geometry; straight lines=geodesics: great circles What is the sum of angles in a triangle? Daniel Czegel Tiling the hyperbolic plane - long version
- 26. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles i) = 3 2 ; A = 2 R2 ii) = 3; A = 2R2 aniel Czegel Tiling the hyperbolic plane - long version D
- 27. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles i) = 3 2 ; A = 2 R2 ii) = 3; A = 2R2 Generally? A = ( )R2 ) KA = aniel Czegel Tiling the hyperbolic plane - long version D
- 28. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles i) = 3 2 ; A = 2 R2 ii) = 3; A = 2R2 Generally? A = ( )R2 ) KA = The sum of angles is size-independent: only if K = 0 (euclidean geometry) aniel Czegel Tiling the hyperbolic plane - long version D
- 29. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Spherical triangles i) = 3 2 ; A = 2 R2 ii) = 3; A = 2R2 Generally? A = ( )R2 ) KA = The sum of angles is size-independent: only if K = 0 (euclidean geometry) elliptic geometry: KA 0 ) Daniel Czegel Tiling the hyperbolic plane - long version
- 30. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Stereographic projection sphere: K6= 0 ) cannot be unfolded: there is no sphere ! plane map that preserves both distance and angle! aniel Czegel Tiling the hyperbolic plane - long version D
- 31. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Stereographic projection sphere: K6= 0 ) cannot be unfolded: there is no sphere ! plane map that preserves both distance and angle! We have to choose; e.g.: preserves angle (conformal), but does not preserve distance: stereographic projection aniel Czegel Tiling the hyperbolic plane - long version D
- 32. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Stereographic projection sphere: K6= 0 ) cannot be unfolded: there is no sphere ! plane map that preserves both distance and angle! We have to choose; e.g.: preserves angle (conformal), but does not preserve distance: stereographic projection unit sphere, equator plane, project from the north pole Daniel Czegel Tiling the hyperbolic plane - long version
- 33. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Properties of stereographic projection aniel Czegel Tiling the hyperbolic plane - long version D
- 34. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Properties of stereographic projection northern (southern) hemisphere7! outside (inside) of the unit circle (north pole7! 1) aniel Czegel Tiling the hyperbolic plane - long version D
- 35. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Properties of stereographic projection northern (southern) hemisphere7! outside (inside) of the unit circle (north pole7! 1) circle7! circle aniel Czegel Tiling the hyperbolic plane - long version D
- 36. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Properties of stereographic projection northern (southern) hemisphere7! outside (inside) of the unit circle (north pole7! 1) circle7! circle special case: geodetic (straight line)=great circle7! circle Daniel Czegel Tiling the hyperbolic plane - long version
- 37. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Metric induced by stereographic projection The spherical geometry can be modeled on a plane aniel Czegel Tiling the hyperbolic plane - long version D
- 38. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work 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) 4 (1 + x2 + y2)2 aniel Czegel Tiling the hyperbolic plane - long version D
- 39. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work 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) 4 (1 + x2 + y2)2 metric tensor: g = (x; y) 1 0 0 1 ) isotropic scaling ) conformal (angle-preserving) aniel Czegel Tiling the hyperbolic plane - long version D
- 40. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work 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) 4 (1 + x2 + y2)2 metric tensor: g = (x; y) 1 0 0 1 ) isotropic scaling ) conformal (angle-preserving) distance on the sphere, if (0; 0) ! 1 on the plane? aniel Czegel Tiling the hyperbolic plane - long version D
- 41. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work 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) 4 (1 + x2 + y2)2 metric tensor: g = (x; y) 1 0 0 1 ) isotropic scaling ) conformal (angle-preserving) distance on the sphere, if (0; 0) ! 1 on the plane? Z 1 0 ds = Z 1 0 2 1 + x2 dx = Daniel Czegel Tiling the hyperbolic plane - long version
- 42. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work The Earth The image of the Earth under stereographic projection from the south pole: Daniel Czegel Tiling the hyperbolic plane - long version
- 43. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic geometry Is K 0 (negative curvature) possible at a point? aniel Czegel Tiling the hyperbolic plane - long version D
- 44. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic geometry Is K 0 (negative curvature) possible at a point? Saddle point: min 0; max 0 aniel Czegel Tiling the hyperbolic plane - long version D
- 45. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic geometry Is K 0 (negative curvature) possible at a point? Saddle point: min 0; max 0 Hyperbolic plane: saddle points everywhere! (e.g. K 1) aniel Czegel Tiling the hyperbolic plane - long version D
- 46. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic geometry Is K 0 (negative curvature) possible at a point? Saddle point: min 0; max 0 Hyperbolic plane: saddle points everywhere! (e.g. K 1) Can you imagine it? Is it possible to embed it into a 3 dim euclidean space? Daniel Czegel Tiling the hyperbolic plane - long version
- 47. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic paper Cut equilateral triangles aniel Czegel Tiling the hyperbolic plane - long version D
- 48. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic paper Cut equilateral triangles At every vertex, glue 7 (instead of 6) triangles to each other! aniel Czegel Tiling the hyperbolic plane - long version D
- 49. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic paper Cut equilateral triangles At every vertex, glue 7 (instead of 6) triangles to each other! aniel Czegel Tiling the hyperbolic plane - long version D
- 50. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? aniel Czegel Tiling the hyperbolic plane - long version D
- 51. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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 - long version
- 52. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? aniel Czegel Tiling the hyperbolic plane - long version D
- 53. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? elliptic euclidean case: KA = aniel Czegel Tiling the hyperbolic plane - long version D
- 54. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? elliptic euclidean case: KA = good news: it is also valid for K 0! aniel Czegel Tiling the hyperbolic plane - long version D
- 55. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? elliptic euclidean case: KA = good news: it is also valid for K 0! A 0 ) KA 0 ) aniel Czegel Tiling the hyperbolic plane - long version D
- 56. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? elliptic euclidean case: KA = good news: it is also valid for K 0! A 0 ) KA 0 ) special case: K = 1 A = aniel Czegel Tiling the hyperbolic plane - long version D
- 57. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic triangles Sum of angles in a triangle? elliptic euclidean case: KA = good news: it is also valid for K 0! A 0 ) KA 0 ) special case: K = 1 A = Despite the hyperbolic plane is in
- 58. nite, no triangle can have larger area than !! Daniel Czegel Tiling the hyperbolic plane - long version
- 59. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane aniel Czegel Tiling the hyperbolic plane - long version D
- 60. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane Poincare model: angle preserving, but not distance preserving (like the stereographic projection) aniel Czegel Tiling the hyperbolic plane - long version D
- 61. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane Poincare model: angle preserving, but not distance preserving (like the stereographic projection) in
- 62. nite hyperbolic plane ! unit disc aniel Czegel Tiling the hyperbolic plane - long version D
- 63. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane Poincare model: angle preserving, but not distance preserving (like the stereographic projection) in
- 64. nite hyperbolic plane ! unit disc in
- 65. nity7! edge of the unit disk aniel Czegel Tiling the hyperbolic plane - long version D
- 66. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane Poincare model: angle preserving, but not distance preserving (like the stereographic projection) in
- 67. nite hyperbolic plane ! unit disc in
- 68. nity7! edge of the unit disk geodesics (straight lines)7! circles that meet the edge of the unit disc at 90 aniel Czegel Tiling the hyperbolic plane - long version D
- 69. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane K6= 0 ) There is no distance-preserving AND angle-preserving map to the euclidean plane Poincare model: angle preserving, but not distance preserving (like the stereographic projection) in
- 70. nite hyperbolic plane ! unit disc in
- 71. nity7! edge of the unit disk geodesics (straight lines)7! circles that meet the edge of the unit disc at 90 parallel lines7! not intersecting such circles Daniel Czegel Tiling the hyperbolic plane - long version
- 72. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Poincare model of the hyperbolic plane Daniel Czegel Tiling the hyperbolic plane - long version
- 73. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Ideal triangles Triangles with largest area? aniel Czegel Tiling the hyperbolic plane - long version D
- 74. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Ideal triangles Triangles with largest area? A = aniel Czegel Tiling the hyperbolic plane - long version D
- 75. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Ideal triangles Triangles with largest area? A = largest: if = 0 ( , =
- 76. = = 0): How does it look like? aniel Czegel Tiling the hyperbolic plane - long version D
- 77. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Ideal triangles Triangles with largest area? A = largest: if = 0 ( , =
- 78. = = 0): How does it look like? Figure 1 : Ideal triangles having all verices at in
- 79. nity. Note that these triangles are congruent (having the same area A = )! Daniel Czegel Tiling the hyperbolic plane - long version
- 80. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic metric Metric? aniel Czegel Tiling the hyperbolic plane - long version D
- 81. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic metric Metric? ds2 = (dx2 + dy2) 4 (1(x2 + y2))2 (compare with stereographic projection!) aniel Czegel Tiling the hyperbolic plane - long version D
- 82. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic metric Metric? ds2 = (dx2 + dy2) 4 (1(x2 + y2))2 (compare with stereographic projection!) again: isotropic scaling ) angle-preserving aniel Czegel Tiling the hyperbolic plane - long version D
- 83. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic metric Metric? ds2 = (dx2 + dy2) 4 (1(x2 + y2))2 (compare with stereographic projection!) again: isotropic scaling ) angle-preserving Near the edge (x2 + y2 1): ds2 dx2 + dy2 aniel Czegel Tiling the hyperbolic plane - long version D
- 84. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic metric Metric? ds2 = (dx2 + dy2) 4 (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: 1 distance in the hyperbolic plane! Z 1 0 ds = Z 1 0 2 1 x2 dx = 1 Daniel Czegel Tiling the hyperbolic plane - long version
- 85. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Figure 2 : Hyperbolic man takes a walk to in
- 86. nity Daniel Czegel Tiling the hyperbolic plane - long version
- 87. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Schlafli-symbol Regular tiling: tiling by i) congruent regular poligons (n-gons), ii) at every vertex, m poligons meet aniel Czegel Tiling the hyperbolic plane - long version D
- 88. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? aniel Czegel Tiling the hyperbolic plane - long version D
- 89. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? aniel Czegel Tiling the hyperbolic plane - long version D
- 90. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? Schla i-symbol: fn;mg aniel Czegel Tiling the hyperbolic plane - long version D
- 91. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? Schla i-symbol: fn;mg Any relationship between these three Schla i-symbols? aniel Czegel Tiling the hyperbolic plane - long version D
- 92. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.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? Schla i-symbol: fn;mg Any relationship between these three Schla i-symbols? 1 n + 1 m = 1 2 There is no other such n;m 2 N , no other regular euclidean tiling Daniel Czegel Tiling the hyperbolic plane - long version
- 93. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Regular tilings on the sphere? (assume n;m 3, i.e. nondegenrate cases) aniel Czegel Tiling the hyperbolic plane - long version D
- 94. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Regular tilings on the sphere? (assume n;m 3, i.e. nondegenrate cases) n = 3; m = 3 ? aniel Czegel Tiling the hyperbolic plane - long version D
- 95. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Regular tilings on the sphere? (assume n;m 3, i.e. nondegenrate cases) n = 3; m = 3 ? A blown tetrahedron! Daniel Czegel Tiling the hyperbolic plane - long version
- 96. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Regular tilings on the sphere? (assume n;m 3, i.e. nondegenrate cases) n = 3; m = 3 ? A blown tetrahedron! Daniel Czegel Tiling the hyperbolic plane - long version
- 97. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere f4; 3g ? aniel Czegel Tiling the hyperbolic plane - long version D
- 98. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere f4; 3g ? A cube: aniel Czegel Tiling the hyperbolic plane - long version D
- 99. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere f4; 3g ? A cube: f3; 4g; f3; 5g; f5; 3g ? aniel Czegel Tiling the hyperbolic plane - long version D
- 100. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere f4; 3g ? A cube: f3; 4g; f3; 5g; f5; 3g ? Octahedron, icosahedron, dodecahedron. Daniel Czegel Tiling the hyperbolic plane - long version
- 101. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Any more? aniel Czegel Tiling the hyperbolic plane - long version D
- 102. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Any more? No. Why? aniel Czegel Tiling the hyperbolic plane - long version D
- 103. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Tiling on the sphere Any more? No. Why? 1 n + 1 m 1 2 only for these
- 104. ve! Figure 3 : The
- 105. ve Platonic solids Daniel Czegel Tiling the hyperbolic plane - long version
- 106. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling Hyperbolic tiling? aniel Czegel Tiling the hyperbolic plane - long version D
- 107. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling m n Hyperbolic tiling? If 1+ 1 2 1aniel Czegel Tiling the hyperbolic plane - long version D
- 108. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling m n Hyperbolic tiling? If 1+ 1 2 1How many such tilings? aniel Czegel Tiling the hyperbolic plane - long version D
- 109. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling m n Hyperbolic tiling? If 1+ 1 2 1How many such tilings? In
- 110. nite! aniel Czegel Tiling the hyperbolic plane - long version D
- 111. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling m n Hyperbolic tiling? If 1+ 1 2 1How many such tilings? In
- 112. nite! Figure 4 : f3; 7g Figure 5 : f7; 3g Daniel Czegel Tiling the hyperbolic plane - long version
- 113. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling n or m can even be in
- 114. nite! aniel Czegel Tiling the hyperbolic plane - long version D
- 115. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling n or m can even be in
- 116. nite! Figure 6 : f3;1g, for every triangle: =
- 117. = = 0 ) A = Figure 7 : f1; 3g, aperiogon Daniel Czegel Tiling the hyperbolic plane - long version
- 118. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Hyperbolic tiling Or both! Figure 8 : f1;1g Daniel Czegel Tiling the hyperbolic plane - long version
- 119. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Classification of regular tilings Figure 9 : Classi
- 120. cation of regular tilings Daniel Czegel Tiling the hyperbolic plane - long version
- 121. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Figure 10 : M.C. Escher Figure 11 : H.S.M. Coxeter Daniel Czegel Tiling the hyperbolic plane - long version
- 122. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Figure 12 : Escher's Circle Limit I. (1958) Daniel Czegel Tiling the hyperbolic plane - long version
- 123. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Figure 13 : Circle Limit I.: nonregular tiling of the hyperbolic plane (m = 4 and 6) Daniel Czegel Tiling the hyperbolic plane - long version
- 124. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Any regular tiling? aniel Czegel Tiling the hyperbolic plane - long version D
- 125. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Any regular tiling? Figure 14 : Escher's Circle Limit III. (1959). aniel Czegel Tiling the hyperbolic plane - long version D
- 126. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Any regular tiling? Figure 14 : Escher's Circle Limit III. (1959). Schla i symbol? Daniel Czegel Tiling the hyperbolic plane - long version
- 127. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work Figure 15 : Schla i symbol of Circle Limit III.: f8; 3g! Daniel Czegel Tiling the hyperbolic plane - long version
- 128. Gaussian curvature Elliptic geometry Hyperbolic geometry Tiling M.C. Escher's work References Weeks, J. R. (2001). The shape of space. CRC press. Dirnbock, H., Stachel, H. (1997). The development of the oloid. Journal for Geometry and Graphics, 1(2), 105-118. 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 - long version