# Hyperbolic tiling in a ring

 Slides from the talk "Artistic Models of Hyperbolic Geometry" by Vladimir Bulatov Conformal Stretching of the Poincare Disk Periodicity of the Band model Bending the disk into a ring Infinite series of rings Examples presented on March 25, 2010 at the Gathering for Gardner 9, Atlanta, GA.

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# Poincare Disk Model

 Poincare disk model of hyperbolic plane. Geodesics in this model are segments of circles orthogonal to the unit circle (circle at infinity). The model is conformal - the angles between intersecting geodesics are equal to the euclidean angles between the tangent lines of the circle. This tiling with right angled regular pentagons is hyperbolic analog of simple square tiling of the euclidean plane.

# Band Model of Hyperboic Plane

 Band model of the hyperbolic plane is obtained by applying to the disk model a special conformal map of the complex plane: w =(4/\pi)tanh\^-1(z); z, w \in CC, |Im(w)| < 1, |z| < 1 The band model is conformal, because it is obtained from disk model using conformal transformation. [animation]

# Periodicity of the tiling

 The same hyperbolic tiling in the band model may have different euclidean periods. The period depends on the particlular location and orientation of the tiling in the disk model before stretching the disk into the band.

# Bending Band into Ring

 Ring can be obtained from the infinite band using conformal mapping w = exp(z/a), z \in CC. In order to make the tiles match around the ring, the real parameter a \in R should be divisible by the euclidean period of the tiling in the band model. There is infinite series of different value such parameters, which makes infinite series of different rings from the same hyperbolic tiling.

# Series of Rings

 Ring with 8 periods.

# Series of Rings

 Ring with 6 periods.

# Series of Rings

 Ring with 4 periods.

# Series of Rings

 Ring with 3 periods.

# Series of Rings

 Ring with 2 periods.

# Series of Rings

 Smallest ring with only 1 period. The hole in the center is really tiny.

# Another series of rings

 Another infinite series of rings can be generated by different orientation of tiling in the hyperbolic plane. This is a ring with 15 periods.

# Another series of rings

 Ring with 10 periods.

# Another series of rings

 Ring with 6 periods.

# Another series of rings

 Ring with 5 periods.

# Another series of rings

 Ring with 4 periods.

# Another series of rings

 Ring with 3 periods.

# Another series of rings

 Ring with 2 periods.

# Another series of rings

 Ring with 1 period.

# Examples

Tiling with 4*3 symmetry. 2 periods.

# Colored Ring

Tiling with 4*3 symmetry. 3 periods.

# Colored Ring

Tiling with 4*3 symmetry. 10 periods.

# Colored Ring

Tiling with 4*3 symmetry. 3 periods.

# Colored Ring

Tiling with 4*3 symmetry. 10 periods.

# Colored Ring

Tiling with *832 symmetry. 4 periods.

# Colored Ring

 Ring with *24 3 2 symmetry tiling and one period along the ring. Colored symmetry with 3 colors.

# Colored Ring

 Ring with *642 symmetry tiling and 2 periods along the ring. Colored symmetry with 2 colors.

# Colored Ring

 Ring with *642 symmetry tiling and 4 periods along the ring. Colored symmetry with 2 colors.

# Colored Ring

 Ring with *642 symmetry tiling and 6 periods along the ring. Colored symmetry with 2 colors.

# Colored Ring

 Ring with *642 symmetry tiling. 2 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 3 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 4 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 2 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 3 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 4 periods along the ring.

# Colored Ring

 Ring with *642 symmetry tiling. 5 periods along the ring.

# Colored Ring

 Ring with *642 symmetry colored tiling. 3 periods along the ring. 3 colors colored symmetry.

# Colored Ring

 Ring with *642 symmetry colored tiling. 6 periods along the ring. 3 colors colored symmetry.

# Colored Ring

 Ring with *642 symmetry colored tiling. 9 periods along the ring. 3 colors colored symmetry.

# Conclusion

• A conformal mapping of the hyperbolic tiling of Poincare disk into a ring generates an infinite family of different visual representations of the tiling.
• One parameter of the family is orientation and location of the original tiling in the disk.
• Each particular orientaion yelds infinite series of rings with periods divisible by euclidean period of the tiling in the band model. The integer multipier of the period is the second parameter of the family.