November 25

The NonEuclid Simulation with Hyperbolic Tessellations

Beautiful tessellations are being created using hyperbolic geometry. One such artist is Yoshiaki Araki's site, a Japanese artist, who is working with hyperbolic tessellations in a new way (http://www.sfc.keio.ac.jp/~aly/) or where you can click to view a number of his tessellations (http://www.novel.mag.keio.ac.jp /~aly/TESSELLATION/98/)   The theory of hyperbolic tessellations states: that a hyperbolic tessellation {p,q} exists for every p,q such that (p-2)*(q-2) > 4. Such tessellation's are represented by a Schlafli symbol of the form {p,q}, which means that q regular p-gons surround each vertex.

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Example of a non-regular Hyperbolic Tessellation

 

The image on the right is the start of a non-regular hyperbolic tessellation of four quadrilaterals. The original quadrilaterals were centered around the point A and then reflected and reflected and reflected.

Required Exercises:

Activities: Adjacent Angles, Angles, Parallel Lines, General Triangles,
Isosceles Triangles, Equilateral Triangle, Right Triangles, Congruent Triangles,
Rectangles & Squares, Parallelograms, Rhombus, Polygons, Circles, Tessellations of the Plane.

Webliography on Hyperbolic Tessellations

Applet to draw hyperbolic tessellation using a triangle
(http://www.math.utah.edu/~deraux/tessel/)

Hyperbolic and Spherical Tiling Gallery
(http://bork.hampshire.edu/~bernie/hyper/)

Hyperbolic Tessellations
(http://aleph0.clarku.edu/~djoyce/poincare/poincare.html)

and more from this same site
(http://aleph0.clarku.edu/~djoyce/poincare/other_tess.html)

Many, many Hyperbolic Planar Tessellations by Don Hatch
(http://www.hadron.org/~hatch/HyperbolicTesselations/HyperbolicTesselations.html)

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