Hyperbolic Tilings

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Hyperbolic Tilings
Fields: Geometry and Topology
Image Created By: Jos Leys
Website: Mathematical Imagery

Hyperbolic Tilings

This image is a hyperbolic tiling made from alternating two shapes: heptagons and triangles.


Contents

Basic Description

In addition to tessellations found in the traditional Euclidean geometry, there are also tilings that can be created in non-tradition or non-Euclidean geometry. Hyperbolic tiling is a type of tessellation that is visualized in a non-Euclidean geometry called hyperbolic geometry. This particular tessellation is viewed in the Poincaré disk model, where straight lines are arcs that form right angles at the boundary of the disk.


A hyperbolic tiling
A hyperbolic tiling


The polygons in a hyperbolic tiling grow smaller as they reach the edge of the disk. This is due to the representation of the tiling in the Poincaré disk model that we use to visualize hyperbolic tilings. The Poincaré disk model does not show traditional distances, so that as we get closer and closer to the edge of the disk, the distances increase and the boundary of the disk is in fact infinite.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Hyperbolic Geometry

Schläfli symbol

Hyperbolic Tiling {3,8}

The Schläfli symbol is a notation that can be used to describe the properties of polygons in a tessellation. The symbol is in the form {n , k}, where n refers to the number of edges of a regular polygon and k refers to the number of regular polygons that meet at each vertex.

Thus, if the summation of all the angles at a vertex is 360 degrees:

the angle at each vertex is \frac{360}{k}
and the polygon has a summation of angles \frac{n360}{k}.


Regarding the image shown to the right, the tessellation is made up solely of triangles, so that n = 3. At each vertex in the tessellation, there is always a meeting of six triangles, giving us k = 8. Thus, the Schläfli symbol for this tessellation is {3,8}. Also, the angle at each vertex is 35 degrees, and the summation of angles of the polygon is 135 degrees (which is less than the traditional 180 degrees, a characteristic of hyperbolic tilings).


Types of Tilings

Hyperbolic Tiling {5,4}

Making Quasi-{5,4}

Quasi-{5,4}

Dual {4,5}

Most tessellations are made from regular polygons, which are polygons with edges of equal side length and vertices of equal angle. This allows the tessellation to have the same number of polygons meeting at each vertex.


Quasi-regular Tilings

You will notice that some hyperbolic tilings utilize two polygons instead of one polygon. These types of tilings are called quasi-regular tilings. The two polygons will alternate meeting at each vertex and are usually colored differently for emphasis. To describe a quasi-regular tiling, the Schläfli symbol is still used with an prefix quasi and with n being the number of edges of one polygon and k being the number of edges of one polygon: quasi-{n,k}.


It's interesting to note that every regular tessellation can be made into a quasi-regular tessellation. You simply connect the midpoints of all the edges of the polygons in the tessellation. In the image gallery below, the first tessellation shown is a regular tessellation {5,4}. The second image shows how to connect the midpoints of the edges to create a quasi-{5,4} tessellation like the one shown in the third image of the gallery. The n in the quasi-regular tessellation now refers to the the four-sided squares, while the k now represents the five-sided pentagons alternating within the tessellation.


Dual Tessellations

There is another type of tessellation that occurs when the Schläfli symbols {n,k} are switched to {k,n}. This is called a dual tessellation, and an example can be found in the final image in the gallery: {4,5}. (Note, although the Schläfli symbols for the quasi- and duel tessellations are the same for this particular set of examples, they are very different tessellations).


Hyperbolic Tiling {5,4}

Making Quasi-{5,4}

Quasi-{5,4}

Dual {4,5}



Tilings in Different Geometries

There are various geometries that can be used to draw tilings, including in the Euclidean, hyperbolic, and elliptic planes. It is interesting to note that Euclidean geometry only allows for three regular tilings (using the regular polygons squares, triangles, and hexagons), while non-Euclidean geometry allows for an infinite amount of regular tilings. A simple way to determine which geometry a tiling can be draw in is by using its Schläfli symbol. Every regular tessellation can be completely broken up into triangles, because every regular polygon is made up of n-2 triangles, where n is the number of sides of the polygon.

The Three Regular Euclidean Tessellations
The Three Regular Euclidean Tessellations

In Euclidean tessellations,

the summation of angles in all triangles is 180 degrees, so the summation of angles of a n-gon is (n-2)180\,.
and \frac{n360}{k} is the summation of the angles of the polygon in a tilling
then the following must be true: \frac{1}{n} + \frac{1}{k} = \frac{1}{2}

\frac{n360}{k} = (n-2)180
\frac{n360}{k} = n180 - 360
\left( \frac{n360}{k} = n180 - 360 \right)\frac{1}{n360}
\frac{n360}{k (n360)} = \frac{n180}{(n360)} - \frac{360}{(n360)}
\frac{1}{k} = \frac{1}{2} - \frac{1}{n}
\frac{1}{n} + \frac{1}{k} = \frac{1}{2}


Elliptic Tilings
Elliptic Tilings

In hyperbolic tessellations,

triangles always have a summation of angles less than 180 degrees, so that: \frac{1}{n} + \frac{1}{k} < \frac{1}{2}


In elliptic tessellations (also called spherical tessellations),

triangles always have a summation of angles greater than 180 degrees, so that: \frac{1}{n} + \frac{1}{k} > \frac{1}{2}



How the Main Image Relates

This image is a quasi-regular hyperbolic tiling with Schläfli symbol of quasi-{7,3}, created using heptagons and triangles. This tessellation was created in the hyperbolic plane, so it must be visualized using the Poincaré disk model.

Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

About the Creator of this Image

Jos Leys creates images from mathematics using programs such as Ultrafractal and Povray. He has created numerous tessellations, fractals, and other images.


Related Links

Additional Resources

Jos Ley's hyperbolic tilings gallery at http://www.josleys.com/show_gallery.php?galid=262
Also, Jos Ley's page on M.C. Echer's artistic works http://www.josleys.com/show_gallery.php?galid=325.
Make Your Own Hyperbolic Tilings Applet at http://aleph0.clarku.edu/~djoyce/poincare/PoincareApplet.html

References

David E. Joyce, Hyperbolic Tessellations

Wilhelm Magnus, Non-Euclidean Tessellations and their Groups (www.cims.nyu.edu/vigrenew/ug_research/JohnAdamski05.pdf)

Wolfram MathWorld, Wolfram MathWorld

Wikipedia, Uniform Tilings in Hyperbolic Plane

Future Directions for this Page

  • The Tilings in Different Geometries section should be expanded so that separate section for each geometry should be created and perhaps merged with the Tessellations page.
  • An applet for the allowing users to make their own tilings by specifying the Schläfli symbol (similar to this applet)


If you are able, please consider adding to or editing this page!

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