Hyperbolic Tilings
From Math Images
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Hyperbolic Tilings
- This image is a hyperbolic tiling made from alternating two shapes: heptagons and triangles.
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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.
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
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
- and the polygon has a summation of angles
.
Types of Tilings
 Hyperbolic Tiling {5,4} |
 Making Quasi-{5,4} |
 Quasi-{5,4} |
 Dual {4,5} |
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 summation of angles in all triangles is 180 degrees, so the summation of angles of a n-gon is
.
- and is the summation of the angles of the polygon in a tilling
- then the following must be true:
In hyperbolic tessellations,
- triangles always have a summation of angles less than 180 degrees, so that:
In elliptic tessellations (also called spherical tessellations),
- triangles always have a summation of angles greater than 180 degrees, so that:
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)
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