On this page I will show some of my research into tiling patterns.
Executable jar file (1610 KB)
Source files (947 KB)
My G4G10 exchange gift paper (2661 KB)
My G4G11 exchange gift paper (1034 KB)
Send any comments or suggestions to tilings a t jaapsch d o t net.
I have written a Java applet that allows you to see lots of different tilings. If your browser still allows java applets to run, you can click the link below to launch the applet in a new window. It is about 1.5 MB in size, so may take a moment to load.
Launch the Tiling Viewer Applet
Unfortunately most modern browsers no longer run applets. In that case you should install the latest version of java, and then download and run the executable jar file from the link on the right (usually double-clicking the saved jar file will start it running).
The left section of the screen shows a tree structure. Expand any categories in the tree, until you can click on
the tiling you want to see.
Below the tree is a series of check boxes where you can filter out the tilings based on the symmetries they have.
The main centre panel shows the tiling.
The centre bottom left panel shows a single tile, with edges and corners labelled.
The centre bottom right panel shows the relations that the tile angles and edge lengths must satisfy to be able to form that tiling.
On the right is the control panel where you can change how the chosen tiling is shown.
Show Info: determines whether to show the information panels at the bottom.
Fill plane: determines whether to show the whole tiling or just one primitive cell.
Show unit parallelogram: determines whether to transparently show the unit parallelogram over the tiling.
Colourfill tiles: determines whether to colour the tiles or leave them white.
Copy: (Executable jar only) Places an image copy of the tiling in the copy/paste clipboard.
Various sliders: Change the size/shape of the tile. The effect depends on the tiling.
It is well known that any triangle can tile the plane, as can any quadrangle, convex or not.
The situation with pentagons is much more complicated. A regular pentagon does not tile the plane, but various non-regular convex pentagons do. In 1975 Martin Gardner wrote an article in Scientific American, reporting the results of Richard B. Kershner about which types of convex pentagons can tile the plane [MG]. Richard had attempted to enumerate them, and thought that his list of 8 types was complete until in a reaction to Gardner's article Richard E. James III wrote in with another tilable convex pentagon (now known as type 10). Soon after, Marjorie Rice [MR] found four others (types 9, 11-13), bringing the total to 13. One further type was found by Rolf Stein in 1985 (type 14), and lastly, a 15th type was found in 2015 by Casey Mann, Jennifer McLoud-Mann and David Von Derau. In May 2017, Michaël Rao announced a proof that the list of 15 types is complete. The proof relies on a computer program to eliminate all cases, but it has now been verified by Thomas Hales so it is likely to be correct.
This uncertainty around pentagons inspired me to do my own research into tilings. In 2010 I wrote a program that generates all possible tilings that use k copies of a single polygon tile as its fundamental unit. With this program I tried to find new types of tilable convex pentagons, but that quest was not successful. It has nevertheless generated many fascinating tilings, some that I think have not been seen before, most of which I have incorporated into the applet.
Isohedral tilings with convex polygons are well-studied and can be found in [G+S] and [TM]. There are:
14 isohedral tilings with a triangle tile
56 isohedral tilings with a quadrangle tile
24 isohedral tilings with a convex pentagon tile
13 isohedral tilings with a convex hexagon tile
Convex n-gons with n>6 cannot tile the plane. This is because for n>6, the average of the tile's angles is more than 120 degrees, but if at every vertex in the tiling at least three or more tiles meet then the average angle must be 120 degrees or less. Therefore there are vertices in the tiling where only two tiles meet, and either one of the angles is more than 180, or they are both equal to 180. So either the tile is not convex, or it has redundant vertices. This is also fairly easy to prove by using Euler's characteristic.
All the isohedral tilings with a convex tile can be found in the applet. Some of them are special cases of others, where the settings are such that the tiling has extra symmetries, and those have not all been implemented separately.
There are no isohedral tilings that are specific to non-convex quadrangles, though many isohedral tilings with convex quadrangles can be used with non-convex ones too. There are however isohedral tilings with pentagons where the tiling forces the pentagon to be non-convex. The same is true for hexagons, but I have not (yet) implemented all the isohedral non-convex hexagon tilings in the applet.
Most 2-isohedral tilings have a fundamental unit consisting of two tiles, one from each isohedrality class. In the applet the tiling will then have an equal amount of the two colours. Some examples are shown here:
My search program has exhaustively generated all of the tilings for which the fundamental unit consists of two identical convex polygons, and assuming no mistakes or oversights, the applet contains a complete set of these types of tiling. No new types of tilable pentagon were found. Most of the tilings used type 1 pentagons, seven of type 2, one of type 4, and there is one alternative way to tile the type 6 pentagon tile.
I have also tried generating tilings with non-convex polygons. There were no new tilings with non-convex quadrangles, as they were all variations of tilings with convex tiles already found. There were many tilings with non-convex pentagons, all of which have now been included in the applet.
My program has generated all edge-to-edge tilings non-convex hexagons, and I have implemented many but not yet all of them in the applet.
There are also 2-isohedral tilings where the two types of tile occur in a 2:1 ratio. In these cases the fundamental unit consists of 3 tiles, and has an extra symmetry that maps two of the tiles to each other. These will be found by my program when it searches for 3-isohedral tilings, possibly as a special case of a more general 3-isohedral tiling.
I have not yet searched for all 3-isohedral tilings, as that will take a lot of time. I have therefore limited myself to the edge-to-edge 3-isohedral tilings for triangles, quadrangles and pentagons, all of which have been implemented in the applet. Nevertheless, some tilings that are not edge-to-edge have been implemented too.
Searching for k-isohedral tilings with k>3 is probably infeasible with my current search program. I did start it searching for edge-to-edge 4-isohedral tilings, and it found the following neat one using a type 8 pentagon tile.
Here are the two articles I wrote as exchange gifts for the Gathering for Gardner conference.
My G4G10 exchange gift paper (2661 KB): Similar to the contents of this page.
My G4G11 exchange gift paper (1034 KB): Some geometry problems arising from tilings.
Here are some interesting links about these kinds of tilings:
Version 1.0: First published online, March 2012, in time for Gathering for Gardner 10.
Version 1.1: 2012/07/22. Added filtering based on symmetry groups.
Version 1.2: 2012/11/04. Added all 2-isohedral non-convex pentagon tilings, and many non-convex hexagon tilings.
Version 1.3: 2013/07/01. Added many 3-isohedral tilings with triangles, and with pentagons.
Version 1.4: 2015/07/30. Added the 15th convex pentagon type (Mann, McLoud, Von Derau)
Version 1.42: 2015/09/29. Added info panel with current angles and side lengths.
Version 1.43: 2015/10/11. Fixed some tilings that showed incorrect angles in the info panel.
Version 1.44: 2015/12/13. Centred unit parallellogram, added missing equation to NC5-14 info.
Version 1.45: 2016/01/13. Fixed N5-8b info, and rearranged pentagon tiling menu to sort by tiling type.
Version 1.46: 2016/09/04. Added two 4-isohedral convex pentagon tilings by Dave Smith.
Version 1.47: 2016/09/05. Added two 6-isohedral convex pentagon tilings by Dave Smith.
Version 1.49: 2017/02/23. Improved Unit Parallelogram - reduce to most rectangular version, and display its stats in info window.
Version 1.491: 2017/07/06. Display more decimals in the info window.
Version 1.495: 2017/07/25. Added the 4-isohedral tiling (N5-1bf2) found in this math.stackexchange post.
Version 1.5: 2017/09/09. Instead of Copy to clipboard (did not work on all platforms), it now saves the tiling image to file in BMP, GIF, PNG, or JPG format.
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