An interesting problem in combinatorial geometry involves “self-surrounding tiles.”  Such a tile is a shape that can be completely surrounded by congruent copies of itself, but which cannot tile the plane.  Branko Gruenbaum and G. C. Shephard describe them in “Some Problems on Plane Tilings” in Mathematical Recreations:  A Collection in Honor of Martin Gardner, formerly titled The Mathematical Gardner.  They give an example of such a tile discovered by H. Heesch, shown above.  The tile is a square joined to an equilateral triangle and a 30-60-90 triangle, as shown, and you can surround it with copies of itself, so that each point on the central tile’s perimeter is a positive distance ( > 0 ) from any point exterior to the area covered by the tiles.  The tiles that so surround the central tile can be called the ring.  But you will find no way of using copies of this tile to cover the entire plane without gaps or overlaps.

An interesting unsolved question which Gruenbaum and Shephard mention is:  Can a tile surround itself twice, and still be unable to tile the plane?  “Surround itself twice” means that, after a ring of tiles has surrounded it once, the ring is itself surrounded in the same way by a second ring, so that every point of the ring is a positive distance from any exterior point.  If you can find such a shape, or prove its existence impossible, you will make a name for yourself; no one has ever found such a tile, but for all anyone knows, a tile might surround itself any finite number of times, and still be unable to tile the plane.

After writing the above paragraph I learned from Ed Pegg Jr. (whose mathpuzzle.com site I recommend) that such a tile has indeed been found, by one Robert Ammann.  This shape surrounds itself as many as three times!  Take a hexagon and put an outward-dimple on three edges and a corresponding inward-dimple on two edges.  This clearly cannot tile beyond three layers, since four layers would require a 61-cell hexagonal array; each of the 61 hexagons has a “trade imbalance” of one outward-dimple, which would require 61 outward-dimples on the perimeter of the array.  But the perimeter of such an array is only 54 edges.  Finding an arrangement that fills an array three layers deep is not difficult, however.  But there are still many open questions in this field; for example, how much further a self-surrounding tile can go -- four layers? five layers? a hundred layers? -- and still be unable to surround itself infinitely many layers, i.e. tile the plane?  Is three layers a maximum?  Also, is Heesch’s single layer the limit for a convex tile?  Also, is there a self-surrounding tile that can surround itself to a depth of two layers and no more?  (Below I give a link to a website where I later learned that tiles that surround twice have been discovered.)

But tiles that surround themselves once seem to be not too hard to find, at least if you don’t require convexity.  I’ve found several, though none of mine is convex like Heesch’s.  I’ll describe mine here.  Alas, it seems to be impossible to fit a second ring around any of them!  But some of them can make some interesting patterns.

I was delighted recently to hear from Peter Raedschelders, a Belgian engineer and amateur mathematician who came across this site.  Peter is also interested in self-surrounding tiles (which he appropriately calls Heesch tiles), and also has a webpage devoted to his discoveries.  If you are interested in the present page, you would also like Raedschelder’s.  He has also discovered an infinite family of Heesch tiles based on odd-order regular polygons; like mine, his are concave, but his are bilaterally symmetrical and resemble heart-shapes.  What other infinite families of Heesch tiles may be lurking in the odd-order regular polygons, I wonder?  Also, I was interested to find that Raedschelders and I discovered the Comma independently!

I’m honored that many distinguished Heesch experts seem to be visiting this page.  Prof. David Eppstein of UC/Irvine wrote with corrections and new information, and directed me to his excellent website on Heesch tiles.  He notes that shapes are indeed known that surround themselves exactly twice, so that the new frontiers in this research would be a shape that surrounds itself a finite number of times, more than three.  Glenn C. Rhoads of Rutgers University writes that there are an infinite number of polyominoes that surround themselves exactly twice, and tells me that he recently discovered the smallest polyomino with this property!  I wonder how many squares it contains?

Casey Mann, a Ph.D. candidate at the University of Arkansas at Fayetteville, happened across this page and wrote me that in 1999 W. R. Marshall discovered a shape that would surround itself exactly four times, and this year Mann himself found a shape that surrounds itself five times and no more!  Mann’s illustration, which he kindly gave me permission to publish here, is shown below.  The construction is based on Ammann’s dimpled hexagon, with several of them joined in a line.  When I first saw the figures it seemed possible to me that the process could be extended indefinitely, perhaps allowing the construction of figures that might surround themselves any finite number of times, but Mann writes that he has proved that “long, skinny” polyhexes won’t continue to give big Heesch numbers; more compact polyhexes have a better shot.

Casey Mann’s tile, surrounding itself five times

Questions, corrections, comments:  Send me e-mail at  markthom@flash.net