14 VERTICES YIELD 16383 COMBINATIONS.
SOME COMBINATIONS WORK AND SOME DON’T.

by Paul Hofmann

1. THE STATE OF THE ART

From Wolfram Alpha, slightly condensed:

“A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematic. Some authors define it as an orderly composition of the 3 regular tessellations and the 8 semiregular tessellations (which is not precise enough to draw any conclusions from), while others define it as a tessellation having more than one transivity class of vertices (which leads to an infinite number of possible tilings).

“The number of demiregular tessellations is commonly given as 14 (Critchlow, Ghyka, Williams, Steinhaus). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by ‘demiregular tessellation.’

“A more precise term of demiregular tessellation is 2-uniform tessellation (Grunbaum and Shepard). There are 20 such tessellations (Krotenheerdt)”

IN OTHER WORDS, IT’S A MESS !

2. THE BIG LIST

Leaving out the octagon (which will tile only with the square), there are 14 ways that the 4 useable regular polygons (triangle, square, hexagon, and dodecagon) will fit around a vertex. In dual tilings, these 14 vertices become 14 distinct dual polygons. For every tiling of regular polygons there is a tiling of dual polygons and vice versa. They are each other’s skeletons.

Let’s say that we label each of these 14 dual polygons (or, if you prefer, each of these vertices of regular polygons—it amounts to the same thing) with a letter of the alphabet, A through N. How many ways can these 14 letters be combined? The answer is: 16383 combinations in 14 Groups of from 1 to 14 letters each. This is the Big List.

What would happen if we tested every one of these 16383 combinations in order to find out which combinations will tile the plane, without gaps or overlaps, edge-to-edge and corner-to-corner, in at least one way?

AS FAR AS I KNOW, THIS QUESTION HAS NEVER BEFORE BEEN ASKED, LET ALONE ANSWERED.

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The 14 Groups of the 16383 candidate-combinations break down like this: 14, 91, 364, 1001, 2002, 3003, 3492, 3003, 2002, 1001, 364, 91, 14, and 1.

Successful combinations in the 14 Groups are as follows: Groups 1, 2, 3, and 4 yield 10, 16, 44, and 82 successes respectively. Groups 13 and 14 yield 12 successes and 1 success respectively. Groups 5, 6, 7, 8, 9, 10, 11, and 12 each yield many successes, but I do not expect to check anywhere near all 14958 candidates in these 8 middle groups in my lifetime.

All possible tilings (an infinite number) of the 4 regular polygons and the 14 dual polygons have a place in the Big List system. This is the global rather than the myopic view of tilings of regular polygons. It seems to me that the emphasis on regular, semi-regular, and “demi-regular” (whatever that means) tilings in the literature, along with the grotesque terminology, tangled prose, and awkward systems of notation, has discouraged the exploration of a potential cornucopia of beautiful and interesting patterns.

3. WORKING WITH THE BIG LIST

The Big List may be written out by hand on graph paper. Two columns of 50 combinations each will just about fill a page. If you start a new page for each of the 14 Groups, you will need 173 pages in all. Putting the 14 letters across the top of each of the two columns and then placing each letter of each combination in its proper stack will make it easier to read the combinations.

If you can’t find appropriate graph paper, it’s easy enough to draw your own with a fine-point pen on printer paper and then print up as many as you need.

Of course, if you know how to write code, you could probably generate the list that way. I don’t, so I wrote out the whole list myself, 114,688 letters in all. Twice, to get it right. (See sample pages below.)

Note that usually a capital letter without brackets, such as A, refers to a particular dual polygon, while a capital letter (or group of capital letters) with brackets, such as [A], [AB], or [ABC], refers to a candidate-combination, successful or otherwise. On graph paper, however, the brackets are not necessary.

Once you have the Big List you will be ready to try out all 16323 candidate-combinations to see which ones will tile the plane. This will take you a while.

Or you might want to just try a few combinations at random from each of the 14 Groups.

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4. TESTING THE COMBINATIONS

The Vertex Chart shows the 14 possible arrangements of the 4 regular polygons around a vertex and also shows the 14 dual polygons that these 14 arrangements generate.

There are 3 ways that you can test a combination:

  1. By drawing. Sketching the regular polygons is easy because all the sides are the same length. You just have to get used to the angles. Then draw in the dual polygons with a different color line.

  2. By manipulating the tiles. Each shape of tile should get its own color. You can get various colors of thin cardboard at art supply stores. Availability of colors will vary from store to store and time to time. Tiles of regular polygons will require 4 colors and tiles of dual polygons will require 14 more colors. Finding the lengths of the sides of the dual polygons will require a little figuring. But for testing combinations by manipulating tiles, I find it easier to work with the dual polygons than with the regular polygons. Which ever you choose, don’t forget to draw in the complementary lines (dual lines or regular-polygon lines, as the case may be) before cutting out the tiles.

  3. By computer program. There are of course various drawing programs available. That’s fine if you already know what tiling you want to draw. But to move tiles around, a dedicated drag-and-dock computer program would be very handy.

There are 2 shortcuts you can use to eliminate candidates just by looking at the letters. These shortcuts seem to eliminate about half of the candidates. But the survivors will still have to be tested. The shortcuts are to eliminate candidates, not to confirm them.

SHORTCUT #1

The dual polygon pair AM will only tile the plane in a combination having a total of 4 or more distinct dual polygons. And so on, as below:

  • Pairs which will only work in combinations of 4 or more: AM, AN, BG, BN, CE, CF, DN, EJ, EL, FJ, FM, IL, and LM.

  • Pairs requiring combinations of 5 or more: AH, BH, CH, CK, CL, CM, EK, EM, EN, GL, HI, HJ, IK, IN, and MN.

  • Pairs requiring combinations of 6 or more: CN, FK, GK, GN, and HM.

  • Pairs requiring combinations of 7 or more: FN, HK, and HL

  • Pair requiring combinations of 8 or more: HN


SHORTCUT #2

H will not tile the plane without F, and so on, as below:

  • H requires F.

  • C requires A or I.

  • N requires K or L.

  • G requires E, F, or I.

  • M requires I, J, or K.

  • D requires A, B, E, F, or K.

  • L requires A, B, J, K, or N.

  • J requires A, B, I, K, L, or M.

  • E requires A, B, D, F, G, I, or K.

  • F requires A, B, D, E, G, H, I, or K.

  • B requires A, D, E, F, I, J, K, or L.

  • A requires B, C, D, E, F, I, J, K, or L.

  • K requires A, B, D, E, F, I, J, L, M, or N.

  • I requires A, B, C, E, F, G, J, K, L, or M.

5. ORGANIZING THE SUCCESSFUL COMBINATIONS

Successful combinations are identified by numbers. For instance, [A] is No. 1, [DE] is No. 17, [FGH] is No. 63, and so on. Each successful combination gets only one identification number whether it is singular or infinitely variable. Of course this means that a successful combination in the latter part of the list cannot be assigned a number until all combinations have been tested up to that point.

The 16383 candidate-combinations can be organized not only by Group but by Class and by Family.

There are 15 possible sets of the 4 regular polygons but only 10 of these sets yield tilings. These are the 10 Classes. The 14 Groups and the 10 Classes can be combined in a Big Grid of 140 Slots. Only 43 of the slots will contain successful tilings and these comprise the 43 Families of tilings of regular polygons. Each Family uses a different combination of regular polygons and dual polygons.

LIST OF CLASSES

  • [3] Class 1

  • [4] Class 2

  • [6] Class 3

  • [12] -will not tile-

  • [3-4] Class 4

  • [3-6] Class 5

  • [3-12] Class 6

  • [4-6] -will not tile-

  • [4-12] -will not tile-

  • [6-12] -will not tile-

  • [3-4-6] Class 7

  • [3-4-12] Class 8

  • [3-6-12] -will not tile-

  • [4-6-12] Class 9

  • [3-4-6-12] Class 10

6. SPECIAL TILINGS

At least 31 successful combinations are singular. That is, they will tile in only one way. Most likely all other combinations will tile in an infinite number of ways. I have never found a combination that will tile only in a finite number of ways above 1.

At least 28 combinations will tile only periodically, while at least 7 combinations will tile only aperiodically. These 35 tilings occur in the first few Groups and I doubt that any more will be found. Probably all other successful combinations will tile both periodically and aperiodically.

At least 3 combinations will tile only chirally. That is, in both right- and left-handed versions.

The known (by me, at this time) special tilings are as follows:

  • Singular: [A], [B], [C], [D], [E], [G], [H], [J], [M], [N], [AJ], [DF], [DK], [IJ], [ABK], [ABL], [BDI], [BDK], [BEI], [BFI], [BIJ], [BKL], [ABIM], [AFIJ], [AFIM], [BCFI], [BDFI], [BDKL], [BEFI], [BEIL], and [BFIM].

  • Periodic only: [A], [B], [C], [D], [E], [G], [H], [J], [M], [N], [AJ], [DF], [DK], [UJ], [BDI], [BDK], [BEI], [BFI], [BIJ], [BKL], [AFIJ], [AFIM], [BCFI], [BDFI], [BDKL], [BEFI], [BEIL], and [BFIM].

  • Aperiodic only: [ABI], [ABK], [ABL], [ABCI], [ABGI], [ABIM], and [ABKL].

  • Chiral only: [E], [ABL], and [BKL].

  • Singular, periodic only, and chiral only: [E] and [BKL].

  • Singular, aperiodic only, and chiral only; the rarest of the rare: [ABL].

7. A FEW EXAMPLES OF NOTABLE TILINGS

[ABL]. Singular, aperiodic only, chiral only. The only tiling of regular polygons that falls into all 3 of these categories. (Below)

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[CIJ]. Most combinations will tile in an infinite number of ways. Variations are usually achieved by such simple operations as rotating, shifting, repeating, or alternating. [CIJ] is unusual in that it has variations which are structurally distinct. (Below)

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[ABCJ]. An example of a combination that will tile both periodically and aperiodically. (Below)

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[ABDE]. Another example of a combination that will tile both periodically and aperiodically. (Below)

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[BKL]. Singular, periodic only, chiral only. (Below)

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[ABCDL] Twelve infinitely long expanding spokes, one infinitely long uniform-width spoke, and an infinitely expanding whirlpool-like spiral. (Below)

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[BDJKM]. There are 12 ways that sets of infinitely long parallel lines can be configured in dual tilings, considering both the number of different sets and their angles relative to each other. [BDJKM] can be rearranged to show 10 of those ways. This particular version shows 4 sets at 30, 30, 30, and 90 degrees. (Below)

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[ABCDEFGHIJKLMN]. This sole member of Group 14 will tile in an infinite number of ways. Here’s one of them. (Below)

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And here are some random successful combinations, one from each of the 14 Groups: [E], [AB], [JKN], [DEFH], [ABDKN], [BGIJKLN], [DEFGHIJ], [ABEFHIKN], [ABCEFGHIJ], [ABCDEFGHIJ], [ABCDEFGHIJN], [ABCDEFGHIJMN], [ACDEFGHIJKLMN], and [ABCDEFGHIJKLMN].

The possibilities are endless. Better to explore the big picture and see what’s there than to decide in advance what to look for.

8. MISCELLANEOUS NOTES

1. Leaving the regular octagon in the System would give us 16384 additional candidate-combinations, only one of which would work. And we already know about that one.

2. Counting the 3 chiral dual polygons (I, K, and M) as 6 distinct polygons would increase the Big List by 114,688 candidates without adding a single success, since those 3 dual polygons always occur in right- and left-handed pairs in tilings of regular polygons.

3. One standard way of labeling a particular combination is with a string of numbers, superscripts, and punctuation marks. For instance, using this method, [BDJKM] would be notated as [3(2).4.3.4;3.4.6.4;3(2).4.12;4.6.12]. (The two numerals in parentheses represent superscripts—no thanks to Squarespace!). Imagine this kind of notation being used for combinations in Groups 12, 13, or 14. Worse yet, imagine the entire Big List of 16383 combinations notated in this way!

4. The literature on tiling states that a combination of polygons, such as [ABK] (for example), can be said to force aperiodicity only if the subsets [AB], [AK], [BK], [A], [B], and [K] also each force aperiodicity. With that requirement, any subset can be a spoiler. I don’t buy it. When I say that the combination [ABK] forces aperiodicity, that’s exactly what I mean. The subsets each have their own properties as candidate-combinations and must be considered separately.

5. We have seen that there are 14 ways that the 4 regular polygons will fit around a vertex and that these 14 types of vertex yield 16383 testable combinations. So how many ways will the 14 dual polygons fit around a vertex? The answer is 648. And how many testable combinations will those 648 vertexes yield? The math is beyond me, but the answer must be truly astronomical.

6. Each of the 14 dual polygons may be found embedded in a dodecagram (12-pointed star) of configuration 1-6-11-4-9-2-7-12-5-10-3-8-1. A circle may be drawn in the center of the dodecagram in such a way that it is tangent to each side of each of the 14 dual polygons. The diameter of this circle is the same as the lengths of each side of each of the 4 regular polygons.

7. Historically, there seems to have been some puzzlement as to why no other regular polygons are compatible in tilings of regular polygons. The explanation of course is that all the angles of all the regular polygons (and all the dual polygons) in this system must be divisible by 30. This requirement also explains why the regular octagon is a dead-end and doesn’t belong in the system despite the fact that it will tile with the square.

8. Convex polygons having 5, 8, 9, and 10 sides, all sides of equal length and all angles divisible by 30, do exist, but, because in each case their angles are not equal, they are not regular polygons. They will, however, tile with the regular polygons.

9. For combinations that will tile in only one way, I use the word “singular” rather than the word “unique” because the latter word has been rendered meaningless from misuse.

10. I was a B student in high school algebra and geometry. There are no mathematical or geometrical breakthroughs here. What I have done is to devise a system of organization that includes all possibilities—something that should have been done a long time ago. After all, it has been well over 2000 years since Archimedes gave us the regular and semi-regular tilings.

11. My thanks to Robert Williams, whose book The Geometry of Natural Structures introduced me to tilings of regular polygons. This book, first published in 1970, is still in print and is available from Eudaemon Press in San Francisco.

12. Questions, comments, additions, suggestions, or corrections? Email me at: paulhofmann@earthlink.net. (Please note: one F and two NNs.)

13. See also: www.phsculpture.com