Tessellation Symmetry:
The Heesch System

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Professor Heinrich Heesch (June 25, 1906 – July 26, 1995) was a German mathematician. His special interest was tessellations of tiles that are all the same size. He developed a system of describing tessellations. In 1932 as a graduate student, Heesch proved that there are 28 ways to tile the plane in a regular manner with asymmetric tiles, He did it by listing all the possible types. That is why it is named the Heesch type..In 1963, Professors Heesch and Kienzele published a complete classification of 28 types of asymmetric tiles which tessellate (i.e. which are the one tile in a monohedral tessellation of the plane), and named the classification sustem after Heesch.

Why did they name it after Professor Heesch, and not Keinzele? Probably because nobody can spell Keinzele twice the same way, and because people so often name their discoveries after themselves. Except when they don't. ..Like when hotdogs were named after a kind of meat you don't want in your hotdogs, or when turkeys and American Indians are named after a country they're not in by stupid european explorers. ...Or "arabic" numbers, which actually came from India (like tech support for your @(*#&$ broken computer), not Arabia. True stories. ...But irrelevant. Fun anyway. So, the Professors named this tessellation description system after Heesch, rather than calling it "the needlessly obscure, complicated system", a name that was already taken by the I.R.S..

The Heesch system is simple:

  1. figure out how many sides a tile has. (The "tile" is the basic shape that repeats and fits together in a tessellation...like simple jigsaw puzzle pieces.)

  2. Moving in a clockwise direction, for each side, figure out which kind of symmetry it has:

  3. Realize that it's pretty dumb to abbreviate "rotation" as "C". It begins with an R, not a C, dunnit?! Anyway, it makes sense if you think of C-is-for-Circle, and C actually looks a little like a circle. I'm also told that rotation is sometimes called "cyclic symmetry", so there's another reason why it's called "C".

  4. If a side uses a 180° rotation, just call label it "C". If the side is rotated only 120°, call it C3 because 360° divided by 120° = 3. Likewise, if the side is rotated only 90° then call it C4 because 360° divided by 90° = 4. If the side is rotated by 60° then call it C6 because 360° divided by 60° = 6.

  5. Decide which side is the simplest. (T is simpler than R, R is simpler than G, G is simpler than C, C3 is simpler than C4, and so on.) Start with that side when you write down the letter-name (T, R, G, C, C3) for each side. Many tessellation experts don't think this step is necessary, but it does at least make comparisons easier. Otherwise, it takes a little more thought to realize that TCTGG is am equivalent description of the same tessellation as CTGGT, TGGTC, TGGTC, GGTCT, and GTCTG. (I just moved each side's description from the beginning of the pack to the end of each pack to get an equivalent Heesch description, but it's not immediately obvious, is it?)

  6. Write down your Heesch description. It should look something like this: TCTC or TGTG or TCTGG or C3C3C3C6C6.

Strangely, if we limit ourselves to just tessellations in which the tiles don't change size, built on a flat 2 dimensional plane, and in which there's only one tile shape, then there are only 28 possibilities that make tessellations. Here they are:

For triangular tiles, 5 possibilities:
  1. CCC example
  2. CC3C3 example
  3. CC4C4,
  4. CC6C6, and
  5. CGG
For quadrilateral tiles, 11 possibilities:
  1. TTTT example
  2. CCCC,
  3. TCTC,
  4. C3C3C3C3,
  5. C4C4C4C4,
  6. C3C3C6C6,
  7. CCGG,
  8. TGTG,
  9. CGCG, example
  10. G1G1G2G2, and
  11. G1G2G1G2 example
For pentagonal tiles, 5 possibilities:
  1. TCTCC,
  2. TCTGG example
  3. CC3C3C6C6 example
  4. CC4C4C4C4 example
  5. CG1G2G1G2
For hexagonal tiles, 7 possibilities:
  1. TTTTTT example
  2. TCCTCC example
  3. TG1G1TG2G2,
  4. TG1G2TG2G1 example
  5. TCCTGG,
  6. C3C3C3C3C3C3, and
  7. CG1CG2G1G2
a shape with 3 sides and 3 corners. The word triangular is made from the words tri (meaning "3") and angular (meaning "of angles").
a shape with 4 sides and 4 corners. Sometimes, they are called quadrangles or tetragons. The word quadrilateral is made of the words quad (meaning "4") and lateral (meaning "of sides").
a shape with 5 sides and 5 corners. The word pentagon is made of the words pent (meaning "5") and agon (meaning "angles" or "angular").
a shape with 6 sides and 6 corners. The word hexagon is made of the words hex (meaning "6") and and agon (meaning "angles" or "angular").

Have a look at these tessellations, each one with my guess at its Heesch type under it. Try to figure out the Heesch type yourself. To reveal my guess, just pass your mouse over the black rectangles.

2 goldfish tessellation: Heesch CCC
3 goldfish tessellation: Heesch CC3C3
forcepsfish tessellation: Heesch TGTG TGTG
Escher Horsemen tessellation: Heesch TGTG TGTG
Sheashell tessellation: Heesch C3C3C6C6 C3C3C6C6
Dinosaur tessellation: Heesch TCCTCC TCCTCC
shark tessellation: Heesch TCCTCC TCCTCC