Irregular Hexagon Tessellations

A tessellation is the tiling of a plane so that there are no gaps or overlaps. Usually only one geometric shape is used, like a triangle, square, or hexagon. Of course the easiest ways to do the tessellation is to use regular geometric shapes, ones where all sides are the same lengths, and to align the corners to maximize the symmetries. There is a class of tessellations that use irregular shapes. There are at least three families of irregular hexagons that can tessellate if the proper constraints are met.

Type 1

Constraints:

  • a = d
  • A + B + C = 360°

That is fairly terse mathematical language, so what does it mean?

First of all the sides of the hexagon are enumerated as a through f. and the angles are enumerated as A through F. The first condition, a = d, just means that the two opposite sides have equal length. The other lengths are free to change. The second condition, A + B + C, means a couple of things. It could mean that if the angles, A, B, and C, were placed in such a manner as to share a common point and not overlap, they would consume all 360° about that common point. In traversing the hexagon, it also means that sides a and d are parallel.

It also implies something else. The sum of the interior angles of a hexagon, enumerated as A through F, will always be 720°. This is from the formula:

degrees = (points in polygon - 2) * 180°

for a hexagon this is:

degrees = (6 -2) * 180° = 4 * 180° = 720°

A way to remember this is that you can decompose any polygon into a number of triangles. The first triangle take three points and every other triangle just uses one more point. This is the (points in polygon - 2) part. Then remember that the sum of the interior angles of a triangle come out to 180° for any shape of triangle.

So if A + B + C = 360° and A + B + C + D + E + F = 720°, it must also mean that D + E + F = 360°, and that the three angles also complete the angles about a point.

All of this results in the following observation:

Style 1:  Style 2:
   ^         ^···
  /B\       /E\··
 /b c\     /e f\·
|A   C|   |D   F|
|a 1 d|   |d 2 a|
|F   D|   |C   A|
 \f e/     \c b/·
  \E/       \B/··
   v         v··

These get layed out as with each type on alternating rows:

  F|C   F|C   F|C   F|C   F|C   F|C   F|D   F|D··
 b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\ c
 /b c\B/b c\B/b c\B/b c\ /b c\B/b c\ /b c\B/b c\·
|A   C|A   C|A   C|A   C|A   C|A   C|A   C|A   C|
|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|
|F   D|F   D|F   D|F   D|F   D|F   D|F   D|F   D|
 \f e/E\f e/E\f e/E\f e/E\f e/E\f e/E\f e/E\f e/E
 f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e·
  F|D   F|D   F|D   F|D   F|D   F|D   F|D   F|D··
  a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2
  A|C   A|C   A|C   A|C   A|C   A|C   A|D   A|D··
 b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\ c
 /b c\B/b c\B/b c\B/b c\ /b c\B/b c\ /b c\B/b c\·
|A   C|A   C|A   C|A   C|A   C|A   C|A   C|A   C|
|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|a 1 d|
|F   D|F   D|F   D|F   D|F   D|F   D|F   D|F   D|
 \f e/E\f e/E\f e/E\f e/E\f e/E\f e/E\f e/E\f e/E
 f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e f\E/e·
  F|D   F|D   F|D   F|D   F|D   F|D   F|D   F|D··
  a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2 a|d 2
  A|C   A|C   A|C   A|C   A|C   A|C   A|D   A|D··
 b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\c b/B\ c
 /b c\B/b c\B/b c\B/b c\ /b c\B/b c\ /b c\B/b c\·
|A   C|A   C|A   C|A   C|A   C|A   C|A   C|A   C|

figure of type 1 hexagons in a tessellation (please forgive old school ASCII art)

Note that this figure shows regular hexagons, which satisfy the type 1 conditions, other shapes are possible. In this orientation, each row is rotated 180° from the row above or below it.

Type 2

Constraints:

  • a = d
  • c = e
  • A + B + D = 360°

What does this mean? a = d is that the opposite sides a and d are of the same length, like in Type 1. Additionally sides c and e have the same length. Now angles A, B and D add up to 360° and by implication so do angles C, E and F.

This results in a different layout of the hexagons in a tessellation with the adjacent hexagons being rotated 180°:

There are four styles of polygon:

Style 1:   Style 2:   Style 3:   Style 4:

   ^          ^          ^          ^
  /B\        /E\        /B\        /E\··
 /b c\      /f e\      /c b\      /e f\·
|A   C|    |F   D|    |C   A|    |D   F|
|a 1 d|    |a 2 d|    |d 3 a|    |d 4 a|
|F   D|    |A   C|    |D   F|    |C   A|
 \f e/      \b c/      \e f/      \c b/·
  \E/        \B/        \E/        \B/··
   V          V          V          V

These get layed out as shown below:

  D|A   C|F   D|A   C|F   D|A   C|F   D|A   C|F·
 e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f
E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\
|C   A|D   F|C   A|D   F|C   A|D   F|C   A|D   F
|d 3 a|d 4 a|d 3 a|d 4 a|d 3 a|d 4 a|d 3 a|d 4 a
|D   F|C   A|D   F|C   A|D   F|C   A|D   F|C   A
 \e f/E\c b/B\e f/E\c b/B\e f/E\c b/B\e f/E\c b/
  c\E/f \B/b c\E/f e\B/b c\E/f e\B/b c\E/f e\B/b
  C|F   D|A   C|F   D|A   C|F   D|A   C|F   D|A·
1 d|a 2 d|a 1 d|a 2 d|a 1 d|a 2 d|a 1 d|a 2 d|a·
  D|A   C|F   D|A   C|F   D|A   C|F   D|A   C|F·
 e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f
E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\
|C   A|D   F|C   A|D   F|C   A|D   F|C   A|D   F
|d 3 a|d 4 a|d 3 a|d 4 a|d 3 a|d 4 a|d 3 a|d 4 a
|D   F|C   A|D   F|C   A|D   F|C   A|D   F|C   A
B\e f/E\c b/B\e f/E\c b/B\e f/E\c b/B\e f/E\c b/
 c\E/f e\B/b c\E/f e\B/b c\E/f e\B/b c\E/f e\B/b
  C|F   D|A   C|F   D|A   C|F   D|A   C|F   D|A·
1 d|a 2 d|a 1 d|a 2 d|a 1 d|a 2 d|a 1 d|a 2 d|a·
  D|A   C|F   D|A   C|F   D|A   C|F   D|A   C|F·
 e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f e/B\b c/E\f
E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\E/c b\B/e f\
|C   A|D   F|C   A|D   F|C   A|D   F|C   A|D   F

figure of type 2 hexagons in a tessellation (please forgive old school ASCII art)

Note that this figure shows regular hexagons, which satisfy the type 1 conditions, other shapes are possible. There are four orientations of hexagons. The hexagon to the right or left of the current hexagon is a mirror image of the current hexagon (angles and sides are in the opposite direction, clockwise vs. counter clockwise and vice versa). Every row is a 180° rotation of the previous row (or a mirror image of the previous row).

Type 3

Constraints:

  • A = C = E = 120

  • a = b

  • c = d

  • e = f

What does this mean? A, B and C are all 120°, so any three of these angles can come together about a point. The further constraints on the lengths of the sides limit which angles can actually be used. a = b limit the coming together of three hexagons at the point with angle A. c = d limit the coming together of three hexagons at the point C and e = f limit the coming together of three hexagons at the point E. Since A + C + E = 360°, B + D + F = 360°. This means that B, D and F can also come together at a point (as long as the sides match up). Not that in this type of hexagon, there is no requirement that any of the sides be parallel, although they might be.

This results in a different layout of the hexagons in a tessellation:

The hexagons for type 3 are as follows:
Style 1:  Style 2:  Style 3:··
   ^         ^         ^······
  /B\       /F\       /D\·····
 /b c\     /f a\     /d e\····
|A   C|   |E   A|   |C   E|···
|a 1 d|   |e 2 b|   |c 3 f|···
|F   D|   |D   B|   |B   F|···
 \f e/     \d c/     \b a/····
  \F/       \C/       \A/·····
   V         V         V·····

These get layed out as:

  F|D   B|F   D|B   F|D   B|F   D|B   F|D··
 a/B\d c/D\f e/F\b a/B\c c/D\f e/F\b a/B\c·
 /b c\C/d e\E/f a\A/b c\C/d e\E/f a\A/b c\·
|A   C|C   E|E   A|A   C|C   E|E   A|A   C|
|a 1 d|c 3 f|e 2 b|a 1 d|c 3 f|e 2 b|a 1 d|
|F   D|B   F|D   B|F   D|B   F|D   B|F   D|
 \f e/F\b a/B\d c/D\f e/F\b a/B\d c/D\f e/F
 e\E/f a\A/b c\C/d e\E/f a\A/b c\C/d e\E/f·
  E|E   A|A   C|C   E|E   A|A   C|C   E|E··
  f|e 2 b|a 1 d|c 3 f|e 2 b|a 1 d|c 3 f|e 2
  F|D   B|F   D|B   F|D   B|F   D|B   F|D··
 a/B\d c/D\f e/F\b a/B\d c/D\f e/F\b a/B\d·
 /b c\C/d e\E/f a\A/b c\C/d e\E/f a\A/b c\·
|A   C|C   E|E   A|A   C|C   E|E   A|A   C|
|a 1 d|c 3 f|e 2 b|a 1 d|c 3 f|e 2 b|a 1 d|
|F   D|B   F|D   B|F   D|B   F|D   B|F   D|
 \f e/F\b a/B\d c/D\f e/F\b a/B\d c/D\f e/F
 e\E/f a\A/b c\C/d e\E/f a\A/b c\C/d e\E/f·
  E|E   A|A   C|C   E|E   A|A   C|C   E|E··
  f|e 2 b|a 1 d|c 3 f|e 2 b|a 1 d|c 3 f|e 2
  F|D   B|F   D|B   F|D   B|F   D|B   F|D··

figure of type 3 hexagons in a tessellation (please forgive old school ASCII art)

This results in three different orientations of the hexagons. each one rotated 120° from the adjacent hexagon.

Movie

So now that we know the constraints on the irregular hexagons, what are the different shapes that are possible? This could be shown as a series of images, but it is more interesting to show a moving of the shapes changing dynamically and then moving between the three types. Remember that a regular hexagon tessellation satisfies the constraints of all three types. Additionally if an angle is 180° or has length of zero, that side is effectively eliminated, so some polygons may appear as pentagons, squares or triangles, but in this generator, they all have the requisite six sides. Normally tessellations would be constrained to convex polygons, meaning that a segment between any two non-adjacent points would be entirely within the polygon. This really constrains all interior angles to be less than 180°. This generator does not have that constraint, so some interesting contortions are possible. Negative interior angles and some angles greater than 180° result in overlaps which violate one of the contraints of tessellations in general.

Movie Generator

This movie was generated by a Turtle Graphics program that saves a series of .png images which are then assembled into a movie. Each image is a frame that is shown one or more times. The program has three main parts:

  • a hexagon solver
  • an image renderer
  • an interpretive manipulator.

The hexagon solver just solves the sides and angles for a particular hexagon type. Since everything is dependent on every thing else, the location of the points of the hexagon are specified and then the sides and angles can be computed. The points are specified as a base and height just like a carpenter would specify for a staircase or roof. They working in run and rise, but that translates directly into base and height.

The image renderer takes the computed sides and angles and draws individual hexagons until the turtle graphic canvas is filled. It has a set of rules based on the hexagon type as to the orientation and direction (clockwise or counter clockwise) or each individual hexagon.

The interpretive manipulator has a data table that lists a set of instructions for manipulating the points of a hexagon. Each instruction consists of one or more steps, each one controlling the height or base of individual points of the hexagon. Individual steps or instructions may take several frames or images to complete as they specify a particular morphing from one hexagon shape to another.

The instructions are interpreted to change the location of the points in a hexagon, the sides and angles are computed for the new hexagon shape and finally the hexagons are rendered on the Turtle Graphic canvas.

Are There More? – The Unknown

There are only three known types of irregular hexagons that tessellate. There may be more. Marjorie Rice, as housewife and amateur mathematician discovered not one but four new pentagonal tilings (find out more about her at Wikipedia or Quanta Magazine). Regular pentagons will not tessellate because the angles do not add up to 360°. Irregular pentagons can tessellate because the adjoining angles can be manipulated to add up to 360°. Up to the point of her discovery, mathematicians thought that there were only eight types of convex pentagonal tilings, now there are fifteen. Even though there were proofs that there were only the eight tilings, Marjorie came up with a different sent of assumptions that broke the proof and opened the door for her discoveries and the discoveries of others.