There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.
Martin Gardner, "The Mathematical Magic Show"
This site is partly inspired by (and takes its name from) the geometry of the
Spiral Honeycomb Mosaic. I will now show how the spiral honeycomb mosaic with 343 tiles is constructed. There is really nothing to it. First, we take a generator:
From the inside out, we have a brown hexagon, which is the SHM of order 0. Then there is a blue flag which has one point fixed at the centre, around which it is free to move. This is the key linkage in understanding how points from the centre of the SHM are projected towards more outwards points, and the key to efficient triangulation of points outside the central hexagon. Outside this again is the boundary of the SHM of order 1. Benoit Mandlebrot gave this boundary the nomenclature "
Gosper Island". I will come back to talking about fractal construction of this "limit cycle" later. But back to the generator... the final shape is a larger hexagon, which is a copy of the original hexagon which has been rotated by arctan sqrt(3)/2 and scaled by sqrt(7).
Given this generator, we can construct all points in the spiral hexagonal lattice. The flag shape includes 5 points:
- the centre
- two point on the boundary of the SHM of order 0
- one point in the centre of a hexagon neighbouring the central hexagon (this is inside the SHM of order 1, but outside the SHM of order 0)
- one point on the boundary of that same hexagon (this is on the boundary of the SHM of order 1)
By repeated rotation of the flag by 60 degrees we can address the centres of all hexagons neighbouring the central hexagons (or, more generally, as you shall see, we can replace "hexagon" with "spiral honeycomb mosaic"). We can also address some of the points of the boundary. The remainder of the points on the boundary are constructible from a simple fractal construction method, or, alternatively, by translation or central symmetry of fractional parts of the original SHM boundary.
Construction of the SHM of order 2 proceeds by first constructing two similar copies of the original flag:
The red flag has been rotated by 30 degrees and scaled by sqrt(3), while the green flag has been rotated by arctan(sqrt(3)/2) and scaled by sqrt(7). We should note:
- the green flag will have similar properties to the original blue flag, with the point in the centre staying at the centre, points on centres mapping to other centres, and points on boundaries mapping to points on other boundaries
- if a generator of order n is extended in this manner, the green flag will map out points in the SHM of order n+1, so
- the green flag finds points in the centre and on the boundary of the surrounding SHM, while
- the red flag finds a hexagon neighbouring the current SHM
- the green flag measures distances in terms of powers of sqrt(7)
- the red flag measures distances in terms of powers of sqrt(3)
While the red flag is not necessary for constructing the boundary of the shape, it is necessary if you plan on constructing a set of hexagons interconnected into an SHM shape. If you're using a compass and ruler construction, you will be thankful of the red flag since it allows you to propagate the more useful metric of sqrt(3) rather than straining to reconstruct numbers in terms of sqrt(7).
The construction can be shown via induction to tile the plane. Here is a partial construction of the SHM of order 3 (343 cells) using some short-cuts for generating some points:
I haven't completed the boundary, but it should be clear from the foregoing how it should go. This was down to laziness on my part, as all the images were created by hand using the excellent
Dr. Geo software.
This description wouldn't be complete without defining the fractal corresponding to the SHM boundary, or "Gosper Island". It is most easily specified as an
L-System with the following parameters:
- Start String: F+F+F+F+F+F+
- Production Rule: F → F-F+F
As a final note, there are similar constructions possible with triangular and square tiles. I intend to write a follow-up article on the shapes and
numbers arising from examining the three tilings together.
