The Ammann bars are an alternative and ingenious way to enforce the adjacency constraints
Whenever the constraints are met, the bars join into straight lines
Ammann decorations must continue straight from a tile to an adjacent one,
this is an alternative way to enforce the adjacency constraints
The decorations have to perfectly align
to form straight lines
The "star pattern" is now constructed with the addition of the Ammann bars
The decorations form five families of parallel lines
and in each family the lines can have "long" or "short" distances.
The sequence of such distances is related, once again,
to the Fibonacci Rabbit sequence
and the Golden Ratio appears, again and again!
We can give fantasy names to some interesting configurations of tiles
One of the most fascinating ways of tiling the plane with kites and darts
is achieved by starting from two half-kites
with the usual deflation/inflation procedure.
We obtain the so-called "cartwheel"
In its center we find a decagon containing a peculiar shape
called "batman"
Emanating from the decagon we find stripes of long and short bows
If we "turn over" a single one of these (infinite) stripes
the decagon gets modified
and the "batman" can no-longer fit.
The remaining "hole" cannot be filled
This configuration is called "Asterix"
By reversing in all possible ways the stripes of "bows"
we obtain a total of 62 distinct "holes", they are called DECAPODS
Some of them have nice shapes to which we can assign a name
As you can see, Asterix and Batman are two of these decapods
All of the decapods can be obtained
placing 10 golden obtuse triangles around the origin
oriented in two ways, shown with a + and a -
This is the "hut" decapod
Around the "hut" we can construct a decagon
and infinite stripes of long and short bows
The stripes (worms) must be suitably oriented
Finally the tiling can be completed in the same way as for the cartwheel
The black lines are the Ammann bars, showing that the adjacency constraints are met
We can change the decapod, e.g. into a "buzzsaw"
by suitably repositioning the tiles in the central decapod
and the "worms" of long and short bows
and still we can fill the whole plane out of the decapod
The Ammann bars again show that the constraints are met
Here we have all 62 decapods
Among them, the "batman" is the only one
that can be filled with kites and darts
obtaining the beautiful cartwheel that we encountered before
We can enforce adjacency constraints also for the fat and thin rombs
similarly to a jigsaw puzzle
We break the rhombs in two golden triangles...
FAT RHOMBUS: two obtuse triangles
THIN RHOMBUS: two acute triangles
We can use the subdivision idea shown for kites and darts
Darts and kites can be subdivided in smaller copies of fat and thin rhombuses
Similarly we can subdivide fat and thin rhombuses
in smaller copies of darts and kites
Let us apply the semideflation process twice to part of a cartwheel
and transform it into a finer tiling
... made of rhombuses
Another "semideflation" produces kites and darts again
organized in a finer tiling, but otherwise equal to the original one