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The Ammann bars are an alternative and ingenious way to enforce the adjacency constraints

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Whenever the constraints are met, the bars join into straight lines

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Ammann decorations must continue straight from a tile to an adjacent one,

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this is an alternative way to enforce the adjacency constraints

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The decorations have to perfectly align

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to form straight lines

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The "star pattern" is now constructed with the addition of the Ammann bars

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The decorations form five families of parallel lines

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and in each family the lines can have "long" or "short" distances.

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The sequence of such distances is related, once again,

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to the Fibonacci Rabbit sequence

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and the Golden Ratio appears, again and again!

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We can give fantasy names to some interesting configurations of tiles

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One of the most fascinating ways of tiling the plane with kites and darts

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is achieved by starting from two half-kites

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with the usual deflation/inflation procedure.

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We obtain the so-called "cartwheel"

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In its center we find a decagon containing a peculiar shape

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called "batman"

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Emanating from the decagon we find stripes of long and short bows

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If we "turn over" a single one of these (infinite) stripes

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the decagon gets modified

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and the "batman" can no-longer fit.

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The remaining "hole" cannot be filled

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This configuration is called "Asterix"

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By reversing in all possible ways the stripes of "bows"

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we obtain a total of 62 distinct "holes", they are called DECAPODS

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Some of them have nice shapes to which we can assign a name

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As you can see, Asterix and Batman are two of these decapods

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All of the decapods can be obtained

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placing 10 golden obtuse triangles around the origin

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oriented in two ways, shown with a + and a -

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This is the "hut" decapod

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Around the "hut" we can construct a decagon

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and infinite stripes of long and short bows

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The stripes (worms) must be suitably oriented

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Finally the tiling can be completed in the same way as for the cartwheel

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The black lines are the Ammann bars, showing that the adjacency constraints are met

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We can change the decapod, e.g. into a "buzzsaw"

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by suitably repositioning the tiles in the central decapod

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and the "worms" of long and short bows

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and still we can fill the whole plane out of the decapod

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The Ammann bars again show that the constraints are met

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Here we have all 62 decapods

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Among them, the "batman" is the only one

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that can be filled with kites and darts

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obtaining the beautiful cartwheel that we encountered before

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We can enforce adjacency constraints also for the fat and thin rombs

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similarly to a jigsaw puzzle

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We break the rhombs in two golden triangles...

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FAT RHOMBUS: two obtuse triangles

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THIN RHOMBUS: two acute triangles

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We can use the subdivision idea shown for kites and darts

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Darts and kites can be subdivided in smaller copies of fat and thin rhombuses

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Similarly we can subdivide fat and thin rhombuses

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in smaller copies of darts and kites

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Let us apply the semideflation process twice to part of a cartwheel

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and transform it into a finer tiling

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... made of rhombuses

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Another "semideflation" produces kites and darts again

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organized in a finer tiling, but otherwise equal to the original one