The result of a level-6 subdivision process starting from a conglomerate of 3 clusters, one H7
and two H8 (see below), becomes the Land of Oz.
The "infinite" worm is the yellow brick road, the path that Dorothy takes to reach the Emerald City.
Many departing roads lead to various places; a randomly selected one leads to the castle of the Wicked Witch of the
West...
[Click here for an augmented reality animation showing the striking connection of
the infinite worm with the golden ratio]
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Starting with a conglomerate of three clusters as shown in the image and performing six levels of subdivision we
obtain the "Land of Oz", where Dorothy starts at the origin, indicated by the black dot and moves along the
infinite worm that forms through the origin in the left, slightly upward, direction.
By performing one level of substitution we can ensure that the resulting finite tiling is actually an extension of
the starting conglomerate and this property remains true so that we can take the limit to get a complete tiling of the plane.
Click on the two images below to show an animation of Dorothy walking along the path displayed on a "level-6" set of super-clusters
(the two images below are obtained at level-4).
The first (straight) path connect the starting position to the Emerald city, whereas the second image show Dorothy leaving
the yellow brick road and arriving through a second crossing point at the Wicked Witch of the West castle.
At time 0:50 an "earthquake" temporarily rotates 180 degrees all clusters of the yellow brick road (resulting
in a legitimate tiling of the plane).
Explanation: successive subdivisions starting from an H7 with the pivotal point taken halfway on the
right portion of the border lead to a tiling of a vast portion of the plane.
We get a complete tiling by adding two more subdivided clusters around the pivotal point.
The resulting tiling of the whole plane contains a bi-infinite worm, i.e. a ``straight'' sequence of adjacent clusters
all with the same orientation (the yellow brick road).
A remarkable feature of the successively subdivided H7
and H8 superclusters is the existence of the
straight worm of equally oriented hat tiles
(and upside down hat tiles) running roughly from east-south-east to west-north-west.
We can imagine the tiles of this worm as painted yellow and follow
Dorothy Gale walking along the resulting yellow brick road.
The following animation does just that in a partial tiling obtained by six successive
subdivisions of an H7:
The sloping line has slope the golden ratio (≈ 0.618) through the point at (0,1).
An important REMARK: a subdivided H8 supercluster is NOT an extension of the former supercluster if the blow-up is performed
with respect to the position of the H7.
As a consequence the final (say 1/3 approx) sequence of clusters along the worm is not the same as the limit semiinfinite worm.
Starting with a conglomerate of three clusters as shown in the image and performing six levels of subdivision we
obtain the "Land of Oz", where Dorothy starts at the origin, indicated by the black dot and moves along the
infinite worm that forms through the origin in the left, slightly upward, direction.
By performing one level of substitution we can ensure that the resulting finite tiling is actually an extension of
the starting conglomerate and this property remains true so that we can take the limit to get a complete tiling of the plane.
