Rigid squares of two sizes form the third hinged tiling.
In figure 3
, when the angles
of the empty prallelogram are 90˚, we
have rectangles separating the
When the angle is 0˚ we have a tiling of squares of two different sizes.
Centers of squares around a parallelogram
In figure 4
we can see that the centers of the four
squares on the sides of a parallelogram form a square for any angle of
tiling of hexagons and triangles that can
be transformed to a tiling with only hexagons. In this case every other
vertex of the hexagons is hinged. When non-hinged vertices of two
hexagons touch each other, we get equilateral triangles.
transforms a tiling using only triangles into a
tiling of empty hexagons surrounded by triangles.
In figure 8
consisting of yellow hexagons can be
transformed into a tiling of yellow hexagons surrounded by empty
hexagons. An interesting intermediate case is when the empty spaces are
equilateral triangles. If we subdivide these into four smaller
equilateral triangles we obtain a regular tiling of hexagons and
In figure 9
linked to other square with a rigid
rod. A tiling consisting only of squares can be transormed into a
regular tiling of squares and regular octagons. For an angle of
60˚, we can imagine subdividing
the four pointed star into four triangles surrounding a square. An
intermediate case is when the empty spaces form bigger
Triangles with linkages
In figure 10
equilateral triangles can be
transformed into a regular tiling of equilateral triangles and regular
empty dodecagons. What are some of the intersting intermediate special
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