Planar 'jitterbugs' can be derived from a square tiling
and a triangle tiling of the plane. The tiles are oriented alternately,
and connected by vertices to their neighbours, with the choice of
vertex depending on orientation.
At maximum expansion the triangle transformation requires four times
the area of the base tiling, and the square transformation requires
double the area of the base tiling.
These tiles may also represent faces or sections of a polyhedron,
and the transformation can then control a transformation of vertex
bonded polyhedra in space. The ratio of the space required
at the maximum and minimum expansion is the same as for the
corresponding plane tiling
Octahedra have their faces attached above the triangles of one
orientation and below the triangles of the other orientation. The
transformation of the triangle tiling transforms the octahedra in
their layers. At the maximum expansion the empty space forms
cuboctahedra, and these cuboctahedra close down to form a pair of
tetrahedra at the minimum expansion.
Octahedra have their faces attached above the triangles of one
orientation and below the triangles of the other orientation. The
transformation of the triangle tiling transforms the octahedra in
their layers. At the maximum expansion the empty space forms
cuboctahedra, and these cuboctahedra close down to form a pair of
tetrahedra at the minimum expansion.
Rhombic dodecahedra have a triangular section attached to have a single
vertex above the triangles of one orientation or below the triangles of
the other orientation. The transformation of the triangle tiling
transforms the rhombic dodecahedra in
their layers.
view animation of transformation with rhombic dodecahedra (9Mb)
The transformation of the triangle tiling is mapped onto a torus.
The model was made as an aid to visualisation. It has no geometric
significance. The expansion is simulated.
view animation of torus jitterbug (4Mb)
The transformation of a triangle tiling mapped onto a torus. When there
are four triangles around the tube the result approximates a
transforming kaleidocycle. A model made with rigid triangles
would transform like this.
view animation of transforming kaleidocycle (2Mb)
Cubes have their faces attached above the squares of one
orientation or below the squares of the opposite orientation. The
transformation of the square tiling transforms the cubes in
their layers. The empty space at the maximum expansion forms
cubes, and these cubes close down to nothing at the minimum
expansion, which is just a packing of the cubes.
view animation of transformation with cubes (7Mb)
Rhombic dodecahedra have a square section attached to have a single
vertex above the squares of one orientation or below the squares of
the opposite orientation. The transformation of the square tiling
transforms the rhombic dodecahedra in their layers.
view animation of transformation with rhombic dodecahedra (9Mb)
The transformation of the square tiling is mapped onto a torus.
The model was made as an aid to visualisation. It has no geometric
significance. The expansion is simulated.
view animation of torus jitterbug (3Mb)
