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Plane and Space 'Jitterbugs' 11 images
tree :: Geometry > Plane and Space 'Jitterbugs'
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    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

 
Image List

   
triangles  Med    Lrg  
Description : 'Jitterbug' of a triangle tiling of the plane. The different colour triangles have opposite orientations.
view triangle tiling animation (3Mb)
Click on one of the size names above to enlarge this image
 
tri octahedra  Med    Lrg  
Description : 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.

The following animations show a 3x3x3 lattice of octahedra.
view animation with fixed plane (7Mb)
view animation with fixed triangle (7Mb)

Click on one of the size names above to enlarge this image
 
tri octahedra  Med    Lrg  
Description : 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.

The following animations show a 2x2x2 lattice of octahedra, surounding a single cuboctahedron space at maximum expansion.
view animation with fixed plane (3Mb)
view animation with fixed triangle (3Mb)

Click on one of the size names above to enlarge this image
 
tri rhomb dodec  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 
lattice jbug  Med    Lrg  
Description : A lattice jitterbug, with fixed triangle, for comparison
view animation of lattice jitterbug (6Mb)
Click on one of the size names above to enlarge this image
 
tri torus  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 
kaleidocycle  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 
squares  Med    Lrg  
Description : 'Jitterbug' of a square tiling of the plane. The different colour squares have opposite orientations.
view square tiling animation (1.5Mb)
Click on one of the size names above to enlarge this image
 
squ cubes  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 
squ rhomb dodec  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 
squ torus  Med    Lrg  
Description : 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)
Click on one of the size names above to enlarge this image
 

 

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Contact: adrian@antiprism.com      -      Generated on Sat 23 Oct 2010 12:33:01 PM CEST