unorthodox tetrahelix

this page is posted to illustrate unorthodox tetrahelices.

an orthodox or regular tetrahelix is defined as:'.. general definition is that there is a screw displacement (a rotation around an axis followed by a translation along the axis), and this transformation when repeated carries a fixed polyhedron into a position on the helix such that the next transformed polyhedron will share a face with it.

 A limitation could be placed on this so that the helix would not be self-intersecting.

Another limitation could be that translation would not be zero,and the rotation would not be 0 or 180, as these could be considered as degenerate helices.' 

 Adrian Rossiter.

the helices on this page are therefore non regular or unorthodox tetrahelices, they might even be called degenerate but I still like them anyway as they each have very

 particular and unique properties.

the best way to consider unorthodox tetrahelices or polytopes of a tetrahelix is to describe how their construction differs from a regular tetrahelix.

how to construct a regular tetrahelix (1,2,3,4)

i will first describe a simple method to construct a regular tetrahelix.

take a regular tetrahedron and number or colour differently each of the four faces 1,2,3 and 4 and start by mirroring:

transformation 1, mirror the tetrahedron through face 1 and bond the new tetrahedron to that face.

transformation 2, mirror the new tetrahedron through its face 2 (and bond the new tetrahedron generated to that face)

transformation 3, mirror the newest tetrahedron through its face 3 (and bond the new tetrahedron generated to that face)

transformation 4, mirror the newest tetrahedron through its face 4 (and bond the newly generated tetrahedron to that face)

repeat from 1 add infinitum.

*it makes no difference the sequence of how the faces are numbered.  also the first two transformations always makes a 'boat' shape as a compound of three 

face boded tetrahedrons can only be configured as a 'boat' shape.

how to construct a 'loose' tetrahelix (1,2,3,1,4)

the construction of the helix referred to as the 'loose' tetrahelix is surprisingly similar. 

take a tetrahedron and number or colour the faces as before 1,2,3 and 4 and also construct by mirroring:

transformation 1, mirror the tetrahedron through face 1 and bond the new tetrahedron to that face.

transformation 2, mirror the new tetrahedron through face 2.

transformation 3, mirror the newest tetrahedron through face 3.

transformation 4, mirror the newest tetrahedron through face 1.

transformation 5, mirror the newest tetrahedron through face 4.

repeat from 1 add infinitum.

the loose tetrahelix can therefore be classified as a 1,2,3,1,4 tetrahelix.

other forms of spiral polyhedra can be seen at Rinus Roelf's  beautifully

 illustrated site.