cubic helix

these are pictures of a cubic helix constructed from 12 mini Rubik's Cubes.

further images are posted on my blog page.

http://tetrahelix.blogspot.com/

As you can see from the image above, the helix does not sit well on a flat plane, no matter how you rotate it the helix will only presents edges and not faces to the plane.

easiest method to make the helix it is to balance a starting cube on a point so that point B is directly over point A (above).  this creates three axis or directions all moving away from the base point A.  if these are considered as vectors A1, A2, A3 and taken sequentially (in steps of equal unit length) A1, A2, A3, A1 etc. they will create a spiral upwards from A to B around a central axis.  please note that the central axis of the helix so constructed is not axis AB but an axis parallel to it. 

in order to illustrate this more clearly the helix below has been constructed by placing 2* cubes against the blue face (direction A1), then 2 cubes against the yellow face of the top cube (direction A2), then 2 cubes against the green face of the top most cube (direction A3) and so on.

an elevation of the helix showing its construction spiraling upwards from A to B

plan view of the helix viewed on axis (BA) 

*a true cubic helix is constructed from single cubes but for clarity and to make the spiral more obvious I used units of 2 for this illustration.

an alternate method to construct a cubic helix using Cartesian coordinates.

if a block is positioned on a plane with edges along each of the Cartesian axis; x, y and z. the helix can be constructed by displacing a block through each axis in turn.  if the block has edges of unit length 1 then the transformations in sequence would be (1,0,0), (0,1,0), (0,0,1) repeat ad infinitum.