Archex Variation

(An explanation of the Archex Variation follows after the pointers.)

archex.JPEG-six 90 degree arcs forming hexagonal loop (JPEG 73k)
Shows how 90 degree arcs joined at 90 degrees to each other congrue with the plane faces of a cube.

archex_toy.MOV-conversion of axial to linear motion using archex loop (Quicktime Movie 143k)
tripods alternate in an out as they rotate on axis perpendicular to archex loop.

archex_VE.JPEG- 4 archex loops form archex VE (JPEG 51k)
4 hexagonal planes form the VE; similarly, 4 archex loops form the archex VE-each loop is a different color in the JPEG.

Archex Precession Look at this!

archex_tetrakai.JPEG- formed from 8 archex loops (JPEG 65k)
the regualar tetrakaidecahedron has 8 hexagonal and 6 square faces. It is a truncated octahedron. The archex tetrakai is formed fro 8 archex loops which have 6 arc-square forms between them.

archex_tetrakai (in IVM).JPEG- shows archex tetrakai (red) in an allspace array of archex VE (JPEG 27k)

archex_ IVM.JPEG-allspace array further expanded-note arc octa between archex VE (JPEG 43k)

Archex Jitterbug variation of jitterbug with archex VE and archex tetrakai (Quicktime Movie 160k)
The 8 arc-triangles of the archex VE rotate and track the 8 archex loops of the archex tetrakai. Kirby Urner and I likened this animation to an artificial heart.

archex_precession.MOV shows precessional relationshp between arhex loops and arc-triangles of archex VE (Quicktime Movie 458k)
The perpendicular armatures are like the radials of a conventional VE but they are equidistant from every point on the circumferentials of the archex VE. This design is still in the formative stages so if you have any thoughts please forward to me or the list.

precession_array (Quicktime Movie 742k): allspace array of the archex_precession:
Set your player to LOOP and PLAY EVERY FRAME for this animation which shows an array of 27 archex VE arranged into archex IVM with (4) perpendicular radials (as in "archex_precession.MOV")interconnecting and forming other matrices as they go through cycle.

The Archex Variation

The archex variation simply substitutes quadrants (90 degree arcs) for unit edge lengths in the IVM (isotropic vector matrix).

It is my own innovation and one which I have contemplated for many years. The term "archex" was coined by Karl Erickson in our discussions on synergetics-l mailing list, and refers to the six 90 degree arcs which when joined at 90 degrees to each other form a continous loop analogous to the hexagonal grouping of edges in th VE (vector equilibrium). See archex.JPEG for a view of one archex inscribed within a cube.

The archex loop has three peaks and three valleys so its' design facilitates the conversion of axial to linear motion and vice-versa (see archex_toy.MOV).

The VE (without radials) can be viewed as 4 hexagonal planes; similarly, the archex VE is 4 archex loops. The arc hex loop can be used in place of the hexagonal face of other polyhedra in the IVM.

The archex tetrakai is formed from 8 archex loops (see archex_tetrakai).

The archex tetrakai and archex VE can be combined to form a jitterbug variation (archex_jitterbug.MOV).

This design may have some practical application, any ideas?

One of the most interesting discoveries in my exploration of the archex topological forms is the precessional relationship between the archex loops and arc-triangles of the archex VE.

If an armature with its' pivot point at center of gravity is rotated tracking through the archex loops of the archex VE, another armature fixed at 90 degrees to the first will track the arc-triangles (see archex_precession.MOV).

-Richard Hawkins