While looking at the jamming characteristics of differently dressed Figure 8 Bends (which I'll discuss on another board), I came to see that one technique for loosening the jammed bends was not available to me. That technique would be the splaying open of the bend by driving a tapered pin through its center. Likely, others have observed this phenomena, but, I thought it might be of interest to discuss its relation to symmetry. This technique is always available for the many bends which have axial inversion symmetry, such as the Smith/Hunter's Bend, but, not necessarily available for those bends which have central (point) inversion symmetry, such as the Figure 8 Bend.

Axial Inversion Symmetry: R1( x, y, z ) = R2( -x, -y, z ) [ read as the section of the first rope located at point ( x, y, z ) is equivalent (twinned if you like) to the section of the second rope located at point ( -x, -y, z ) ]. The origin ( x, y, z ) = ( 0, 0, 0 ) is not completely defined because z = 0 is undefined. There is an axis of symmetry.

Central Inversion Symmetry: R1( x, y, z ) = R2( -x, -y, -z ). The origin ( x, y, z ) = ( 0, 0, 0 ) is completely defined. There is only a point of symmetry.

The implications of these symmetries are (theoretically) that no rope can reside on the axis or point of symmetry (for a bend, i.e. two ropes). Twinned sections of the two ropes can reside on either side of the axis or point, but, not occupy the symmetry location and still satisfy the above equations. Another way to say this is that if one of the ropes is residing on the symmetry location, the twinned part of the other rope would also have to reside on that same spot, which, of course, can't be so.

This means that for the axial inversion symmetry bends, there is always a STRAIGHT LINE FREE (void) OF ROPE right through its center, suitable for driving a tapered pin. This is not necessarily true for central inversion symmetry and is not true for the Figure 8 Bend. For bends with central inversion symmetry, there is only one point which must be free of rope.

DDK