Introducing the Tetrahedral Complex

We have a formation of skew lines connecting end-points of corresponding identical intervals on a pair of skew ‘rulers’, so arranged (in flagrant breach of projective geometric regulations) as to allow us to rotate the entire thing “rigidly” around one of these skews by dragging the dashed blue line. [‘Vector’ MP, for example, is represented, most un-projectively, as having constant length.]  That is, we take that line as an ‘axis’ – a line bearing the points of incidence of other, ‘transverse’ lines, that all rotate around it.

These transverse lines arise from joining the points of intersection by the original set of skews of regularly-spaced planes, by which we mean planes, all incident in the same line, set at projectively-identical interplanar intervals. For simplicity, we take the “view” to be from infinity through infinitely powerful binoculars. One of the two circles (either at will) must be considered to be in front of the other. If they were not, all the lines would be coplanar, and so not skew.  We know thereby that this structure does not lie in or on a surface.


We have a net of skewed quadrilaterals (skewed, because their opposite sides are not coplanar).

It will be appreciated that the two diagonals (here artificially truncated at the vertices for the sake of visual clarity) of each of these quadrilaterals are also not coplanar, and are nowhere incident, so they, “pass each other by”, and must be skew to each other.

We thereby have a tetrahedron, formed from the sides of the quadrilateral and its diagonals - and, of course, four planes.
One reason (though not by any means the only one) that the Tetracell, in its Complex, attracts interest  is that a tetrahedron is the invariant structure "holding" path curves, and path curves are found to match many real, organic - and some inorganic - forms very closely. knapweed

Many such forms, perhaps most, are composite, like the knapweed on the right, and the components often seem to be path curve structures in their own right, similar in form to the composite form.

It may be that we have these ‘path-components’ in an array of tetrahedra, like the one introduced here.


There are one or two caveats.

Traced from quadrilateral to quadrilateral, across the corners, the diagonals might appear to form two sets of plane curves, perhaps conics, but, at best, the diagonals are chords of curves, not tangents to them. Conics have deliberately been put through vertices below, whether or not they properly belong there (a conic can be put through any five points, as here, but not necessarily through six or more).  Clearly, the diagonals are not, and cannot be, tangent to these conics.  Also, the conics themselves are each skew to every other, so cannot form a surface, appearances notwithstanding.

Bear in mind, too, that because the two diagonals of every quadrilateral in this net are skew to each other, they cannot both be coplanar with the same plane curve, so cannot both be tangent to it.  However, they might both be tangent to a “space” curve, such as a vortical path curve — at any rate, one not confined to a plane.
It is a tetrahedral “cell”,
in a net or array
of such cells.
tetracell tetracell vortex The diagonals of a tetracell could both be tangent to a wave of a water vortex - because it is not a plane curve.