## Introducing the Tetrahedral ComplexWe 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.
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
, 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.appear
to form two sets of plane curvesBear in mind, too, that because the two diagonals of every
quadrilateral in this net are skew to each other, they cannot be
coplanar with the both plane curve, so cannot same
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.
both |
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It is a tetrahedral “cell”,in a net or array of such cells. |
The diagonals of a tetracell could both be tangent to a wave of a water
vortex - because it is not a plane curve. |