One Dimension

When we step a point along a line, we have recourse to two directing points and an intermediate line. The result is a pair of conjugate, linear measures in the line, between two invariant end-points that divide the line (i.e., points that cannot be stepped, or, if stepped, step exactly to where they are) .

The end-points lie at places that would be reached by an infinite number of steps, so they serve as "functional infinities" for the measurement process.

Progression through dimensions-

linear measure

stepping a point along a line

Two Dimensions

When we step a point across a plane, we have recourse to two stepping lines, and our stepping point in the plane is at their intersection. Our lines rotate in points, which are the vertices of a triangle, and are guided by linear measures in the sides of that triangle, and the result is a path curve in the plane. The sides of the triangle cannot be stepped, so are invariant, and are functional infinities for the stepping lines. Accordingly, the entire triangle is invariant.

planar measure

stepping a point across a plane

When we want to step a point across the whole volume of space, we must have recourse to stepping planes, and the stepping point will be at their intersection.

Three planes intersect in a point, so we must have three stepping planes.

three planes can track a point across space

Where for the planar case, we have lines rotated in points between functionally infinite lines, now for all space we must have stepping planes rotating in lines, and their functional infinities will be planes, of which there will be one pair per line.

Now, a pair of planes have a line in common, so there will be two distinct lines-in-common for two, distinct pairs of planes:

If these two lines are skew to each other, then the four planes intersect in the six lines of a tetrahedron. (In fact, this is how four planes generally intersect.)

The vertices of it lie at the four, distinct places where three of the four planes intersect.

Opposite sides are skew, so there are three such skew pairs.

All of these elements, point, line or plane, are real, as opposed to imaginary .

Four general planes form a tetrahedron.

they intersect

• by triples in 4 points (vertices)
• by pairs in 6 lines (sides)

The four planes "enclose" volumes of space. That is, they divide the volume of all space into five volumes. If one of these is not cut by the plane at infinity, the remaining four must be.

We might take these four volumes cut by the plane at infinity as "external", and the fifth, uncut volume as "internal"; that is, as the "inside", of the tetrahedron. (Contrast this with the "enclosure" of a cone.)

We see that any one of these six could be a line in which a stepping plane might rotate, and that while six are available, three will do for tracking our point as it steps through the volume of space. Three such are shown one at a time on the right.

The facets of the tetrahedron intersecting in a given line-of-rotation will be the planes of the corresponding functional infinities for the rotating plane.

The rotations are guided by (independently determined) measures in the sides opposite to, and skew to, the sides in which the planes rotate.

In the construction on the left, the red path curve running through the “volume” of the tetrahedron presents as an intersection of two path cones.

The black path curve is a plane section of a path cone with its vertex at D.

The blue path curve is a plane section of a path cone with its vertex at A.

You may change the black path curve in tetrahedral face ABC by dragging points K and L, and the blue path curve in face BCD by dragging points E and F.

The red path curve adapts to these changes: it appears to “drift” between the blue and black curves.