The animation shows how a plane in counter space moves towards its infinity in equal steps for counter space. These are not equal
angles, as is obvious, and we refer to this measure as turn, which is for counter space the analogue of distance.
Its magnitude becomes infinite if the plane reaches the infinitude (shown as a star). The polar relationship between space and
counter space means that the measure of separation of planes is polar to that of points in space, while the measure of the
separation of points (shift) is polar to that of planes and hence like an angle. Thus pairs of
planes can define vectors, but not pairs of points.
Planes in the vertex only have two degrees of freedom and thus make up a polar area, not a polar volume. This may take some getting used to as our Euclidean consciousness tends to regard the region described by the planes as an infinite volume. The polar area is calculated by integration, which is figuratively illustrated in the following animation:
On the left is the integration of the polar area of a cone in counter space and on the right the dual integration of the area of a
circle in space. As it is difficult to represent the planes involved in this diagram we have taken successive conical segments bounded
by a sphere to represent steps in the progress of the integration. However this is purely representational to help convey the idea,
and should not be misunderstood in a point-wise manner. The spherical boundary is adopted to keep the diagram finite, the actual
polar area extending outwards to infinity, as do the cones.
When dealing with the states of matter we worked with point linkages between space and
counter space. For the ethers (as the more subtle aspects of reality are called by Steiner) we are concerned with planar linkages.
The most suitable linkage tensor for light is the contravariant bivector, represented by a cone in counter space (dual to the
oriented-circle-representation in space). It is suitable for polar affine linkages
characteristic of light. This turned out to be an investigation of actual counter space cones acting as photons, the polar area
of a photon cone being constant. Thus photons are initially neither waves nor particles. Their polar area embraces the whole of
the apparatus and so the "spooky" multi-path type experiments of modern physics may be more comprehensible. Reflection, refraction,
absorption and diffraction are all treated on this basis in Reference 11.
We see two CSIs emitting photon cones (yellow) interacting at their apices. Since the turn T increases inwards while the radial
distance increases outwards, and T is inversely proportional to r, we have r1.T1=r2.T2, so r1/t1=r2/t2=c, a constant which is
clearly the so-called velocity of light. The light represented by the polar area is not moving in this way, but an interaction
forces the cone to adopt a particular configuration instantaneously, which then gives the appearance of a velocity when
An obvious time-invariant field of study is action in the surface of a sphere, noting that this refers to its tangent planes. Surface spherical harmonics provide a suitable approach. They are especially significant when the action is linked to space as then Laplace's equation must be satisfied. They are like standing longitudinal waves in the surface. A standing wave round a circle must consist of a whole number of wavelengths to be single-valued, and a similar restriction exists in the surface of a sphere, but of course in a more complicated manner. The following image shows an example of the distribution or wave pattern for such a harmonic:
This depicts a pointwise distribution for the X(30,11) surface spherical harmonic, red for positive amplitudes and cyan for negative.
For counter space the colour of each point represents the magnitude of a surface turn in a plane tangential at that point.
Life ether is concerned with fully metric counter space linkages, which are the most rigid (compared with
affine linkages) and are seemingly unsuitable for it. But membranes are fundamental
structures in living organisms, being metric in character and yet not rigid. They govern "inside" and "outside" in a most important
way e.g. the plasma membrane of neurones which may be interfered with by drugs. Each cell is surrounded by a plasma membrane which
governs what may enter or leave.
This is what makes an organism an organism.
The synergy of such a system of cells forming an organism in this way will thus govern its growth and healing, and mathematically this kind of synergy can be related to fractals, and may explain why we find fractal forms in Nature. Also the polarities involved invoke path curves which - as explained elsewhere - are ubiquitous in Nature. A particular form of polarity that seems fruitful is the pivot transform.