The term "elements" is used here in the ancient sense: Fire, Air, Water and Earth corresponding in modern terminology to heat, gas, liquid and solid. Plasma is yet to be studied in the present context.

Given the central thesis then we expect that stresses may appear either in space or counter space or both. Should they arise only in counter space then they will manifest as forces which are difficult to explain if counter space is not taken into account (as is the case conventionally). A notable example is gravity which Newton never explained, and Einstein also only described. This was the first subject analysed, and it proved possible to obtain Newton's law of gravity, which encouraged further work. It is explained as the gradient of the stress arising from the linkage of points (see Reference 11 for the details). In counter space points are separated by a different kind of measure which is the dual of angle, and is referred to as shift. Thus gravity arises from the gradient of shift stress.

The analysis of point linkages has been used to treat gravity, liquids and gases. In each case the gradient of stress arising from point linkages is involved which lends a coherence and consistency to the whole subject. The difference between the states of matter lies in the different kinds of geometry lying behind the linkages:

• affine linkages for gases

• special-affine linkages for liquids

• Euclidean metric linkage for solids.

When space and counter space are linked then the calibration or scaling of the two spaces is important. How much shift corresponds to one metre for two points, for example? In the case of planes how is turn scaled to spatial quantities? It has been found that the ideal gas law and the behaviour of liquids is comprehensible if temperature is related to the scaling between the two spaces. This may vary throughout a body in a stochastic manner which gives rise to scaling strain and stress which we relate to heat.

Gases are studied on the basis of an affine linkage between space and counter space. The concept of a point linkage is abstract, and in practice it has proved fruitful to consider a fractal relationship between space and counter space such that every point linkage is a fractal image of the infinitude of the primal counter space involved. Different primal counter spaces are envisaged for different elements. This is particularly suited to shift which is a scale-invariant quantity, as fractals are essentially scale-invariant. A quantity of gas is seen as an assemblage of CSIs (counter-space-infinity images) which suffer affine stress as each CSI "sees" the others from a different perspective. The linkage here is affine. Hence in the primal counter space there is strain and stress, analysis of which gives the ideal gas law. This is based on the chord law:

If we have three CSIs at A B and C, the two CSIs at A and B "see" C in conflicting directions denoted by the angles alpha and beta. Their difference phi is a measure of the strain. The gradient of this strain is shown as the red arrow, which must pass through the centre of the circle because the rate of change of phi is zero along the tangent at C. It can be shown ( Reference 11) that the magnitude of the gradient at C is proportional to AB/(AC.BC), the actual value depending also upon the scaling. This is used to derive the gas law by summing the stresses for all such triangles, as illustrated below for a metric (solid) container containing an affinely linked gas.

A and B are CSIs anchored in the wall of the container and P is a free one, the chord law being applied to all such triangles for all orientations within the sphere, the nett result being PV=kT where k is a constant depending on the scaling, which thus enables the scaling constant to be found from Boltzmann's constant K.

Liquids are studied on the basis of a special affine linkage which conserves volume. A constant volume tetrahedron is taken as the basis, just as a triangle was taken for gases. The affine stress gradient is summed vectorially at the vertices, but the result is considerably more complicated than for the chord law. The following animation shows how a tetrahedron released from a particular shape evolves under the action of the stresses (from a computer model of the equations):

The two points to note are: (1) that it stabilises as a regular tetrahedron, and (2) the base to the left moves towards the apex at the upper right. These tetrahedra are only stable when regular, but the equilibrium is dynamic as the residual stresses are not zero, but in a sensitive balance, giving the fluid its sensitivity. The fact that the base moves to the apex (i.e. where the angles are initially greatest) is significant for surface tension and the way a water drop behaves, as tetrahedra within the drop are in equilibrium but those with bases in the surface are not, and strive to pull the surface inwards. Surfaces are thus the principal source of imbalance.

The next animation shows the behaviour of an initially longer, thinner tetrahedron:

Note how the form evolves slowly and rotates until equilibrium sets in quite suddenly (not fully realisable with this animation). This behaviour may refer to vorticity.

The speed of development in all cases depends upon the scaling between space and counter space (i.e. the temperature).

Flat "tetrahedra" behave chaotically, and have relevance to behaviour in the surface such as Brownian movement and also evaporation. The reason is that they have zero volume and are thus singular for special affine geometry. Again a surface is most significant for such cases.

Thus the behaviour of a volume of liquid is based on the constant volume property of special affine linkages coupled with the action of affine stress.