With introduction of the diamond grid as a complementary alternative to the P-R grid, what the code specifies with regard to 2-dimensional surface design is largely completed. To imbue constructs in these 2 grids with a dimension of depth (or height), guidance comes from information intrinsic to the geocentric cuboda.
As such, the cuboda serve a function analogous to that of a prism. Typically, a prism is regarded as a transparent medium of simple geometry through which white light – the perceptual interpretation of the thoroughly jumbled visible light spectrum – is separated into its constituent wavelengths familiarly recognized as the colors of a rainbow. On the other hand, the terrestrial or bodal prism can be viewed as a geometric form by which random natural terrain is separated into a set of distinguishable ground waves.
Such a set is quantized just as is light emitted from atomic or molecular electrons dropping from a higher energy orbit of whole number (particle) wavelengths to another. In the bodal prism analogy, earth would be the nucleus and the extraneous geometric elements of the outer bode determine the shapes of waves. So regarded, the bodal prism has 4 incarnations corresponding the bode’s square, triangle, edge, and vertex-up positions which are oriented at any specified location by way of primary longitudinal and secondary latitudinal rotations relative to earth.
So positioned, the information abstracted from the bode are the angles of its symmetrically sloped elements (square, edge, and triangle) relative to the local horizontal. That set of angles is small at 7, but fairly evenly spread between 0° and 90°. Of the 7 angles, 30°, 45°, and 60° are exact, while 19°, 35°, 55°, and 71° are inverse tangent approximations of 1: 2√2, 1:√2, √2:1, and 2√2:1 respectively.
Some of the angles (35°, 45°, and 55°) are manifested twice with 2 different sloping elements from 2 different orientations. Within each orientation, any angle employed will have either a steeper inherent or less steep extra-terrestrial angle(s) or both. For example, if focus lies on the vertex-up orientation’s 30° sloping edge, its inherent angles are 35° and 45°. Such distinctions and their interplay find application.
What these angles have to do with waves is grasped by examining the nature of any generic wave. If horizontally oriented (as they are with ground design), the wave is shaped such that lines tangent to both trough and crest are 0° or horizontal.
Proceeding upward along the wave from the trough, the angle of lines tangent to the wave steepen until they reach a maximum exactly at a point exactly halfway between trough and crest in both the vertical and horizontal dimensions. From such a maximal point, the slope grows increasingly less steep until it again reaches zero at the crest. The maximum slope naturally makes any wave divisible into quarters or halves because in each of these possibilities the point and its slope serve as a distinguishing identifier.
As the wave has one unique point of maximum slope, bodal prism angles keyed to shape the wave manifest the bode in a distinguishable way according to its orientation and element. With a quantized aspect of allowing only certain maximum slopes, a kind of yin yang symmetry is reached with the aspect of wave slope continuum which makes the wave the ideal form to transition between one level and another.
In an instructive aside, knowing that the maximum slope is at the halfway point has some practical use. Years ago, on a bicycle trek to Alaska, I came upon a series hills which triggered, in their near perfect sinusoidal wave shapes, recollection of the maximum slope attributes. So regarded, my goal for each new hill became not the top but rather the halfway point toward which I rushed aggressively to find the challenge made much easier.
The maximum slope of course can be expressed as the ratio of the wave’s length and amplitude and a wave so specified also exhibits a corresponding height to length ratio. in the table below, the length refers to the horizontal distance between trough and crest.
The conventional parameters of any wave analysis are wavelength and amplitude. Whether or not their ratio or maximum slope has any deep physical significance is unknown to me. But how such waves are applied to landscaping constructs in their grid contexts comprises most of what remains of Ground Design.