Local and global
geometry
This chapter will examine scale and grouping, the whole relating to the part, and the unchanging rules of existence.
Perspective
A local view is to and from a particular point that views back locally. A global view is to and from all points, which also look back globally. When very close to the surface of an object, the surface appears flat. The greater the distance from the surface, the more curvature becomes apparent. The local view of a wavefront, (even a circular one), is one of no curvature. Local flat observations can be made at any scale with an appropriate observer or probe that can get close enough in size or wavelength (frequency) to observe (interact) locally in harmonic resonance with the observed wave.
The Earth is flat or round, depending on a local or global perspective. Space is flat or curved, depending on perspective.
One can be too ‘close to a topic’ to be objective about it. A more removed global perspective objectifies and diminishes individual objects of projection and reflection, and their influences. Local objects strongly locally reflect each other, and suffer reduced amounts of direct reflection as more distance between the two objects is gained, and a more global indirectly reflective perspective is achieved. For a more global focus, the individuals are all considered constituent parts of a single group. The more global the perspective, the more curvature and a contradiction of perspectives is resolved and embraced. With all local images minimized, a higher global structure emerges as a reinforced interference pattern in the same image as its abstract origins. This higher global pattern and others like it will interfere together on a higher local level and become part of some even greater pattern.
Wave
expansion
The depth and range of an entity’s resonant structures’ abilities to interact and be locally or globally co-observed may become unendingly subtle and complex, but their simplicity is finite. Most objects are complex composite interference pattern objects and are made out of somewhat less complex composite objects.
Each interference pattern object is composed of lower interference pattern objects, themselves interfering to create the higher being. Layer upon layer of lower complexity becomes and supports higher complexity.
After factoring out the interference patterns down through each level of existence, the remaining patterns are simple propagation objects with no interactions or overlap of waveform to create a higher-level interference object.
There are three simple propagation waveforms. They can be illustrated by two-dimensional or three-dimensional models. All three shapes depict an aspect of change and represent different traits of a wave and of being. Their structure is the shape of their different propagations.
In these propagations, the waves continuously actively maintain their specific shape. There is an actual force holding these waves in specific shapes. Waves that hold their shape as they propagate are called autocatalytic waveforms.
The first of these autocatalytic waveforms is that of the expanding sphere or circle, propagating directly out away from a central point in every direction.
This is seen in three dimensions as the shock wave of an explosion in the air or in two dimensions as a ripple on the surface of a pond. This wave fluctuation comes from a single pulse origin. Continuous pulses from a relatively still point creates an unending set of concentric wave circles surrounding that point. The wave pulse travels out and away from the origin in all directions at the natural speed of a wave in that particular medium and environment.
The second waveform is the propagating cone. The source of this wave is a point that continuously moves faster than the wave it creates.
This waveform itself propagates diagonally, perpendicular to its conic surface. The shock wave cone seen in two dimensions is the V-wake of a speedboat. In three dimensions, this cone is seen in the sonic boom of an aircraft traveling faster than sound.
The third autocatalytic waveform in three dimensions is the torus or smoke ring. The torus is generated by a single pulse in a single direction.
The smoke ring travels slower than the natural speed of a wave in its medium. It is also seen in two dimensions as the twin vortices of an oar stroke on the surface of the water.
The difference between two dimensions and three in all these examples is one additional axis of rotation perpendicular to the direction of motion.
While the sphere and the cone expand over time, the torus does not. Instead, the torus propagates only along one dimension. While the waveform of the expanding sphere (expressing position) and cone (expressing momentum) will eventually pass across and transition all observers, the torus, exhibiting both angular and linear momentum, expresses a direct exchange from one point to another.
Cone,
sphere and cylinder
Each of these three autocatalytic waveforms may be seen as a dynamic wave in motion, or as a static object. When a plane is passed though each waveform as a static three-dimensional object, a lower-dimensional section is described.
Passage of a plane through a sphere creates only an expanding and then contracting circle in any case.
Plane passage through the cone and torus are more complex.
A plane will intersect a cone in an ellipse, as long as the plane crosses the rotational centerline of the cone and does not become parallel to the edge of the cone.
Any cone, no matter how narrow or wide, can be sectioned congruent to any describable ellipse, no matter how elongated or round the ellipse may be. An ellipse is a closed curve within which there exists two points such that, from all points on the ellipse, both internal points can see only the reflection of the other point.
If the plane is perpendicular to the centerline of the cone, the section describes a special case of the ellipse, the circle. In this case, the two points of mutually observed reflection in an ellipse become one point.
If the sectioning plane does become parallel to the edge of the cone, then the curve described is the parabola. Of the two reflecting points, one is now infinitely far away from the closed side of the curve, and the open ends of the curved line eventually approach straight and parallel to each other at infinity.
If the sectioning plane cuts the cone at an angle beyond the parallel of the cone’s edge, then the two open ends of the sectioning curve will be straight but diverging at infinity. This is the hyperbola. Here the point of focus is not resolved even at infinity, but is only inferred abstractly as a virtual image on the other side of the reflector.
These three possible ways to section a cone correspond to the three classes of reflector. In the ellipse, the focus of a reflection is within the same closed loop as the object. In the parabola, the focus of the object is at infinity (in both directions). The hyperbola shows the local reflection of an object to be focused on the other side of the reflector.
The three conic sections also define the three types of paths of motion that may occur for two objects moving with respect each other in space, where one relates gravitationally to the other either elliptically, parabolically, or elliptically.
The waveform of the torus needs conceptual clarification. Space can be seen as curved or flat. If one were to look into a strong enough instantaneous telescope, one would see the back of one’s head. A torus is the curved version of the flat expression that is a cylinder. What is a cylinder in a flat space can be a torus in a curved space.
There are two axes that a flat surface can curve through, X and Y. Either axis alone leaves the surface flat if it is unchanged. For a surface to be closed, both axes must be curved.
A surface closed along one axis and flat along the other axis is a cylinder. The torus, curving along two axes at once, is a surface closed in two different directions.
In sectioning a cylinder with a plane, only an ellipse or a circle can be described, depending on the angle of the sectioning plane.
The eccentricity of this ellipse is limited to the length of the cylinder, relative to its radius.
It is more than interesting that the cone, sphere, and cylinder have an integral volumetric relationship. The volume of a cone with a height on one and a base diameter of one plus the volume of a sphere with a diameter of one equals the volume of a cylinder with a height of one and a diameter of one. If the cone has a volume of one, then the sphere has a volume of two, and the cylinder has a volume of three. 1+2=3
If all the common factors are reduced out of the equations for their volumes, the basic relationship between the height and radius of these solids is exposed. H=2R
Conic
sections
A fuller description of conic sections needs additional comment. For the sections of a cone that are the ellipse and the parabola, nothing more is needed, but the hyperbola includes another cone for a full description. In a sectioning plane that is angled beyond parallel to the edge of the cone, the plane will intersect a second cone that is inverted, relative to the first cone and extends up, tip down, from above the first cone. The two cones meet at their points.
A hyperbolic section will cut through both cones, but the curved intersection of the conic and a plane (nearest the apex) may be closer to the mutual apex in one cone than in the other cone.
Both sections are more curved when closer to the apex, and more nearly straight when away from the apex. As the curved closer part of the section reaches the apex, the section curve can only appear to be two diverging straight lines going directly away from the apex point.
When the intersecting plane actually crosses the apex of the cones, the three varieties of curve define either an elliptical (or circular) point, a parabolic pair of straight lines running down the edge of one cone infinitely close together and up the other cone similarly, or a hyperbolic pair of lines diverging from the apex of each cone in opposite directions and at similar angles.
When the section of a cone is visibly away from the apex, the view is a local one of the curvature of the section. When the sectioning plane is infinitely close to the apex of the cone, the view is then one of a more distant global perspective. The sectioning plane being close to the apex is the same as an observer being far from the apex. Local and global differentiates between two straight lines moving away from a point and two curved lines that tangent the point. One thing can be seen as two different things, and two different things can be seen as the same thing.
Ant
Colony
Existence as a single thing on one scale does not preclude existence as part of a greater whole on a more global scale.
An ant is an animal by all counts, doing ant-like activities. A more global view sees the ant as just a part of a larger entity, the ant colony. On this larger scale, the ant is just a part of a greater living entity that exists independently of the existence of any particular ant member. The goals of the ant colony are similar yet different from the goals of any individual ant.
Gaia, a name for the living mother Earth, is seen this way. We current living species are only a temporary part of the greater global creature.
Slime
mold
A slime mold is a single celled creature, much like an amoeba. The slime mold culture shows a complete transition in its member cells from individuality to all cells becoming assimilated parts of a single greater whole.
A culture of slime mold cells will grow and divide in a normal single celled manner at first. Then a transformation begins. At some point, the entire culture of cells will begin a pulsation. This pulse starts in the center and pulses outward to the edge of the culture, moving the cells themselves toward the outer edge and into a ring. This pulsing continues, and then starts to pulse around in a circle through the ring of slime mold cells, around and around over and over. As the pulse circles the ring of cells, a beginning edge and a trailing edge of the pulse become defined. Now the circling pulse has a beginning and an end. The circling pulse then slows and stops. The culture of mono-cellular individuals has now self-organized to the point that they behave as a single multi-celled creature. The beginning and the end of the ring of cells are the head and tail of a new higher creature. It just uncurls itself from a circular position and moves off in an inchworm style, looking for its greener pastures. Finding a good location, the slime mold changes again, this time stretching way up as a lifeless stalk with a spore sac at its top, ready to propagate again.