Bricks are really handy for building houses in a way that water isn't. Bricks maintain their shapes under pressure, much like a bond keeps its constituents in a range, and they have shapes that make building with them so much easier than say building with rough stones.

The shapes and sizes of objects matter to the way they can be used. A chair has to be the right shape for me to sit on, and there is no point building me a house the size of a shoe-box. My house must also keep the rain out and let me in. All these requirements can be satisfied using the firmness and emergence of bonds to create strong, useful aggregates like walls and roofs and doors, having particular shapes and sizes that maintain their arrangements under pressure.

Connectives like water can't be expected to have particular shapes and sizes let alone maintain them. Their extreme flexibility renders them useless for building houses but it makes them spatially useful in a different way: Since a connective distorts under the slightest disturbance, it allows a disturbance to pass through it as a wave. A bond's constraint prevents waves from passing through it. A wave would have to sublimate a bond in order to pass through it.

Figures and Embraces

Being an object in its own right, a bond may have novel properties of its own and utilize them in its interaction with other objects rather than utilize the fundamental forces. Many atoms and molecules, for example, though they are bonds utilizing the fundamental electromagnetic force, display and maintain a definite spatial shape as an emergent property, and use these shapes to interact with each other, for example by having them interlock to form crystals. Crystals have shapes too and can also interact by interlocking with each other, eventually aggregating into large objects like rocks which in turn can aggregate - using their shapes - into mountains.

Objects like bricks have rigid shapes while others, like elastic bands, have flexible shapes. Bricks can interact with each other utilizing their rigid shapes to make rigid walls (even without mortar) while elastic bands can use their flexible shapes to tightly bundle pieces of paper.

Objects cannot move towards each other where their boundaries meet because interacting objects exclude each other from their spatial volumes. The shapes of meeting objects can therefore constrain their movement, as do bricks in a wall, a key in a lock, or pieces in a jigsaw puzzle. When the shapes of objects constrain their relative movement in some way, I call their interaction an embrace.

Connectives, like water, can't interact using their shapes - they don't have any - while objects colliding and bouncing off each other may be affected by but are not constrained by their shapes.

Objects utilizing their shapes to embrace - like bricks and keys and puzzle pieces - I call figures.

The spatial patterning of figures in an embrace is governed by the geometry of their shapes. Square tiles will embrace in a different pattern to triangular tiles, for example. Figures having particularly complimentary shapes - like a peg and a matching hole - are especially suited to a tight, firm embrace.

Embraces share features with bonds because they constrain the movement of their figures, but embracing interactions are not quite the same as binding interactions. Firstly, the participating figures must be in (at least approximate) contact for their shapes to play a role in the interaction, and they must maintain their shapes and contact for the duration of the interaction. Secondly, there need be no forces acting between the figures to hold them in contact, the geometry of their shapes alone dictate the figures' constraint. Thirdly, there may be directions in which embracing figures are not constrained. A peg may easily slide out of a hole, or a piece lifted out of a puzzle, for example.

In the dimensions in which it is constrained, an embrace has the characteristics of a bond, while in its unconstrained dimensions it has the characteristics of a connective. However, an embrace always constrains the movement of its figures in at least one dimension so it will always display the characteristics of a bond, while connective characteristics are optional.

Since there is effectively no movement of figures in the grip of an embrace, the embrace will have a shape of its own and is therefore also a figure, capable of embracing with other figures. Figures can aggregate into more complex figures by embracing in the same way that bonds can aggregate into more complex bonds. And just as a new object with new properties may emerge from a bond, a new figure with a completely new shape may emerge from an embrace. A table emerges from an embrace of four posts and a board.

The similarities between bonds and embraces are many. The shape and size of a figure can be regarded as its figurate properties and these comprise its figurate identity which it can maintain in the face of disturbance. Disruption of an embrace means the loss of its figurate identity. Complex embraces display a figurate hierarchy in which their component embraces are ranked. For example a rock has a higher rank in the figurate hierarchy of a mountain than does a crystal it contains. Embraces may engage in figurate contests decided on tests of their figurate strengths while figures higher in a figurate hierarchy have authority over their internal figures.

The arrival of figures into an embrace is a discrete event. Figures are either in an embrace or they are not. Similarly, the separation of figures from an embrace is a discrete event. An arrival of figures in, or their removal from, an embrace is a step from one figurate architecture to another, as for example when a peg is inserted into or removed from a hole, or a hook is inserted into or removed from an eyelet. Figures also process rather than flow, stepping from one static architecture to another.

Though an embrace may be disrupted in a contest with a superior figurate strength, unlike a bond, an embrace may also be disrupted without challenging its figurate strength, by the motion of its constituent figures in a direction not constrained by the embrace, as a peg may slide out of a hole.

The spatial volume of an embrace may be the sum of the spatial volumes of its constituent figures, but it may also be more than their sum, for example when the embrace takes the shape of a shell that completely encloses an empty space.

A figure may be have a number of facets where each facet is capable of participating in an embrace of its own. For example, a peg may be fitted into holes at both its ends, or a peg may have many ends, of different sizes, all capable of fitting into suitable holes. All the facets of a figure need not have the same shape - an object may have the shape of a peg at one end and a hook at the other.

Embraces are not exactly the same as bonds but they are reliant on their underlying bonds to maintain their shapes and exclude each other. Many items of our material environment - trees, timber, bricks, houses, tables and chairs - are complex embraces of their underlying atoms and molecules.

While electrons and protons are spherical or shapeless, the bigger atoms and molecules display definite shapes and so can embrace as figures. The shapes of these atoms determine the shapes of the crystals they aggregate into. But as the complexity of an embrace increases, a stage can be reached where the shapes of its underlying atoms and molecules no longer play a role in the resulting shape of a figure. We can mold ceramics into any shape that pleases us, for example - a pot, a plate or perhaps a brick. A carpenter's screw can be manufactured to the same shape from iron or brass or any suitable material. Our own bodies are largely built of embracing proteins, extremely complex molecules offering a possibly infinite variety of shapes.

The many similarities between bonds and figures allow me to use the term 'object' to refer to either when I don't need to make the distinction, and to use terms such as 'architecture', 'aggregate' and 'identity' to apply to bonds and figures in general.

Waves and Vibrations

When an external interaction disturbs a connective, it will likely act first and most strongly on the objects in the connective that are closest to the external disturbance. The responses of these closest objects will then disturb their neighbours in the connective, which will in turn disturb their neighbours and so on, so that a wave or pulse of disturbance propagates through the connective. The flexibility of connectives means not only that they are easily disturbed but that disturbances can propagate through them.

An aggregate, on the other hand, can constrain its constituent objects from responding in any significant way to an external disturbance, effectively preventing a wave propagating through it. An aggregate can constrain not only the motions and external interactions of its constituent objects, it can also prevent the propagation of a wave.

But a wave could propagate through an aggregate if it sublimated the aggregate, as sound can travel through a solid object for example. The aggregate as a whole might alternatively respond to the external disturbance as an object in its own right, and if it participates in a larger connective, it will disturb its neighbours in the larger connective so that the wave passes through the larger connective.

The following is a summary of the major features of waves for readers who are not familiar with them:

If an external disturbance cycles with a regular frequency the objects of a connective will vibrate in sympathy with the external frequency as the wave passes through. Some connectives, by their very nature, respond even to a single pulse with a vibration of their participating objects, in this case with a frequency of their own.

The distance or time between the extremities of a cyclic disturbance or vibration is called its wavelength. The strength of a disturbance - the strength of its wave and the size of any vibration it induces - is called its amplitude. The amplitude of a wave describes how strongly each object that it traverses is affected by it.

Sound offers a good example of a wave traveling through a connective. A plucked guitar string disturbs the air around it sending a wave through the air that we hear as a sound. Plucking different strings on the guitar generates waves of different frequencies while the amplitude of the sound depends on how forcefully a string is plucked.

Waves travel smoothly through their host connectives. While an external disturbance may act most strongly on the objects in the connective that are closest to the disturbance, objects in the connective that are a little further away are also disturbed but to a lesser degree, while those even further away are also disturbed, even more slightly, and so on. There is a smooth flow of a wave through a connective rather than a series of sharp, distinct bumps from one object to the next. Any vibrations induced in the objects the wave passes through are also smooth, covering all the positions between the extremities of their vibration rather than jumping from one extremity to another, as is the case in the discrete steps of a procession. Perhaps the equivalent in a procession would be a cascade of discrete reconfiguration events, as when a bottom item in a grocery display is removed and other items shift in a series of discrete events to fill the gaps as they appear.

Multiple waves can propagate through a connective at the same time and even disturb the same objects at the same time. Unlike an object, a wave does not occupy a space to the exclusion of other waves (or to the exclusion of objects).

When multiple waves disturb an object at the same time, the net effect on that object is the sum of the effects of all the waves - no effect is excluded. As well, each of the waves continues on its original way after their meeting at the one object, that is, they pass right through each other. This is known as the ability of waves to interfere with each other. Compare such interference of waves to a collision of objects - which cannot occupy the same space at the same time and cannot pass through each other.

When multiple waves traverse the same object at the same time, the net disturbance to that object is the same as a single wave equivalent to an interference of the contributing waves. Such a single wave whose effect is equivalent to that of multiple waves is called a superposition of the contributing waves. In a way similar to the visages of connectives, it is not always possible to tell if a vibrating object is responding to a single wave or to a superposition of multiple waves.

Waves can have patterns. They can have patterns in time, for example as a sound that gets louder and softer; and they can have patterns in space, like ripples on a pond. When waves interfere with each other, their patterns do too. Their interfering patterns could have a very choppy result as a pond would have on a windy day, or they could have a very pure result like a note from a finely tuned musical instrument.

When the patterns of meeting waves are perfect opposites of each other they cancel each other out as the highs of one are canceled by the lows of the other, so that the result of their interference is no vibration at all, in a phenomenon called destructive interference. On the other hand, when meeting waves have patterns that are perfect matches they will reinforce each other, as the highs of one coincide with the highs of the other, to produce even bigger vibrations, in a phenomenon called constructive interference or consonance as I like to call it.

It can happen that when a wave sublimates an object it will be reflected to and fro inside the object, and should its wavelength match the size of the object the reflections will interfere with each other constructively, allowing the object to amplify the wave. This is known as resonance, and the wave is said to resonate inside the object. If the wave persists for long enough, the resonance in the object may even become strong enough to disrupt the object.

While a wave's pattern travels with it, the interference of many waves can sometimes result in a superposition having a pattern that appears to stand still. These are known as standing waves, when, for example, a guitar string plucked in its middle shows a pattern that does not move even though there are waves traveling up and down the string, always having a zero amplitude of the string at its ends and a large amplitude in its middle.

A combination of effect applies to connectives in general. The effect of multiple waves on an object is the net contribution of all the waves acting on the object. The effect of multiple forces on a participating object of a connective is the net combination of all the forces acting on the object. Compare this to the selection of some effects and a negation of others as happens in the contests of bonds and embraces.

Constellations

Sometimes waves and embraces display simple, clear patterns like ripples on a pond or bricks in a wall. In other situations their patterns can be chaotic, like a river churning through rapids or a jumble of rocks, for example. But sometimes they arrange in patterns that hold a charm for us humans - like the symmetry of a snowflake or the spiral of a galaxy - in which case I call them constellations.

In aggregates a constellation may endure, like a snowflake, or only appear at certain steps in a process, as a fractal picture may become striking with a only a very particular set of parameters.

In connectives a constellation will likely be fleeting, appearing as the connective changes, before evolving into a different constellation or dissolving into chaos again. A cloud may momentarily constellate into the shape of a castle, or flare rainbow colours, or display a fine filigree.

Barriers and Containers

The objects participating in a connective may not all be the same. They may have different sizes or different masses or different charges, for example. Being a connective, the objects are not embracing so their shapes don't matter, but if one object is bigger or more massive than its neighbours it could have a relatively muted response to a wave, while a smaller object will have a relatively exaggerated response. This would slow the wave down or speed it up or change its direction (in processes known generally as refraction and reflection).

An aggregate participating in a connective may be so massive (in comparison to its neighbours) and its response to a wave so muted or reflective, that it would effectively prevent waves from propagating beyond it. Large objects can be barriers to waves or change their direction. Indeed, large objects can be barriers to any motion within a connective, so that a suitably shaped aggregate could direct the motion of an entire connective, as, for example, a flow of water can be directed by a pipe.

A large aggregate could be constructed to take the shape of a closed shell or casing so that it completely encloses the objects of a connective in its interior. In this case, the aggregate not only prevents the connective's participating objects from moving outside the enclosure, it prevents disturbances in the connective propagating beyond the enclosure as well. The connective has been contained - it cannot escape, and any waves in it will not be able to escape either (assuming they do not sublimate the container).

Contrarily, a container may exclude a connective from a space rather than enclose it within one, or prevent waves from disturbing a contained connective. Containers may be used to separate one connective from another - and connectives may be used to isolate one aggregate from another.

A rigid container can hold a contained connective to a rigid figure. An external disturbance that imposes a motion on a rigid container imposes the motion on any contained connectives as well.

Containers may not be rigid. Containers constructed from bonds having some elasticity can serve as membranes, like a balloon for example. Membranes too can separate one connective from another but cannot hold a connective to a rigid figure.

So in addition to bonds, containers can also constrain connective interactions; but in this case the constraint is external to the connective, that is, it is an extrinsic rather than an intrinsic constraint.

If you are interested, there is more discussion of this in Appendix 1.

Connective Influence and Architectural Control

Containers and barriers extend the capacity of aggregates to exercise control.

An aggregate controls the motions of its internal objects by constraining them within definite ranges, it controls the hierarchical allegiances of all its internal objects and their ability to participate in external interactions, and it can also exercise control over waves and connectives as a barrier or container.

Connectives and waves have none of these capacities for control, and while they influence events they are incapable of having any specific control over their effects since they are always open to interference and disturbance.

This does not mean that connectives are not capable of being precisely controlled, only that an aggregate needs to be in place to control them. A light beam can be targeted very precisely but not without the intervention of a rigid collimating mechanism.

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