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Information and The Structure of Worlds: A Speculation. Robert M. Hayes.
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Information and The Structure of Worlds:A Speculation Robert M. Hayes
Before proceeding with this presentation, I want to acknowledge a debt to Professor Marcia Bates, my colleague on the faculty at UCLA. The speculation I will here present is a direct result of an E-Mail dialogue between the two of us during a month-long period in early 2004. It was initiated by Marcia as part of her process in developing a paper that was subsequently published by her. • Our discussion forced me to deal with the issues she raises in that paper, but in my own way. If there are errors in what I present, they are mine, of course, not hers. But undertaking these speculations was due solely to the stimulus of her creative ideas.
Summary • 1. Introduction • 2. A Generic World • 3. Examples of Worlds • 4. The Universe of Worlds
1. Introduction • What I will present here examines a number of separate but related contexts, called Worlds, by applying a fairly simple framework to each of them. The framework consists of the following elements: (1) An Objective or Purpose for a world (2) A Space, with substance and structure, within which everything related to that world occurs (3) Particles that are the substance of the space (4) Dimensions that are the structure of the space (5) Entities, as things that exist in the space, formed from the particles of the space and with a continuity of identity and structure (6) Processes by which entities are either created from particles or are changed by interactions among entities (7) The derived structure of the set of entities that results from those processes • The entities and their derived structure may be treated as the substance and organization of a new, more conceptual space, to become the substance and organization of a new world.
Overview of Structure • Thus, the simple conceptual framework can be seen as a means by which dynamic changes in the structure of a world can lead to the identification of entities that retain their continuity of identity despite those dynamic changes and fall into a new, derived structure. • Throughout that process, structure plays a central role. It appears in the structure of the original space. It appears in the structure of each entity. It appears in the structure among sets of entities. It appears in the structure in identification of the new world. • What is structure? It is the organization of something, or the pattern of organization of something. In some terminology, the pattern of organization of something is the Information in that something. In my own terminology, though, information is a measure of the structure rather than being the structure.
An Aside on Structure and Information • My use of the term “information” is based on a generalization of the traditional Shannon measure of information as applied in systems for communication. It is based on probabilities. • If my usage is to be meaningful, for information to measure a structure, it must be related to the probability of that structure. But what does that mean? • Beyond that, presumably the more complex the structure, the less likely it is and therefore the more information it contains. So how does one define complexity? • I will try to deal with both of those questions.
Summary of Presentation • In the following presentation, this simple framework will be illustrated by applying it to a succession of worlds. First among them will be a “generic world”, number (1) in the list that follows, in which the overall framework is illustrated independent of specifics but with special attention to the definition of symbolism that provides means for talking about elements of the framework in a precise way. That is then followed by examination of each of the specific examples, from (A) to (L) in the list that follows. Each will be discussed, not in detail but as means for exploring and illustrating the several elements.
The following matrix summarizes elements for the several contexts that will be discussed:
Objectives • The objectives of the thirteen spaces certainly are quite different. That of the physical world either is simply existence (its objective being just to be, as perhaps the existentialist would view it) or it is the will of God (as perhaps the religionist would view it) though it is impossible to tell what that is. • For the observer world and the perception world within the observer, the objective presumably is to provide means for the observer to deal with the external world and, from an evolutionary perspective, to be able to cope with it and to survive. • For the worlds of processing, concept, and communication, the objectives are more difficult to characterize, deriving from the contexts in which they occur. All three represent the means by which persons, as human beings, can relate to each other and to the worlds in which they live and interact. • The objectives of the academic and disciplinary worlds relate to larger frames of reference. They each step outside whatever other worlds there may be with the intention of learning about those other worlds.
Objectives • The objectives of the documents world are evident. They are to support the worlds of communication, society, culture, academe, and discipline by preserving and making available for future use the messages by which interactions take place in each of those worlds. • The objectives of the economic world presumably are efficient and effective use of resources for the well-being of mankind, either in general or in self-oriented specifics. • The objectives of the religious worlds are more difficult to characterize, since they may be seen either from the perspective of the driving force behind them – some concept of God – or from the perspective of the religionists themselves. In a sense, the dichotomy is reflected in a basic question, “Did God create mankind in God’s own image or do people create God in their own images?”
An aside on Implicit vs. Explicit Objectives - 1 • It is worthwhile to explore the difference between objectives that are explicit in the sense that they are defined by an external being versus those that are implicit in the sense that they simply derive from the fact of being. • The distinction is most clearly exemplified by the battle between, on the one hand, the “creationists” or those committed to the view of “intelligent design” as the basis for life as well as for the physical world as a whole and, on the other hand, the “evolutionists” who are to one extent or another committed to the view that the basis for life is simply the processes of the physical world itself. • The creationists believe that God created the world and life within it and that, in doing so, God set the objectives of the world and of life within it. The objectives are therefore explicit as God’s objectives. • The evolutionists believe that the world simply exists and that life within it is a result of the natural processes that occur within the world. The objectives are therefore implicit, deriving from what happens.
An aside on Implicit vs. Explicit Objectives - 2 • Later, we will return to this distinction when we consider the differences between entities that are externally defined, explicit, versus those that are internally defined, implicit. The effects of those differences are exhibited especially in conflicts between the two if there are explicit entities. • The issue of implicit vs. explicit objectives, though, arises from the very reason for looking at a world. One may do so simply because it is there and one is trying to understand it. • But it is possible that, in a very real sense, one can create a world for the purpose of examining it, exploring it. A world of fiction is in that sense as valuable as a world of fact and in some respects may be even more valuable. • In such a context, it is important to note that the person, in creating such a world and then examining it, plays a double role. And there may be difficulties in separating the two roles. • In a very real sense, the religionists, especially when they base their world upon the Bible, are creating their world. Of course, they view the world as God’s creation and that may well be. But this illustrates the difficulties in confusing the two roles of creator and examiner.
Particles • In each of the worlds, the substance of the space consists of what will be called particles, that is as fundamental units of substance that are individually not divisible. • Of course, particles may change over time in their locations and attributes, but not in their individual identity. • The nature of the particles in each of the spaces will be quite different and, indeed, that is probably the most basic defining characteristic of each space.
Dimensions of Time & Location The several spaces involved in the twelve contexts also can be quite different. • Dimension of Time. Time has a very different character in each of the worlds. Of course, in the physical world and, to some extent the observer world, time is simply time as we generally think of it. But in much of the observer world and in each of the other worlds, time is really quite different from simple physical time. The two are likely to be related, of course, but not necessarily so. As we look at each of the thirteen worlds (not in detail, but as examples), it will be necessary to explore the different meanings of time and the relationships between those meanings and simple physical time in each of them. • Dimensions of Location. The dimensions of location for the physical world, the observer world (in part, at least), and the societal world (again at least in part) are clearly Euclidean space. Physical things and persons are located in the physical world; they interact with each other in the physical world. So the dimensions of the physical world are central to determining their physical locations as they interact. • However, having said that, it must also be said that, in other than the purely physical world, things may be located by other than merely physical locations. Indeed, in some worlds, physical location may be irrelevant to some of the processes in the world.
Dimensions of Attributes • To deal with dimensions other than location, we need to consider the dimensions of substance (what are called attributes) because they become not only means for characterizing the substance but for locating particles of the substance in space. Simply to illustrate, within the societal world, persons interact not simply because they are physically located nearby to each other, but because of their relative positions in their attribute dimensions – their age, sex, social position, cultural background, personal wealth, etc. In fact, for some interactions among persons, being physically located nearby may be irrelevant to the interaction (though, of course, it may be relevant). • The key point is that the dimensions of attribution may serve two roles. One is to characterize the nature of particles as the substance of the world. The other may be to locate the pieces of substance within the world of attributes so that the interactions among pieces can occur within the context of those attributes.
Sub-Spaces • In some worlds, it may be desirable or even necessary to identify sub-spaces whose structures are sub-sets of the total set of dimensions. This arises if the particles fall into categories to which only some of the dimensions apply. • To illustrate, when we examine a Perception World, the particles will be sensations, such as those occurring in the senses of vision, hearing, smell, tactility, balance, pressure and temperature. Each of those senses has attributes that are quite distinct from the others. • Of course, the entire set of attributes applies to the entire space, but it is useful to be able to sub-divide the total space into sub-spaces, one associated with each of the senses.
Selections of Particles • Throughout the subsequent discussion, “selections of particles” will play a central role. A selection of particles is simply that; it could consist of all of the particles (the universal selection); it could consist of a single particle (an atomic selection); it could consist of no particles (the null selection). • And also throughout the subsequent discussion, “similarity of selections” will also play a central role. The measure of similarity is a function of the attributes of the particles in the selections involved. The nature of the measure can vary widely, including consideration of the physical locations of particles, the attribute locations, the nature of the individual particles as defined by their attributes, the numbers of particles, the physical or conceptual arrangements of particles, the stability or change of any of those aspects.
Structure and Similarity of Selections • There are two considerations and related measures that will play significant role with respect to selections. The first is Structure and the second is Similarity. • Structure, at least initially, will relate to individual selections. • Similarity, at least initially, will relate to pairs of selections.
Measures of Coherence & Continuity • A specific and rather crucial example of structure and similarity is that representing physical cohesiveness or coherence. It embodies the fact, first, that a selection “holds together” and, second, that it continues to do so over time. • This applies to the things we tend to identify in the physical world – the chair or table in a room, the person we are talking to, the sun or moon in the sky. • How can we define entities that we so readily identify in the real world, like those just listed? And the answer to that question is by no means simple or trivial; it is a very complex question, the answer to which will be presented when the generic and physical worlds will be discussed. • In a sense, the answer, as I will present it, is perhaps the simplest example of structure and similarity. The structure might be characterized by what will be called contiguity or adjacency of particles. The similarity might be characterized by the percentage of particles that two selections have in common.
Entities • So, among the possible selections of particles are those that will be called “entities”. The definition of an entity in each world is determined by the nature of structure and similarity as they are applied to sets of selections of particles of substance. • The relationship between structure and similarity and the entities that they define is, of course, very close. Indeed, it is sometimes difficult to tell which comes first. In the definition of an entity, it is assumed that structure and similarity are defined first, so that the entity then arises from those definitions. But the fact is that one may want to talk about a kind of entity and must then determine what structure and similarity would lead to such an entity. That having been done, of course, the definition then becomes the basis for defining the entity as it was wished to be. • There are many possible criteria for structure and similarity. The most obvious ones, perhaps, are those related to the dimensions of location in the physical world. But those related to the dimensions of attributes are also important. In particular, they may be especially applicable in those worlds that involve people – those that lead to identifying communities because of structure and similarity in the attributes of people.
Processes • A crucial point about each world is that it is dynamic. It changes over time. And those changes result in interactions among selections of particles within the world and, indeed, may be caused by those interactions. Those interactions are caused or created by processes, and those processes differ from world to world. • In the physical world, of course, the primary processes are the physical and chemical interactions in the substance of physical space. They operate at both a gross level and an atomic or sub-atomic level. The study of them is the domain of the natural sciences, physics and chemistry especially. • In the world of the observer, as in perception and in processing, those processes are neural (in animals and perhaps even in plants) or electronic (in computers and similar man-made devices). They are embodied in the built-in structures of logic that serve as the building blocks for the more complex programs that either grow out of experience or are inserted de novo.
Processes • In the worlds based on people – communication, society, and culture – they are the bases of inter-personal interaction, both physical (as in mating and warring) and conceptual (as in arguing, joining together, convincing, encouraging, inciting). • In the academic and disciplinary worlds, they are the very processes of research and teaching – the tools by which objects of investigation are brought together into new configurations that identify issues of importance in understanding the relationships among entities. They are the paradigms of research. • In the world of documents, they are the means by which documents are created, acquired, organized, and, most importantly, used. Two interesting examples of those processes are “citation”, as the reference from one document in another, and “co-citation”, as the joint reference of two documents by another document. But that is simply one pair of examples.
An aside on Limits on my Objectives • It is important to note that there are significant limits on my objectives in developing this model and they arise specifically with respect to the Process component of the model. • It is not my objective to deal with the specifics of the processes. This arises, in particular, when I examine the real physical universe. The physicist is concerned with the specifics of the dynamic process, as represented by the equations that govern them and measure their effects. • Obviously, this limitation for my objective makes the model, as such, valueless for the physicist. • In the same vein, I do not try to deal with the specifics of biological processes or political processes, as they reflect the dynamics of biological, political, or economic systems. • Obviously, this limitation for my objective makes the model, as such, valueless for the biologist, politician, or economist. • Given that, one might ask what value the model has. And the answer simply is that I am trying to provide means for examining the similarities among the several worlds rather than the differences among them.
The Derived Structure of Space • I turn now to the final element in the framework, the derived structure for the space. Specifically, I will define it as follows: “The derived structure of the space is the decomposition of the lattice of similar selections of particles under the relationship of containment into the direct product of sub-lattices”. Let’s briefly examine what that means. • Containment Selections of particles in the space are characterized by several levels of “containment”. Selections may be components of other selections, and selections may be grouped into categories based on some criterion of similarity.
The Derived Structure of Space • Meet and Join of Selections We can then consider the sub-selections that two selections may share in common, called the meet, and the selection that results from combining two selections, called the join. Of course, for some pairs of selections, the meet may be the zero selection because they have no sub-selections in common. • Lattices As a result of the containment and the meet and join of selections, the most important result of the dynamics of processes operating on selections is that the selections relate to each other within the structure of a standard mathematical concept of a lattice.
The Derived Structure of Space • Decomposition of a Lattice The resulting lattice can be represented as the “direct product” of sub-lattices within it. That direct product constitutes a “derived structure” which provides a new way of looking at the substance of the world and thus a new world at a higher level of abstraction. The component sub-lattices constitute the dimensions of the space in that new world. • The Dynamics of the Derived Structure Note that the selections of particles in the world, the relationships among the selections, the lattice of those relationships, and the resulting dimensions for the selections all arise from the dynamics of what happens in the world. They are not pre-defined; they are not inherent in the original space or its dimensions; they are the result only of the interactions that actually occur among the selections, as they are created and as they interact.
2. A Generic World • A generic world will be used as the means for describing the general pattern and for defining symbolism that will be universally applied in each of the more specific worlds. • Each of the illustrative worlds, as previously listed, provides an example to which the generic world can be applied.
Overview of what follows • Generic Objective • Resulting Space and its Structure • Particles • The Matrices of the Space • Selections of Particles • Structure of a Selection • Selections in Time • Sets of Selections • Similarity of Selections • Dynamics of Particles • Continuity of identity of Selections in Time • Entities • Structure of an Entity • Information and Complexity • Induction from Continuity of an Entity over Time • Identification of Equivalence Classes • Structure of the set of Equivalence Classes • Derived Structure of Space • Measurement of Space
Generic Objective • There is an Objective or reason for existence of a world. The crucial point about the objective is that it determines what the space for the world will be, what the structure of it is especially as represented by its dimensions, what things (substance) will be included in the world and what their form will be, what criteria identify entities within it, what the processes are by which entities are created and interact, how the output from the world will relate the world to whatever is outside of it. • The objective for a world may change over time both as a result of a change in the reasons for identifying it and as a result of its own operation, the latter especially for an objective that is self-defining. • The objective for a world may be simple or it may be complex, including sub-objectives, in which case the objective will have a structure relating those sub-objectives.
Particles • A given space will contain Substance, located within the space and made up of particles. It is assumed that there is a finite number of particles, even though that number is likely to be exceptionally large. Each particle, when present in the space, has a specific location in space at a given point in time. However, particles may appear and disappear at different points in time. • A given particle will be permanently identified by a number determined, in sequence, from the time of its initial appearance in the space and the location of it in space at that specific point in time. It is assumed that only one particle can make its initial appearance at a given location at a given point in time. Therefore, that initial time and initial location can serve as the unique identifier for it at any subsequent point in time. • It is assumed that, at the origin of time, only one particle can be at a given location so that the initial location at the origin of time is a unique basis for identifying each starting portion. However, at future points in time, varying numbers of particles, from zero to the totality of particles, may be at a specific location.
Structure of the Space • The space will have dimensions that represent its basic, underlying structure and provide means for identifying the location and form of substance (i.e., particles) within the space. The distinction between location and form is meant to be evocative of counterparts in the real physical world, but it is not an essential distinction, since the concept of dimension includes both without essential difference between them. • The world is to be seen as dynamic, changing over time. Therefore, there must be a means for measuring time in the space so that, as change occurs in the space, it can be related to the point in time at which that change occurs. There is a specific point in time, called the origin of time, at which the world is taken as beginning to exist for all subsequent times. That will be the dimension of time. • Among the dimensions of space are those that provide the means for locating particles within the space. The dimensions may be physical, in the usual Euclidean sense of dimensions, but they can be other than physical. These will be the dimensions of attributes.
An aside on Einstein and Time • Albert Einstein is reputed to have said, “The only reason for time is so that everything doesn’t happen at once.”
Structure of Space • The dimensions of time, location, and attribution provide the basis for the structure of the space. • However, although a given world can be treated as a uniform space, there is frequently value in subdividing it into components, each of which may have different properties, and indeed different dimensional structures, within the overall context of the space. • This kind of subdivision creates a structure for the space in which the several components can each be treated separately or the entire set of them treated as one space, with proper means for reconciliation of different dimensional structures.
Dimension of Time • While, in principle, time could be measured on a continuum, it will be assumed that time is measured in finite units of time, the size of a unit of time defining the resolution of the measurement of time. Let that unit of time be 1. The origin of time will be taken as 0, so that all times are positive integers. • A set of times, A = {t1,t2, … ,tk}, is in time-sequenced-order if, for 1 < i < k – 1, ti≤ ti+1. If the set of points, A = {t1,t2, … ,tk}, is in time-sequenced-order and ti+1 = ti + 1, for 1 < i < k – 1, it is called a time interval and symbolized by (t1,tk). Note that, for a time interval, A, if any point in time, t, is such that t1≤ t ≤ tk, then t is included in A. A set with a single point in time, t, would be the time interval (t,t). • A set of points in time-sequenced-order, A = {t1,t2, … ,tk}, ti≤ tj for i < j, can be divided into m discrete time intervals (t1i,t2i), t11 = t1, t2i + 1 < t1(i+1) and t1(i+1) = t2i+1, 1 ≤ i ≤ m - 1, t2m = tk.
Dimensions of Location • Among the dimensions of space are those that provide the means for locating particles within the space. The dimensions may be physical, in the usual Euclidean sense of dimensions, but they can be other than physical. The dimensions for a given space must be identified and the resolution for measurement of location, as measured by the size of the smallest unit for measurement, must be identified. The smallest unit of measure for each dimension of location will be taken as 1. • To some extent, the minimum resolution depends upon the maximum size of a particle, since the initial location of particles is determined by the location, so the minimum size of a location must be greater than or equal to the maximum size of a particle. • There is a point in space, called the origin, from which all locations are measured. The origin will be identified by the value of 0 for each dimension of location, so all points in space are identified by x-tuples of integers (positive or negative), where x is the number of dimensions of location.
Dimensions of Attribution • Beyond the dimensions of location are those of form, which will be called attributes. These are the means for characterizing the nature of a specific particle of substance. To illustrate, substance in the real world consists of mixes of matter and energy. Recognizing that the one may be converted into the other, at a given point in time the form of any portion of the substance of the real world is determined by its position on the two dimensions of matter and energy. The values of the dimensions of form, the attributes, will be taken as positive integers. • Some attributes are categorical, in which case the integer values identify the categories involved in the attribute. Other attributes may be sequential in which case the integer values identify the position in the sequence. Some attributes may be regarded as variables measured at a resolution; for such attributes the measure will be taken from an origin with the value 0 and the resolution as 1 so that all values are integers. • Each particle has a form or nature, as identified by its positions on the dimensions of attribution, that is specific at a given point in time.
An aside on Variables and Scales • It is traditional in dealing with variables, which attributes are, to identify the various possible sizes of the set of values and the various possible scales for measurement. • Specifically, the set of values for a variable can be finite, countably infinite, or uncountably infinite. A finite variable might be as limited as a dichotomy, with simply two values. A countable infinite variable might be represented by the integers. An uncountably infinite variable, by the continuum of real values. • I am taking the set of values for all attribute variables as finite, though potentially very, very large. • The set of scales for measurement can be nominal (i.e., simply names, with no other relationships among values), ordinal (i.e., in a sequence), interval (i.e., differences between values are measurable), and ratio (i.e., an interval scale with a zero value). • I am taking the scales for measurement of an attribute as any of the traditionally defined ones, as enumerated above.
Adjacent or Contiguous Locations or Attributes • Within the dimensions of location and attributes, there are what will be called adjacent or contiguous locations or attributes. Two locations or attributes are adjacent or contiguous if they differ by one unit in just one dimension and are equal in all other dimensions. • Note that the concept of adjacency or contiguity applies to the dimensions of both location and attributes (with the exception of nominal attributes for which adjacency cannot be defined). For ordinal scales, adjacency of two locations would require that there by no value between the two values.
The Matrices of the Space • The entire structure of the space can be represented by a set of matrices, one matrix for each point in time, the rows of which are the particles, the columns of which are the dimensions of location and form, and the values of which are the values for the particles at the given point in time. The values in the matrices are all integers. • The matrix for the origin of time will include, at its end, a set of rows that have zero values and that represent particles that will appear at subsequent points in time. The prior rows are sequenced in inverse lexical order by the values in the succession of dimensions. The number that identifies each particle is that for the sequence in which it appears in the rows. • The following display illustrates the set of matrices, with m being the number of dimensions of locations, n being the number of dimensions of attributes, and t being the number of points in time.
Selections of Particles • A selection is a subset, S, of the N particles taken at points in time: S = {Pi1ti1, … ,Pijtij}, where if p < q, tip≤ tiq and ip < iq if tip = tiq. Thus, the order is by time and then by particle number within time. • Taking a selection of particles is clearly simply taking a set of rows from the set of matrices, and that can be done in any way desired. For example, they could be selected quite at random – in time, in location in space, in attributes – or they could be selected for specific times, locations, and/or attributes. • One might focus on a specific location and consider all particles that appear in that location over time; one might focus on a particular particle and consider it at various points in time; one might focus on an attribute and consider particles that have a specific value for it; one might focus on a specific time (i.e., one of the matrices) and select particles within that time. • A selection with no particles is the null selection. A selection of all N particles is the universal selection.
Structure of a Selection • The structure of a selection is defined as the array of k*(m + n) values for the (m+n) dimensions of the k particles in the selection. This structure may be called the organization or perhaps the pattern of organization exhibited in the selection, as determined by the positions in the dimensions of location and form of the particles in it. • The structure may be represented in ways other than the array, though always dependent upon it and derived from it. For example, one might have a selection of three particles and represent the structure by the triangle of their locations.
Structure of a Selection • Of course, although the structure is indeed based on the array of attributes, that is much too simplistic. One has the feeling that there needs to be means for recognition of various types of structure (such as the example of the triangle exhibited by a set of three particles). • The problem, though, is that recognizing a type of structure implies that that type has been, in some way, pre-identified. It thus pre-supposes much of what I am trying to have arise from the dynamics of the space itself. • The task I have set for myself, therefore, is to develop means by which the structure based on and embodied in the attributes can arise from the dynamics of the space rather than being pre-defined.
Structure of a Selection • I want the set of structural elements to reflect simple things that come directly from the set of particles themselves, without imposing any externally identified patterns (such as “triangle”). • The following structural elements seem to satisfy that desire: • Number of particles. This seems self-evident • Distance between pairs of particles. This is measured by a metric for the space; the metric could be either the usual Euclidean metric, the “city block metric”, or another metric. • Variance of the set of particles. This is measured by the sum of the squared deviations of each particle from the mean of the set of particles (as determined by the metric) divided by the number of particles. • Adjacency. For each pair of particles in the set, this identifies whether they are adjacent in space or not adjacent. • Connectivity. This is more complex and not yet well defined. One possible measure is the number of paths of adjacencies that are embodied in the adjacencies. • Compactness. This too is more complex and not yet well defined. Basically it measures the extent to which the particles are concentrated in a region of the space. It thus relates to density as reflected in variance, adjacency, and connectivity.
Structure of a Selection • Of course, the number of particles and the distance between pairs is the basis for all of the others, so we should start by examining the nature of those two. • The number of particles involved is unequivocal, but the measure of distance is not. It can be based upon any selection of dimensions and any of the means for measuring distance within them. For simplicity in description and because it is in a sense the most natural, I will use the dimensions of location as the dimensions involved and the “city block” measure as the measure within them. But that is simply for the purposes of illustration and example. • I reiterate that the measure of distance need not be limited to the one being used for purposes of illustration. In many worlds, in fact, the dimensions of physical location may be totally irrelevant and the dimensions of attribution may be critical.
Structure of a Selection • It turns out that “cluster analysis” provides a set of methods, with associated structural elements as identified above, that are needed for the objectives here. • In particular, the “one-dimensional hierarchy linkage” method examines pairs and groups those that are closest, iterating the process until done. • Hence, given n particles, there are n*(n-1)/2 pairs. Let {p1, …, pn} be the set of particles and let dij = d(pi,pj) be the distance between the pair pi and pj. • If two particles, pi and pj, are contiguous (i.e., in adjacent locations) then dij = 1. • In the same vein, if there are two selections, S1 and S2, let g1 and g2 be their respective centers of gravity. Then d12 = d(S1,S2) =d(g1,g2) is the defined as the distance between the two selections.
Structure Examples • I am going to use selections of 16 particles as examples to illustrate the elements of structure as I will try to develop them. The space, for purposes of illustration, will be limited to simply two dimensions of location. Further, particles will be located at most one to a location value (x,y). • It is important to note that, though I use as my example only two dimensions and those just of location, the principles apply to any number of dimensions and to those of attribution as well as to those of location.
Structure Examples: Density • The first and probably the most evident element of structure is the density of a selection of particles in the space. For example, consider the two selections of 16 particles shown to the right. As the top one shows, they could be sparsely distributed in the space. • Alternatively, as the second example shows, they could be concentrated, with an average density of 1. • Subsets of the selection could vary in their density. • The following displays will show, first, the values of locations for the first example and then the resulting pair distances.
Structure Example 1: Density • For the top figure, let’s suppose that the 16 particles are located as follows • (4,10) • (2,9), (5,9), (9,9) • (3,8), (6,8), (8,8), • (6,6), (10,8) • (1,5), (7,5) • (4,4) • (2,3), (8,3) • (6,2) • (9,1)
