3rd Lecture – Appendix

In response to students’ request or warning that there is a lack of study material on character variables, I am posting here an excerpt from my PhD thesis. That much I can do quickly.
I have written this text for a completely different purpose, so I ask for your understanding and patience as you read it.

Elements of Space

Thirty crossings converge in a fist,
in the empty space between them is the essence of the chariot.
They knead the clay to make it a vessel,
the hollowness in them makes the essence of the vessels.
windows and doors pierce the walls,
the essence of the house is made of the voids themselves.
So then:
the benefit is in the material,
the essence is in the substancelessness.

(Lao Zi: Dao de jing, 11)

Space is the relationship between the positions of bodies and volumes.

(Laszlo Moholy-Nagy)

Space is a system of relations between things.

(O. F. Bollnow)

It makes sense to consider definitions that follow directly from the relations between bodies. That is, the relations are those fundamental properties to which I refer in the individual examples and proofs.

As an example, let us take several different floor plans of rooms. Each room has an opening which, depending on the length of the perimeter walls, may be narrow, very wide or somewhere in between. Let us consider the area of the opening. Where would you feel you could draw an imaginary boundary between the interior space defined by the perimeter walls and the exterior space? We can see immediately that as the width of the opening increases, it becomes more and more difficult to define the dividing line between inner and outer space. In addition, it also becomes more difficult to demarcate the space as the parts of the perimeter walls that face each other along the opening become shorter and shorter. Usually, we can only determine the dividing zone by feeling, whereas the exact position of the dividing line is a matter of reason. If a room has a large aperture, only the N zone can be described as the inner space and only the Z zone as the outer space.

A drawing can be used to clearly represent the direct relationship between the ends of the walls by connecting them with an imaginary straight line. In this way, we have defined the boundary between the outer and the inner space in a rational way. In the same way, we can represent the relationships of the wall sections with all the intermediate points of the defining lines of space (the perimeter of the space). In the floor plan, we divide the perimeter walls into sections, or into bodies standing close together. They are connected by solid lines to graphically establish the relationship of each individual body with all the other bodies. This gives a pattern of lines of different densities.

In the floor plans of rooms, which are interspersed with “relations” between individual objects, there are also areas through which no object-pair continuum (no relation) passes.

Let’s ask ourselves two questions:

1. How would you specify the relationship of a pair of wall sections to each other, or the relationship of a pair of bodies to each other?

2. How is the (varying) density of relationships, which is in any way perceptible in the floor plans, characteristic of each individual space?

While we do not yet have an inkling of the answer to the second question, we can quite quickly find the answer to the first question. The wall sections are three-dimensional. From the implications of the definitions of space, we can conclude that a pair of opposite objects form a space. If we can call the projections of the peripheral bodies of space the elements (basic parts) of these bodies, then the relation of the pairs of projections gives us a space in the form of an “element of space”. Graphically, this relationship is represented as a belt. In practice, two objects form only the intervening space, while the real space is defined by at least three objects.

The pairs of individual wall sections are connected by bands of relations of different lengths. An observer looking at and judging the relationship between a pair of sections A1 and A2 knows by the size of their distance from each other that they stand close together or far apart. Based on this, it judges that the relationship is strong or weak. He therefore forms a judgement on the intensity of the relationship based on perception, according to the size of the distance.

For any observation location in space, the number of connecting bands is characteristic. Once these are evaluated, we can determine the eigenvalue of each location with respect to the number of (bands of) relations. This value expresses the specificity of the site in space. For now, we can only say that the value depends on the observed elements of space from the observation site. But for more we need to process other data and parameters.

In addition, we also know that the observer judges the sections A1 and A2 in relation to himself. In particular, the magnitude of the distance between him and the cuts A1 and A2 is important. From the point of view of the whole space, this means that the characteristics of the observer’s relation to the perimeter of the space are already determined by the place of observation. Therefore, this relation can also be evaluated similarly to the mutual relation of a pair of snippets before.

It seems that we can use the relationships between the walls and between them and the observer to capture the action (effect) of the space in relation to its shape and size. Therefore, we can develop a useful model for evaluating the performance of a space based on these relationships.

Formal parameters of architectural space

The term architectural space is mainly used to describe everything that is surrounded by walls and floors. Whether or not there is a ceiling or a roof, a rough distinction is made between external and internal space. We know that there are many other notions of space besides architectural space. For us, at this point in time, it is important to define the concept of architectural space more precisely. In his classification analysis, Hartmann makes a meaningful distinction between ideal, real and perceptual (visible) space.
Ideal space (also geometric space) is a pure dimensional system. It is homogeneous, continuous and unbounded. Mathematically, it is defined as a space with n dimensions. Three-dimensional, i.e. Euclidean, space is just a special case of such a space.

Real space is the space where things exist, where form and type are determined by the outside world. This is the space in which we live. It is the earthly space (world space); the space where man works and manages.
Perceptual space is not visible in itself. It is only the form of visible content. In it we see the exterior of objects. Only objects in their corporeality are visible, i.e. optically perceptible. Corporeality is determined by the three-dimensional dimension. This is perceptible only through the distances that bodies have between themselves. The distance between bodies is also “that between bodies” which we perceive as space. We use dimension and distance to define (describe) visible space.

Visible space can be perceived, pictorially experienced and represented. Here Hartmann distinguishes between perceptual space, the space of performance (representation) and the space we experience. Here the latter is closely related to the other two.

Among these forms, the visual space is not homogeneous, because the observer judges the relations between bodies differently depending on his position. Proximity is determined in the field of vision by foreground, by intrusion, by empty distance, by disappearance. The visual space is therefore a system of arrangement where all places (all points) are not equal (organisational – topological hierarchy).

The three basic types of space are not clearly distinguished from each other. Certain designations of one form often meet with others. Thus, visible space, with its three-dimensionality, also has the characteristics of Euclidean space and can be realistically described.

What characterises architectural space as a form of visible space? Visible space can be evaluated in terms of meaningful associations and semantic content. Each space of a specific meaning association is characterised by its structure, function and meaning. However, since these features are only perceived through bodies, it is bodies that constitute architectural space. They form it in an architectural way, according to the way in which they surround it.

In our case, it makes sense to consider as space only the formations that follow directly from the relations between objects. However, we call everything that we describe here as “that between bodies” space.

We proceed from the following:

Take a pair of bodies. The bodies stand at a distance from each other. This means not only that they are separated from each other, and therefore occupy different positions, but also that they are connected to each other in terms of direct (shortest) connections.

We connect new bodies to a pair of bodies. Given the mutual distance of pairs of opposite peripheral bodies 1, 1′ and 2, 2′, the mutual relations O1 and O2 of these bodies are determined. The relations O1 and O2 are called the perimeter relations of the region P. The region P is defined as the space formed by the two types of bodies and the relations O1 and O2. If we let only the corner solids stand, instead of the previous two types of solids we get the relations O3 and O4. This preserves the area P. We know it as the space formed by four bodies (four pairs of bodies!) or four circumferential relations. The interrelation of the perimeter bodies becomes the perimeter relation of the given area. This means that the area formed by the bodies is space. Its dimension is determined by the relations between the peripheral bodies.

In this respect we have two extreme possibilities: first, in the case of only one pair of bodies, the circumferential relation is assimilated to the area itself, and second, in the case of a continuous series of bodies, the circumference itself, in the form of the bodies, takes the place of the circumferential relation.

From what has been shown it is clear why only the formations which follow from the direct relation between the bodies can be regarded as space. It should be noted here that the perception of space is connected with the three-dimensional dimension of the field. It is also important in what relation (relationship) the horizontal and vertical dimensions are. It is very difficult to call an area defined by a pair of very slender pillars a space. Space is not perceptible at all if it is not defined by the horizontal dimension of the bodies. It is only the larger horizontal dimension, or the relation of the two pillars to the third, that we perceive as space.

Definition of the factors determining space

We have learned that we perceive space by the bodies that fill it. Let us therefore take a closer look at what are the signs of bodies that are essential for perception.

First, a few remarks.

The following propositions can only be deduced without reference to optical perception. All influences of other senses are excluded.

We will call the observer who perceives space the subject, and corporeal things the object. We will use the terms observer and body interchangeably. What we have called a relation so far, we shall henceforth call a relationship.

Object

An object perceives space by identifying the relationships between objects; we say that it classifies objects in relation to each other. The number of all objects it can classify with respect to some individually selected object is unlimited. The only condition that applies is that the subject can only classify objects between which it establishes a visual association. It may also classify itself among the objects.

The dimension of an object and its distance to other objects or to the subject is perceived as a size relation. We speak of the size of an object, of its shape and of its distance. This is how space is constituted. Quality factors such as colour, texture and optical tightness realise the space. All these factors together determine the uniqueness (particularity) of an object; this is what we use to judge the classification of objects in relation to each other or to the subject. This peculiarity, this quality, should be the quality of the object. If we see in judging the expression of the action of the indicated possibilities of classification, we can conclude that the action depends on the quality of the object.

Realistically, we can evaluate (measure) the size of the object and the distance. If factors such as shape, colour, etc. can be evaluated in this way, the quality of the object can also be expressed realistically. The object quality value should be denoted by vk.

To make things clearer, let us say the following:

It is certain that we will not get the same results from measurement as from perception. However, the results of measurement are in exact relation to each other because they have the same basis. The object being observed must always have the same position in the same surroundings. Once all the factors by which the subject judges the classification of objects have been assessed and realistically evaluated, the measured value will correspond to the value of the perception. The results are therefore not of the same kind, but they are equivalent. Therefore, we can write Zv Rv (Zv – perceptual value, Rv – real value). However, if we see Rv as the quantitative value of Zv, we can write Zv = Rv. This makes the result of the perception real. Practically, this means that the action of space can be expressed in real terms.

In our implementation, when we have the actual value of the perception in mind, we will talk about judgement. But if we think of its real value, we will talk about evaluation. Evaluation is therefore the quantification of judgement.

The factors of the object expressed by the quality of the object are of a different nature. They cannot be completely defined in the context of this work. We will limit ourselves to listing and describing only the main ones that are of interest from the architect’s point of view.

Unfortunately, we cannot realistically evaluate all the factors. These can only be quantified statistically, as the results of research with experimental subjects. Such research is beyond the scope of this thesis.

Elements of buildings

The first step on the way to evaluating a classification is to decompose the (overall) relationship between objects of arbitrary shape and size, or between them and the subject, into individual classification elements.
So far we have divided the perimeter wall of the space (i.e. the object) into sections. From the relation of a pair of sections (“that between bodies”) we have defined an element of space. Following the same procedure, we can break down objects into parts of an object, into elements of objects. We know that we see an object only by its surface. So the elements of objects are surface elements. Let us denote them by df. Elements have all the characteristics of an object, which are determined by the quality of the object. Thus, we can realistically evaluate the overall classification in terms of the quality of the individual elements.

Classification – coherence

Take two objects O1 and O2 with different positions. The entity classifies them or connects them to each other . The classification is detected by the distance between the objects. Distance implies direction and anti-direction with respect to a predefined position. We denote it as the classification of the objects. Based on the direction and the reciprocal direction, we determine the amount of connectivity and thus establish a relationship to the other objects.

Applying this to the elements of the objects df1 and df2, we can see that the ranking of df1 and df2, or df2 and df1, is similar to that of the objects, expressed in terms of a partial amount of connectedness. For simplicity of expression, let us refer to it as connectivity. Thus, the unique classification of df1 and df2 is determined by two connectivities. This is what we called earlier the element space. It follows that the whole classification can be realistically evaluated through the object elements of df.

Object pair connectivity

We denote the associations resulting from the classification of objects as object pair coherence. We have explained how this type of association is defined in the previous paragraph.

Subject-object association

If a subject S is associated (classified) with an element of an object, we speak of a subject-object association. The subject is both the observer and the one who establishes the relations. It is understood as standing in front of the object. In this form of classification, there is no ‘in-between’ for the subject, as we have seen in the object-pair relation. The classification is only in the direction of the object element df towards the subject S. There is no relation in the opposite direction. The entirety of the classification of df and S is formed by only one association.

Exposed association

A connection that is realised but not evaluated is called an established connection. As we have just seen, the existence of a connectedness is based on the mapping of one element of a df object to some other surface element or to an entity. We denote the connectivity that arises from an element of object df by dp.

Evaluated Connectivity

To be clearer, let us equate the classification of object elements, or the classification of object and subject elements, with a state similar to tension. The fields that give rise to the stresses are the object elements, which also determine the magnitudes of the stresses. If we judge that two objects are very distant, it is also clear that their classification is weakly perceptible. There is therefore little tension. It can also be said that the classification has a low intensity. The intensity is therefore used to tell how the subject judges the classification.

When we speak of distance or proximity, this judgment depends, on the one hand, on the distance between the elements of the objects, and, on the other hand, on how the elements of the object are composed; whether they are of intense or of indistinct colour, whether one can see through them well or badly, and so on. When all the factors influencing the judgement and determining the intensity are taken together, the quality of the object mentioned above is obtained. Intensity is therefore a function of the quality of the object. The quality of an object can be realistically evaluated. If the value of the quality of the object elements is equal to vk, then the (partial) intensity of the individual association is the same:

dI = vk . dp

We use dp to give the existence of the association and vk to give the association weighting factor.
Next, we will process the factors that determine the quality of an object, describe and define them. In particular, we will describe factors that are geometric in nature, such as size, distance and shape. These will be followed by factors which are conditioned by the surface quality and the optical tightness of the object. Finally, we will describe the meaning factor. The values of the first three will be defined in terms of partial qualities and the values of the others in terms of additional values.

Object quality factors

Size

The df denotes the object elements as infitemal size values. The partial quality qV, which is obtained from the size of the surface element df, is therefore equal to df itself:

qV = df

Distance

As we have already said, two elements of objects are distant from each other if their mutual distance is large; conversely, they are close to each other if their mutual distance is small. The intensity of the connection is then either small or large (assuming that we observe the intensity only in terms of distance). The value of its magnitude is therefore inversely proportional to the distance. If we denote distance by l, then the partial quality ql obtained from distance is:

ql = 1 : l

Shape – position

Imagine an element of object df as a flat surface. Thus, its shape is defined. The shape of the object is determined by the relative positions of the elements of the object. Given the relative position of the object elements, we detect the object elements at a specific angle with respect to the direction of the connection. If we want to determine the quality of an object element, this angle is important. Let us consider an example. The plan element df is detected at an angle of 90° in its perfection. However, when we rotate it by 90°, we can no longer detect it at all in theoretical two-dimensionality. The share that each element has in determining the quality of an object in the direction of coherence with respect to the size of its own surface is equal to the value of its projection measured in that direction. This means that the smaller the projection of the surface area of the object element that is detected, the smaller the intensity of the coherence.
Let us denote the angle at which we see the object element by φ. In the direction of the coherence, the magnitude obtained differs from the actual value by the sine of this angle. Thus, the partial value of q with respect to the position of the object element is:

q = sin φ

Colour – illumination

It is well known that surfaces of certain colours appear closer to the observer than surfaces of other colours, although in fact they are all equidistant. For example, a glowing red plane appears closer than a light blue plane. Colours affect the optical perception of distance in such a way that the result of the perception is not the same as the result of the measurement. It seems likely that this phenomenon can be determined for each colour separately in the form of a value relation. The value expresses a relationship which shows how the results of perception and measurement are related. The relationship is given as a distance factor Just as colour influences the determination of distance, it also influences the determination of plot size. We know, for example, that black and white colours make a surface appear smaller and larger, respectively, in relation to its measured size. This phenomenon can also be determined by a factor for the different colours. The optical change in the size of the surface due to colour is denoted by p(v).

This phenomenon is due not only to the colour itself, but also to the brightness obtained through the incident light. In strong light, the colour appears brighter than in weak light. In addition, the colour of the incident light affects the colour tone of the coloured surface. Since colour (or rather colour perception) is as much related to light as light is to colour, we perceive the colour of the surface at the same time as the light. Colour and light are inextricably linked. Therefore, we perceive the power and colour of the light itself together with the colour of the surface.

Therefore, the illumination of a plane is expressed by the colour factors p(l) and p(v).

Texture

Surface patterns, such as raster and grain of different sizes, cause apparent changes in the size and spacing of surfaces, much as paint does. Thus, a surface with a finer texture appears farther away from the observer than one with a coarser texture at the same measured distance. On the other hand, a finer textured surface appears larger than a coarser textured surface.
As with colour, the values of the factors can be determined. Let the apparent distance change factor due to texture be st(l) and the apparent size change factor due to texture be st(v).

Optical tightness (transparency)

We have a colourless, perfectly flat, polished glass surface with no reflections. Ideally, it is undetectable. But if there is even a very thin layer of paint or dust on the glass, that is enough to be able to detect it optically while still being able to see through it. The denser the coating, i.e. the more resistance the surface offers to vision, the harder it will be to see through it. The optical tightness, or optical resistance, of perfectly clear glass is infinitesimal and so has a value of 0. As the tightness of the cladding increases, the resistance increases. When it is no longer possible to see through it, it takes the value 1.

We know from experience that the lower the optical tightness of a surface (for example, glass or a wall made of transparent material), the smaller the perimeter and therefore the smaller the space-forming effect. The less the boundaries are perceived, the “bigger” the space looks.

This realisation should certainly be taken into account when determining the quality of the building elements. The magnitude of the optical tightness should be expressed as a factor t.

Importance

We have already established that architectural space is defined by the objects of an architectural type. Therefore, objects of this type are likely to be of primary importance in the evaluation of architectural space, while objects of a different type are of lesser importance.

In the evaluation of an urban space, such as a square, the buildings (objects of architectural type) are of primary importance, while the trees in the square (objects that cannot be classified with certainty as architectural type) are generally seen as elements that co-determine the square. Parked cars should not play any role in the evaluation.

We therefore see the buildings in a certain order (rank) according to their type. The architectural type itself may have different meanings depending on the perspective from which the space is being evaluated. For example, the interior space of a church can be seen in its entirety, where the individual parts of the perimeter of the space have (roughly) the same meaning. It can also be seen in terms of the rhythm created by the sequence of columns and arches. In this sense, the columns, pilasters, niches, etc. are more important than the other elements of the space. The ranking of the primarily observed objects is always more important than the ranking of the less observed objects. The intensity of the connection will be greater or lesser because of the corresponding importance of the ranking.

The intensity of the relationship is not only determined by the size, distance, position and material properties of the elements of the objects, but also by the importance or significance that these elements have in relation to the architectural whole (composition). The place, i.e. the value of the place, which the elements of the buildings have in relation to their importance in the evaluation of the space is denoted by the factor a.

Evaluating the coherence of pairs of objects and the relationship between the subject and the objects

Together the individual values known so far into a common value. If the quality of the objects is the same, we get a different value for the object pair relationship than for the subject-object relationship. This important difference arises because we evaluate the df elements of the objects according to the ranked element in the pair. In the first case, this is the object element, which has the same quality as the element from which the association is derived. In the second case, it is an entity that acts as a “neutral”. To make it simpler, let us first clarify the relationship between the subject and the objects.

Take an object element df and a subject S. The element has (initially) neutral surface properties. Thus, the distance detected by subject S between itself and df is equal to the measured (1). The angle is 90°.

Now let us vary the individual properties. Let the object feature df be expressed by a glowing red colour and a coarse texture. This makes the apparent distance between S and df no longer equal to the measured distance. The element appears closer to the subject by the values of the factors p(l) and st(l). Furthermore, the angle (<90°) may also change. Therefore, the subject does not see the element in its direction at full size (sin φ).

In addition, colour and texture cause the size of the planar element to change by a factor of p(v) and st(v). In addition, the optical tightness t and the weight of the significance a of the object element also affect the subject’s perception.

Thus, the correlation between S and df:

dIs = ((sin φ. p(v) . st(v) . t . a) : (l . p(l) . st(l))) . dp

The value of colour and texture, as well as the values of optical tightness and meaning, i.e. the values attached to the object, can be captured together in the value of the object V. Thus the intensity of the association between subject and object:

dIs = ((V . sin φ) : l) . dp

.
The evaluation of the connectedness of pairs of objects can be described in the same way. Let P1 and P2 be the positions of the elements of object df1 and df2. Let the measured distance between them be equal to l. Let the position of the elements of the armour with respect to the direction from P1 to P2 be 90°.

The surface properties of the elements of the object should (initially) be neutral here too. Thus, the positions perceived by the subject correspond to those actually measured.
Now let’s change the individual properties. Let df1 be a glowing red colour and a coarse texture. This apparently reduces the distance between the two elements by the factors p(l) 1 and st(l) 1. When df2 is given similar properties (p(l) 2, st(l) 2), the distance is further reduced. The colour and texture also cause a change in the size of the faces of the pair of object elements (p(v) 1, st(v) 1 and p(v) 2, st(v) 2. We also vary the angles of the two elements (sin φ1 and sin φ2) and their optical tightness (t1 and t2), as well as the significance factors (a1 and a2). If we combine the individual factors, we get the values:

l : V1, sin φ1 : V2, sin φ2.

This describes how the elements of the objects df1 and df2 are ordered. We see that, for the evaluation of the coherence arising from df1, it is only important what relation the values of df1 have with respect to the element df2. Thus, the intensity of this association:

dI0 = ((V1 . V2 . sin φ1 . sin φ2) : l) . dp

Take that, in quite general terms, the value of V and the sin of φ of an element of an object df correspond to the value of the object V and the sine of the angle φ. After that, the intensity of the connectedness of the pair of objects is:

dI0 = ((V . V. sin φ. sin φ) : l) . dp

Evaluation of the coherence system

General assumptions

We have so far defined factors to describe space as a mathematically determinable classification of the whole. Now we will try to use what we already know to capture space as a whole and evaluate it. First, we need to derive some general assumptions.

For all assumptions, we assume that the observed space is simple and transparent.

On perception

From what has been said so far, it is clear that architectural space is a phenomenon that depends on perception. We have realised that we capture space by perceiving connections. These connections, in turn, arise from a classification that does not depend on will, but is carried out spontaneously in the flow of perception. Moreover, the intensity of the association depends on the attention of the subject.

As a rule, we cannot capture a space with just one glance. With single glances we see only snippets, spatial segments, which we associate by changing the direction of the gaze. Therefore, perceiving and judging space from a particular place (the place of observation) is only possible mentally, after we have perceived it in parts. On this basis, we can ask whether this also applies to the connections between objects when the subject does not perceive a group of objects in a certain view. Here we are thinking of perceptual space, and thus undoubtedly of architectural space. We do not deal with perceptions of existing connections in existential space. In our observations we assume that every possible connection exists. We should also note that our attention is primarily focused on the human capacity to see. For psychological reasons, we cannot perceive a connection as well from the vantage point directly in front of the object as we can when the subject is further away from the object. This is particularly important when it comes to understanding the quality factors of an object and thus judging connectedness. How to investigate and identify more precisely the deviations in the perception of connectedness in the area immediately in front of the object is, however, a task for other research. Let us therefore refer to this area as the critical area.

Through the dependencies between objects and between subject and objects, we have obtained a connectivity system: a pair-object connectivity system and a subject-object connectivity system. The interconnections of the two systems are detected simultaneously, and thus there is an overlapping or overlapping of the two systems. For the sake of presentation and evaluation, we will consider the two systems of connections separately.

Let us ask ourselves what we get from the interconnection of these systems.

By linking pairs of objects, we perceive space as a given. It is used to capture and judge space as “that between objects”. That is, with object-pair connectedness we capture the operation of space on the basis of formal qualities as they act on the observer.

In turn, the subject-object connection characterises the subject’s relationship to space, which is conditioned by the place of observation. Each section of space lies, pictorially speaking, in front of the subject. He cannot enter it without leaving the place of observation, nor can he reach it as part of the space in front of him. By perceiving the connection between the subject and the objects, the subject captures the action of the space by classifying himself to the space, or more precisely, to the objects.

These two forms of action, which derive from the point of view of the classification of pairs of objects, or the classification of the subject to the objects in space, are perceived simultaneously in life in space. In order to analytically capture the action of space, the division into the two classifications is necessary.

If the subject is located inside or outside a particular space, it can be experienced both through the association of a pair of objects and through the association between the subject and the objects. If the subject is located “between” objects, then connections are made in the direction of the subject’s place of observation, or the connections intersect when the subject enters the space. Thus, the connectivity of pairs of objects can be used to determine when the subject is located within the space. When we judge his position on the basis of the connections between the subject and the objects, we know him in space when all the connections together enclose an angle greater than 180°. Then we see the subject surrounded by objects.

Evaluation notes

We express a system of connectedness and intensity in terms of connectedness. As we have just seen, the intensity value captures the ordering of pairs of objects, or the ordering between the subject and the objects. This is a tension-like state. It describes the operation of a space with the value of a particular place. So the quantity of connectedness determines the “spatial” basis that is the carrier of this action.

Example:

We have two spaces of the same basic shape and size, but with differently constructed perimeters. One space is to be defined by the glass surfaces between three supporting columns, the other by a series of supporting columns. Both are expressed with approximately the same intensity value. However, their respective coherence quantities are different because of the different perimeter areas. This means that the performance of the two spaces is identical in value, although based on different quantities of coherence. It is clear from this that their actual performance is different.

The amount of connectivity therefore takes on the role of a comparative value

.

For the intensity value, two extreme cases can be constructed:

1. We can imagine that the distance between the objects is such that we can no longer detect the objects and their classification. The intensity value goes towards 0. A void takes the place of the perceptible space. Therefore: when l goes towards ∞ and I goes towards 0, we get a void. By void we denote a space of infinitesimally small intensity. We start from the observation that intensity is a function of distance. It is also possible that objects cannot be visually perceived because they are completely transparent. In this case, too, we get a void from the point of view of visual perception.

We can define absolute emptiness, but we cannot imagine it, nor can we perceive it. In the extreme case, it can be detected by diffuse light reflections. In our representation, a void is a plane without objects, which can be a blank sheet of paper. A void is therefore something that does not contain that by which it can be perceived in itself. The imaginable or perceptible void is always a relative void.

2. Imagine that the distance between objects is so small that the objects are touching each other. We can no longer perceive the distance between the objects. An object, the body itself, steps into the place of “between objects”. The intensity value goes towards infinity. The intensity of architectural space always lies between 0 and ∞. When l goes towards 0 and I goes towards ∞, we get a body. A body denotes a space of infinitely large intensity.

Both extremes are of theoretical interest only. The intensity of architectural space always lies between 0 and infinity.

The quantity and intensity of the connectedness of pairs of objects and of the connectedness between subject and object are interdependent, because both kinds of connectedness are determined by objects of the same group. In the following, we will discuss the system of pairs of objects and the system of subject and objects in space separately.

Evaluating the object pair system

General observation

In order to be able to show the density of connectedness, we first need to build the connectedness system to completion.

A void, for example an infinite plane with no objects, should be the basis for building a group of objects. On the plane, we place an object in the form of a slender column (and an observer). Then place another such object next to it. The two objects are classified relative to each other and the classification creates space. However, as such, it is almost imperceptible due to the small horizontal dimension of the objects. It is only when a third object is placed that space is created. This is the triangular area between the objects. The graphical representation of the connectivity shows that the space is defined by perimeter relationships and that the inner area is not obstructed by any connectivity. The amount of connectedness and the intensity of connectedness of the area O1 both have the value 0. A space is created which is formally indistinguishable from a void in its entire area (within the perimeter relations). We say that the property of this space is emptiness.

By adding new objects, we obtain contiguities that cover this area. Different parts of the area have different densities of connectivity. In addition, there are areas that are not characterised by any connectivity: their quantity and intensity values are equal to 0.

The continuum of objects causes each part of the area to be permeated with connectivity. Space is clearly distinguished from emptiness. Different densities of connectivity and different distances between objects lead, as a rule, to unequal intensity relations ‘within’ the space. Emptiness is an extreme possibility in the structure of a group of objects. As a form of architectural space it is almost impossible. The void only explains how the (public) space differs from the surrounding area. Architectural space is usually based in an already existing system of connectedness.

From the above we can conclude that for the space as a whole as well as for each part of it (the observation site) we can define a quantity of connectivity, which determines the density of connectivity and the intensity value.

To the section On perception, we can say: the total intensity covers the action of the whole space perceived by the subject in space. The intensity of a part of the area covers the activity perceived by the subject in space from his place of observation.

It is the very specific objects that stand opposite each other (looking at the place of observation) that are the cause of the action of the space. The subject is located at the focal point of the constellation of objects. He stands, therefore, at the intersection of connectedness. The quantity and intensity of the connection is characteristic of the particular place of observation. The action (impression) varies from place to place as the subject enters other constellations of objects and thus other fields of connection.

Evaluating space as a whole

The total amount of connectedness

Based on the observation that a relationship is established by classifying an element of an object to its pair, we can calculate for an arbitrary space the total amount of connectedness Kwhole o:

Kwhole o = ∫dp

Specific quantity of connectivity

The specific amount of coherence Ksp o is characteristic of the total volume V of a given space:

Ksp o = (Ktotal o) : V

Since the total amount of connectivity is unevenly distributed over the space, Ksp o gives the average value per unit volume.

Total intensity

The total intensity of a space Itotal o is the intensity determined analogously to the total amount of connectivity:

Iinteger o = ∫ DIo

Specific intensity

The specific intensity of the space Isp o is analogous to the specific amount of connectedness:

Isp o = (Icel o) : V

It gives the average intensity per unit volume.

While Icel o captures the entire performance of the space, Kcel o and Isp o have the character of comparative values.

Places in space

Quantity of connectivity and intensity

For the following analysis, we need to determine each observation site that can be occupied by a subject.

To this end, the following considerations:

The set of objects forming the perimeter of the space should be denoted by G, and the possible place of observation of the subject by Qo. The amount of connectivity that pervades this place in the subject’s body height ho depends on the number of elements of the objects. They are classified according to Qo. The elements contained in the perimeter sections G1 and G2 are opposite to each other with respect to the place of observation, while the others do not have an opposite element of their pair. This is a characteristic of the classification between G1 and G2. For Qo, we obtain the quantity of connectivity:

KQo = G1 ∫ dp + G2 ∫ dp

For the same site, the intensity value:

IQo = G1 ∫ dIo + G2 ∫ dIo

IQo captures the action of space as introduced in the General Observation section.

Subject-object system evaluation

General Observation

Let us observe space in the same way again in terms of the classification between subject and objects. To clarify the density of connectedness, we also establish here a system of connectedness. The basis for the structure of the group of objects should be the empty plane here too. The subject detects connectivity to an object on this plane, and further connectivity to further objects.

The basis of this system of connectedness is the orientation of all connectednesses to one place, the place of observation of the subject.

It is clear from the figure that among the areas that are connected by connectivities, there remain those that have no connectivity and that are indistinguishable from the (empty) plane. The result of a continuum of objects is the addition of connectivity.

Evaluation

Quantity of connectedness – intensity

Take again in this case the set of objects G that forms the perimeter of the space, and the possible observation site of the subject Qs, which is not an element of this set G. Establish all possible connectivities of G to Qs. We see that determining the amount of connectedness and intensity of the whole system is as important as determining them in terms of Qs. When determining the amount of connectivity and the intensity, space cannot be separated into the whole and its parts (observation sites). Moreover, the determination of the specific amount of connectivity and the specific intensity is meaningless here, because these data are only meaningful for each individual observation site separately. The amount of coherence obtained by grouping all the optically perceptible elements of the objects up to the subject (at eye level hs) is:

KQs = G ∫ dp

In terms of evaluating the coherence, it is the intensity value from the observation site:

IQs = G ∫ dIs

Each observation site can therefore be defined by its connectivity value and its intensity. The IQs captures the action of the space on the observer according to the subject-object classification.

Differentiating intensity values

It is interesting to note the explanation of the dependence of intensity on the place of observation provided by the differentiation of intensity values IQs. As we know, the intensity value is obtained by summing all the site Qs connectivities. It is also possible to combine intensity values that have the same direction in the floor projection. This gives, for example, the intensity value of IsA in direction A by summing all the intensity values of the connections lying on A:

IsA = s ∫ dIs

This intensity value can be plotted graphically in the form of a vector with a given direction. The same applies to a value in any other direction. By plotting all IsA, a diagram is obtained in which the intensity relationships of the site Qs are visually represented.

Intensity in the viewing direction

Intensity can also be defined in other ways. Let us see how the optical perception of a subject obtained by looking depends on the direction of looking.

Let us determine the value of intensity and the number of conjunctions that the subject perceives by gaze without moving his eyes to the left or to the right, nor up or down. We know that the binocular field of view of a human being is 180°. The area of sharp vision is limited to an angle of approximately; from here outwards, the sharpness of vision decreases. At a deviation of 90° it is 0. (In any case, no conclusion can be drawn from this as to how great the deceleration of acuity is at a given angle.)

In determining the intensity in a fixed direction of gaze, we assume that all the coherences can be detected and judged within 180°. The accuracy of perception and judgement decreases with the deviation from the direction of view until, at an angle of 90°, final perception is no longer possible.