Glossar-English:Space and Geometry

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Translation of: Raum und Geometrie


Space as the basic category of picture morphology

The set of the syntactic components for perceptoid signs can generally be divided in two groups: On the one hand there is an abstract relational basic structure. It usually spans several coordinated dimensions, in which the elements of the second group can be arranged. The latter are forming systems of perceptual marker values that firstly allow us to perceive the underlying basic structure, which actually is, as Kant has put it, a category.[1] In the case of pictures, the basic structure is two-dimensional space while the marker values are essentially colors and textures that render purely spatial structures visible. Hence speaking about picture syntax implies talking about space.

The rules organizing our talking about space (at least as far as it is supposed to be rationally controlled) have been in the focus of interest at least since the neolithic revolution with its introduction of architecture, farming, and the cooperative spatial manipulations involved in those activities. Today, it is essentially the calculization of geometry of ancient Greece and its algebraic reformulation in the 16th century that influence the conception of space, which also underlies pictorial syntax.[2]


Geometric calculi as formalization of space

Basically, a geometric calculus is a vocabulary with a set of rules originally setting up the options of talking in the abstract – without reference to any concrete objects with contingent non-geometric properties – about “geometric entities”. Those rules form the conceptual determinations of a set of spatial concepts. The concepts for basic geometric entities are essentially determined by the relations they can enter. Those are usually divided in relations concerning contact and neighorhood (topological relations),[3] relations concerning distances and extensions (metric relations),[4] and relations concerning direction and orientation (directional or projective relations).[5]

Having got geometric concepts has essentially three consequences:

  • The “ontological” aspect of the rules appears in that they enable us to describe something as being spatial.[6] Understanding something as an instance of such a concept, i.e., as a geometric entity, is to view it as a kind of purely spatial entity – apart from other characterizations that might hold of it simultaneously (e.g., being colored, being heavy).[7] The world appears spatially organized if we distinguish its parts by means of geometric concepts.
  • The “epistemological” aspect entails that the Gestalts that appear in descriptions of visual perception can be understood as geometric (hence spatial) entities. That is, the calculus formalizes an abstract part of our understanding of visual perception (also ⊳ objects of spatial perception): We see the world as geometrically organized.
  • Finally the “argumentational” aspect means that the rules of the calculus describe how a given description of spatial objects can be transformed without changing the truth of the descriptions: These transformations are usually summarized as ‘Spatial reasoning’:[8] When we rationally discuss about space, we employ the rules of geometric concepts.

The standard approach to geometric calculi is Euclid's axiomatic system based on the concept of an unextended but uniquely located »point«: This concept is, however, quite abstract and rather distant to (spatial) experience. A formal approach to geometry that was primarily developed in the 20th century tries to do justices to cognitive principles. As a result, a family of non-standard geometric calculi has been developed. It offers interesting properties for picture morphology: ‘mereogeometries’.

Euclidean calculus of geometry

More than 2000 years ago, a first axiomatic system of geometry has been proposed by Euclid.[9] Based on a set of basic postulates (‘axioms’), implications can be drawn in that system in order to formally (i.e., by means of logical deductions) prove theorems about geometric objects and their properties or relations. The discussion of this approach and in particular of the independence of the five basic postulates about geometric objects has indeed led to several variants of such a geometric calculus forming the field of ‘synthetic geometry’.[10]

In the 16th century, Descartes (Essais, 1637) developed an alternative formalism based on algebraic formulae – locations are described by numbers encoding distances to a point of “origin” relative to coordinate axes. This ‘analytic’ geometry, which has basically led to the vector calculi used today in most technical approaches to space, has been formally proven (Hilbert) of being completely equivalent to the Euclidean calculus.[11]

Essentially, all geometric entities are viewed as sets of individual locations, called ‘points’, which, as Euclid has put it, is “what does not have parts”. The points can be organized in one or more dimensions – depending on the number of independent (‘orthogonal’) coordinate axes, which are associated with basic directions. Each axis furthermore organizes the points according to the real numbers. This implies that Euclidean points are essentially arranged in a continuum.

The rules of the geometric calculus establish spatial homogeneity (invariance with respect to translations of the point of origin) and isotropy (invariance with respect to rotations of the coordinate axes), which ultimately are the basis for syntactic density of pictorial space. However, the common Euclidean formalization of geometry leads also to the “unpleasant” consequence that the most basic pixemes must be non-extended points – a concept highly abstracted from experience, that is. Any extended region – i.e., any pixeme – then must consist of an infinite number of basic geometrical entities, quite in contrast to the Gestaltian conceptions of (spatial) perception.

Mereogeometric calculi

Some non-standard approaches to geometry offer an interesting way out of this dilemma. Mereogeometries are the result of a formal approach to geometry that was primarily developed in the 20th century, and that tries to do justices to fundamental cognitive principles. If a point is, as Euclid thought, “what has no parts”, part-whole relations should be considered as crucial to geometry. At least, geometric entities that do have parts may after all be more natural candidates for grounding pictorial space.

In contrast to Euclidean-style geometries, the family of mereogeometries is based on the concept of an extended region, which may or may not have (distinguishable) proper parts. Such regions are often called “individuals” as they form the ultimate basic elements of the calculus.[12] They do not have immediate attributes of form or position:[13] Only the relations to other individuals, which in particular could be their parts or the wholes they are a part of, determine form, extension, and (relative) location.

While Euclidean geometry first introduces the continuous range of infinitely many points some of which are then chosen to be relevant (still an infinite number in any practical relevant instance), mereogeometric descriptions start with a (usually finite) number of relevant individuals (regions).[14] Such an individual may quite well be thought of as a visual Gestalt – thus following the principles of perception psychology of the Gestalt school: One has to consider the perceived whole first, and introduce the concepts for perceptual atoms as instruments of the explanations of the former, not the other way round. Human beings do not see sets of zero-dimensional points but extended Gestalts – especially in picture perception. The abstract notion of a spatial entity without extension is secondarily constructed in order to explain some aspects of experienced space, but leads on the other side to severe difficulties.

Since space has been traditionally captured by point-based geometry, it must be recognized that, overall, the properties of Euclidean space (i.e., the concepts of space as described by a Euclidean calculus) fit our commonsense notion of space. Thus, it should not be surprising that most mereogeometries lead to systems ‘equivalent’ to Euclidean geometry [Borgo & Masolo 2010a]Borgo, Stefano & Masolo, Claudio (2010).
Full Mereo­geome­tries. In The Review of Sym­vbolic Logic, 3, 4, 521 - 567.

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. This very fact shows how our cognitive perception of space is quite stable and precise and is not affected by the choice of geometric primitives. Indeed, the properties that commonsense space should satisfy are not an issue. The crucial point is to explain how we cognitively attain this specific notion of space. In this perspective, the first question that mereogeometries try to answer is what primitives apply to extended objects and are expressive enough to generate the commonsense notion of space.
Calculi of geometry and the morphological structures of pictures

Indeed, the research of mereogeometric calculi corroborates a conclusion that has puzzled researchers in pictorial morphology: the lack of constraints on the choice of primitives. In both domains one arrives easily at equivalent formalisms despite starting from quite disparate assumptions. It follows that the choice of primitives cannot rely on purely formal properties. It must be supported by arguments and observations from other perspectives like those embedded into the cognitive, evolutionary, mental, and perceptive views. The development of several geometrical systems up to mereogeometries has led to geometric approaches that, exploiting disparate primitives, naturally result in formal geometries of equivalent expressiveness. On the one hand, the search for grounding pixemes (either as primitives or as prototypical) naturally leads to a discussion that matches the debate on basic geometrical entities. On the other hand, the need of rendering and understanding complex images in a computational setting suggests (at least in theory) the existence of a limited number of basic pixemes that can be combined via a formal calculus of limited complexity (⊳ image processing, digital).[15]

Pixemes as geometric entities

In the context of the morphological structures of pictures, the geometric calculus describes how we talk (rationally) about the basic structure of the picture plane and the pixemes contained there. A basic structure not following the rules determined by the calculus leads to a syntactically invalid picture. With a point-based calculus, pixemes are conceived of as (infinite) sets of points, which are determined by Gestalt-organizing processes operating on the marker values (⊳ color as a category of pictorial syntax). The infinitely many points contained in any pixeme are locations, which merely are potentially interesting: They might become morphologically relevant when comparing the picture vehicle in question with another picture vehicle. Correspondingly, the Euclidean-type calculus of geometry has no need for a concept of resolution: Its virtual value of resolution is always infinit – a God's eye view on space.

With a mereogeometric calculus, pixemes are “indviduals”, i.e., primitive entities of the calculus. Assuming that the Gestalt principles governing visual perception determine exactly those regions that are syntactically relevant, the pixemes can be quite naturally conceived of as being given in perception. There is no need to consider more points than necessary

Points, resolution, zooming, and microscopization

While mereogeometries deal with pure space, pictorial morphology has to take into account other elements like granularity (which may affect very basic properties as connectedness among entities, i.e., the topology itself). In fact, the concept of a minimal region can be introduced in mereogeometry: They are usually called a “point”, but we may well use “pixel” instead. A point in this sense, which is quite obviously rather different from the Euclidean point, is approximately a region that has no proper parts (or rather, a region where no proper parts are considered).[16] When the concept »point« is introduced to the calculus in that manner, there is no need in any concrete instance for using infinitely many point instances: Only the “relevant” points have to be instantiated. This also means that there is always a finite resolution for describing space with such a calculus. However, N. Asher & L. Vieu [1995] have proposed a formal mechanism called “microscopization” covering a kind of zooming operation by means of a modal extension to their mereogeometric calculus: What is a “point” on one level may be a compound of regions with several points on a microscopized level.

The empty picture plane and maximal pixemes
In contrast to space according to standard geometries, pictorial space is usually externally limited: The picture plane consists of one unique maximal pixeme, which is not part of another pixeme of that picture – Saint Martin here uses the espression “basic picture plane” ([Saint-Martin 1987a]Saint-Martin, Fernan­de (1987).
Sémio­logie du lan­gage visuel. Mont­réal: Presses de l’Uni­versi­té du Quebec, Englisch: Semio­tics of Visual Lan­guage. Blooming­ton, India­napolis: India­na Uni­versity Press..

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).[17] The distinction essential to mereogeometric approaches of the closed region – a region that includes its borders – and that maximal part of it not including the border, delivers a direct approach for dealing with frame and picture proper (⊳ frame).
Abb. 1: Rectangular maximal pixeme with “energetic phenomenon” following Saint Martin
If the usual rectangular form of the picture plane is chosen, four regions that are points in the mereogeomeric sense have to be considered in mereogeometric approaches: the four corners. The “energetic load” of the corners indicated by Fernande Saint-Martin in her description of picture morphology (cf. Fig. 1) might be related to the construction of the concept of a point in most mereogeometries: In the calculus of Tarski, for example, points are introduced as the classes of all concentric circles in the situation described ([Tarski 1929a]Tarski, Alfred (1929).
Les fonde­ment de la géo­metrie des corps. Cracovie: Socié­té polo­naise de mathé­matique, Erwei­terte Fassung in Tarksi: Lo­gique, Séman­tique, Méta­mathé­matique. A. Colin, Paris 1972, Vol. 1, 27-34.

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). Therefore in the Tarskian description of a rectangular picture plane, for each corner corresponding circles had to be instantiated as the defining basis for the point. The parts of those circles that are inside the picture plane indeed correspond exactly to those energy lines of Saint-Martin.


Further Aspects
Semantic aspects of geometry

Obviously, the spatial concepts framed in geometric calculi also play a major role in picture semantics: Projecting a three-dimensional scene to a two-dimensional picture plane obeies rules that are in particular (though not only) determined by the calculi of projective geometry. Those considerations are discussed in lemma Perspective and Projection.

Time and Geometry

In physics, temporal extension and order are conceived of as another (i.e. fourth) geometric dimension. Corresponding conceptions may be used in picture philosophy when dealing with film, video, TV or other formats of moving pictures. In the strong sense of physics, the temporal "direction in space" is contrasted to the spatial "directions in space" by taking it as a (mathematically) imaginary axes of a four-dimensional complex vector space, or inversely as the only real axes complementing imaginary spatial components (‘quarternions’).[18] In computer graphics, quarternions are often used to calculate transformations of 3D model. Rotations in particular can then be handled quite easily.


Notes
  1. As a conceptual basis of perception (a ‘condition of possibility of experience’), categories are not perceptible in themselves.
  2. This does not imply that those calculi are sufficent when dealing with semantic and pragmatic aspects; ⊳ theories of picture space.
  3. Cf. also Wikipedia: Topologie (Mathematik).
  4. Cf. also Metrik.
  5. Cf. also projektive Geometrie.
  6. More precisely: Such a description consists of utterances containing a proposition with a predication using the geometric concept determined by the calculus.
  7. That is, ‘being geometrical’ as such does not necessarily imply ‘being spatial’.
  8. In particular, the rules linking geometric concepts with each other, e.g., that »left« is the inverse of »right«, or that »in« is under certain conditions a transitive relation, can be used as "middle terms" in spatial sylogisms relating several propositions about geometric entities. Note however also the components of such deductions exceeding pure geometry as motivated in particular in [Herskovits 1986a]Herskovits, Annet­te (1986).
    Lan­guage and Cogni­tion – An Inter­disci­plinary Study of the Prepo­sitions in English. Cam­bridge: Cam­bridge Univ. Press.

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    or [Aurnague & Vieu 1993a]Aurnague, Michel & Vieu, Laure (1993).
    A Three-Level Approach to the Seman­tics of Space.
    In The Seman­tics of Prepo­sitions – From Mental Process­ing to Natu­ral Lan­guage Process­ing, 393–439.

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    , and the association of those considerations with concept-genetic arguments in [Schirra 1994a]Schirra, Jörg R.J. (1994).
    Bildbe­schreibung als Verbin­dung von visu­ellem und sprach­lichem Raum. St. Augus­tin: DISKI.

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    : in particular Chap. 5.
  9. Cf. also Wikipedia: Euklidische Geometrie.
  10. Cf. Synthetische Geometrie. In particular the "fifth axiom", as it is often called, concerned with the concept of parallelity has been rather fertile, although the classical Non-Euclidean caluli developed in consequence are usually not relevant for picture syntax; cf. Parallelenaxiom.
  11. Cf. also Wikipedia: Analytische Geometrie.
  12. Mereogeometric regions can be conceived of as undividable (‚in-dividuum‘) within the mereogeometric calculus since their properties can not be derived from a composition of more basic elements although they partake in part-whole-relations, and hence could have other regions as their parts.
  13. Here, exceptions substantiate the rule: In some variants, circular regions are considered ...
  14. For a more detailed example of a mereogeometric calculus, see Exkurs:Mereogeometries.
  15. Of course, there is much more in pictorial morphology than this as it was clear from the beginning, see for instance the seminal work of [Goodman 1968a]Goodman, Nelson (1968).
    Lan­guages of Art. India­napolis: Hackett, 2. rev. Aufl. 1976.

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    .
  16. The point in mereogeometry is usually not identical to the minimal region but is defined as a class, namely of all individuals (regions) of which the minimal region is a part.
  17. Of course, this maximal pixeme is still perceived always as a part of its particular spatial surroundings.
  18. Cf. Wikipedia: Quarternionen.
Literatur                             [Sammlung]

[Aurnague & Vieu 1993a]: Aurnague, Michel & Vieu, Laure (1993). A Three-Level Approach to the Seman­tics of Space. In: Zelin­sky-Wibbelt, C. (Hg.): The Seman­tics of Prepo­sitions – From Mental Process­ing to Natu­ral Lan­guage Process­ing. Berlin: Mouton de Gruyter, S. 393–439.

[Borgo & Masolo 2010a]: Borgo, Stefano & Masolo, Claudio (2010). Full Mereo­geome­tries. The Review of Sym­vbolic Logic, Band: 3, Nummer: 4, S. 521 - 567. [Goodman 1968a]: Goodman, Nelson (1968). Lan­guages of Art. India­napolis: Hackett, 2. rev. Aufl. 1976. [Herskovits 1986a]: Herskovits, Annet­te (1986). Lan­guage and Cogni­tion – An Inter­disci­plinary Study of the Prepo­sitions in English. Cam­bridge: Cam­bridge Univ. Press. [Saint-Martin 1987a]: Saint-Martin, Fernan­de (1987). Sémio­logie du lan­gage visuel. Mont­réal: Presses de l’Uni­versi­té du Quebec, Englisch: Semio­tics of Visual Lan­guage. Blooming­ton, India­napolis: India­na Uni­versity Press.. [Schirra 1994a]: Schirra, Jörg R.J. (1994). Bildbe­schreibung als Verbin­dung von visu­ellem und sprach­lichem Raum. St. Augus­tin: DISKI. [Tarski 1929a]: Tarski, Alfred (1929). Les fonde­ment de la géo­metrie des corps. Cracovie: Socié­té polo­naise de mathé­matique, Erwei­terte Fassung in Tarksi: Lo­gique, Séman­tique, Méta­mathé­matique. A. Colin, Paris 1972, Vol. 1, 27-34.

Contributions by: Joerg R.J. Schirra [10] (remark)