TThe Topology ofShapes Made withPoints
SAGE
Alexandros Haridis Abstract
In architecture, city planning, visual arts, and other design areas, shapes are oftenmade with points, or with structural representations based on point-sets. Shapesmade with points can be understood more generally as finite arrangements formedwith elements ( i.e. points) of the algebra of shapes U i , for i = 0 . This paperexamines the kind of topology that is applicable to such shapes. From a mathematicalstandpoint, any “shape made with points” is equivalent to a finite space, so thattopology on a shape made with points is no different than topology on a finitespace: the study of topological structure naturally coincides with the study of preorderrelations on the points of the shape. After establishing this fact, some connectionsbetween the topology of shapes made with points and the topology of “point-free” pictorial shapes (when i > ) are discussed and the main differences between thetwo are summarized. Keywords
Shape with Points, Finite Order Topology, T -space, Structural Description,Mathematics of Shapes Department of Architecture, Massachusetts Institute of Technology, USA
Corresponding author:
Alexandros Haridis, 77 Massachusetts Avenue, Room 10-303, Cambridge, MA 02139, USA.Email: [email protected]
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Mathematics of shapes
In a previous investigation (Haridis (2020)), topology was studied on (point-free) shapesmade with basic elements of the algebras of shapes U i , when i > . What was leftundiscussed is the topology applicable to shapes made with points, i.e. when i = 0 . Thatis the subject of this present paper. Shapes made with points
Before expanding on the main topic of this paper, it is useful to give a clear idea of whatis meant by a point and a shape made with points .A well-known approach is to consider the point as an infinitesimally small entity oran abstract (immaterial) member of a set—this is common, for example, in mathematics.More generally though, a point can be considered as a representation or surrogate ofa concrete (material) object—an object that has physical presentation, can be seen ortouched. We see this many times, for example, in physics when modeling the motionof physical objects and also during composition with shapes in drawings and models inarchitecture and the visual arts.What are the properties of a “concrete/material object” when we say that it behaves likea point? First of, the notion of point cannot be associated with a specific mark or materialfigure. A point is agnostic of how it looks. This is nicely expressed in the following quoteby the Russian painter W. Kandinsky:“Externally, the point may be defined as the smallest elementary form,but this definition is not exact. It is difficult to fix the exact limits of‘smallest form.’ The point can grow and cover the entire ground planeunnoticed, then, where would the boundary between point and plane be?...In its material form, the point can assume an unlimited number of shapes(Kandinsky 1947, pp. 29-31).”
Figure 1.
Redrawn from (Kandinsky 1947, p. 31).
Prepared using sagej.cls lexandros Haridis Any one of the shapes in Figure 1, for example, may equally function as a point. And itis only by convention that a point is often represented as a small circle (void or solid).Thus, an object that behaves as a point must have certain properties that have nothing todo with the way it looks.An object is said to behave like a point whenever certain assumptions have been madeabout how one interacts with the object and how the object is supposed to interact withother objects, when put together in a spatial composition. These assumptions (to beexplained shortly) are formally captured in the mathematical framework of the algebraof points U , which is part of the shape algebras U i ( i ≥ ) invented for purposes ofcalculation with shape grammars .Points are the basic elements in an algebra U —lines, planes, and solids, are the restof the elements in the algebras when i > . The algebraic properties of points, and theirrelation to the other basic elements in the series, are described in detail elsewhere (e.g.Stiny (2006) provides a comprehensive coverage). Intuitively, the basic properties ofpoints can be summarized as follows. A point is an element without proper nonemptyparts. Unlike with lines, planes or solids, one cannot perceptually recognize parts in apoint; a point has one part only, namely, itself. Symbols or indivisible units, such as “0”sand “1”s, or “a”s, “b”s and “c”s, used in linguistics and logic, behave similarly. Justlike symbols in a set, points stay mutually impenetrable when put together in a spatialcomposition. They are meant to stay distinct and exclude one another, unless if they areidentical, in which case (and only in this case) they fuse into one. Figure 2.
Design for cover of exhibition catalogue “H. Matisse,” by Henri Matisse, 1951,gouache on paper, cut-and-pasted, 27.0 x 40.0 cm (© 2019 Succession H. Matisse / ArtistsRights Society (ARS), New York).
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By a “shape made with points”, we understand any finite arrangement formed withspatial elements of any form, size or shape, that function as points. One spatial examplecoming from the visual arts is the concept of the collage . A collage is a type of artwhere one sticks different elements together to form a composition, for example, theone in Figure 2, made with paper cut-outs. The elements in a collage stay distinctfrom one another and do not materially fuse when combined. The elements may fusein observation, when one looks at the resulting arrangement, and give rise to emergingforms and meanings; one may say, for example, that in Figure 2 there is a shape thatlooks like an “eyebrow” which extends downwards to become the “nose” of a partiallyvisible face. Nonetheless, as far as the interaction is concerned, if one tries to touch thecollage, to rearrange its elements in some other way, only the independent compositionalelements can be manipulated.Besides collages, one can think of many other kinds of shapes, used in a wider range ofareas, formed with elements that are made to have point-like behavior. While the materialappearance of the elements is different each time depending on the application and use,their point-like properties are uniformly the same. Some examples are: building models, , , ,, ,, , , , , shape made with points vocabulary elements(points) shape made with points vocabulary elements(points) , , //// / ////// ////// ////// ////// ////// ////// ////// /////////////// ////// ////// ////// ////// ////// ////// ////// /////// &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&& &&& &&& &&& &&& &&& &&&&&&
BBBB BBB BBB BBB BBB BBB BBB BBB BBB BBB BBB BBB BBBBBBBBB //////////////////////////////// .......... ........
Figure 3.
Examples of shapes made with points. Each shape comes with the vocabularyelements (points) out of which it is composed of.
Prepared using sagej.cls lexandros Haridis physical or digital, for architectural design determined by a “kit of parts;” Froebel-like,or similarly, Lego type designs; digital images, ASCII computer artworks and voxeldescriptions of three-dimensional form; hatching systems for filling discrete regions inthe plane; spatial configurations made with primitive components and relations; electricalcircuit diagrams, and other graph-based descriptions of form, and so on. Figure 3 showssome such examples of shapes made with points.Recall that when shapes are treated as “point-free”, i.e. unanalyzed, objects (whenmade with basic elements of algebras U i when i > ), they can be interpreted into partsin indefinitely many ways, independently of what pieces were originally used to makethem (Stiny (2006)). In contrast, shapes made with points cannot be interpreted intoparts in ways other than what their points allow. By a part of a shape made with points,one essentially means a subset of points of the shape. The possible parts that can berecognized and manipulated are only those that can be formed by combining the pointsthat were originally used to form the shape: parts are subsets of points, and no more.This is explained in a more graphical manner in Figure 4. The figure shows a shape madewith points (in this case a motif, extracted from the collage in Figure 2) along with a setof visible and possible parts and a set of visible but non possible parts. While the nonpossible parts are embedded in the shape—we can see them—, it is as if they do notexist from an interpretative standpoint: we cannot combine points to form these parts.(Obviously, one can cut the elements themselves to get a new revised set of points, butthe intended argument here with respect to perceptual interpretation remains the same.)Returning now to the main topic of this paper, the fact that shapes made with pointscan be interpreted into parts only in terms of subsets of points, makes topology on shapesmade with points no different than topology on finite sets. In particular, a topology on ashape made with points is solely a matter of inducing an appropriate structure on the set ofpoints out of which the shape is made of. In this sense, the study of topological structurenaturally coincides with the study of preorder relations on points. After establishingthis fact in the next section, some connections between topologies for shapes made withpoints ( U ) and topologies for point-free shapes ( U i , for i > ) are discussed, and themain differences between the two are summarized.In the remaining paragraphs, S represents a shape made with n number of points, forsome natural number n > , and T a nonempty (finite) set of parts of S , satisfying thethree basic requirements for a topology given in Haridis (2020). The operations of sum(+) and product ( · ) that make up the algebraic structure of T , coincide with union andintersection of sets. Moreover, the part relation for shapes ( ≤ ) coincides with the subset Prepared using sagej.cls
Environment and Planning B: Urban Analytics and City Science XX(X) ( ⊂ ) relation for sets and the statement that p ≤ S , for any point p of S , has essentiallythe same meaning as the set-theoretic containment relation, that is, p ∈ S . Order topology on a shape made with points
A finite topology and a preorder on S are mutually inverse constructions. Start with apreorder relation on the points of S and derive a topology T out of this relation or,conversely, start with a topology T for S (by presenting the open parts) and derivea preorder relation on the points of S based on this topology. In other words, a finitetopology and a preorder on S represent the same combinatorial object, considered fromtwo different perspectives. A technical explanation of this argument now follows.Suppose S is equipped with a preorder (reflexive and transitive relation), denoted withthe symbol (cid:22) . For each point p of S , the minimal open part that contains it is given by, U p = (cid:88) q, for all q (cid:22) p. (1) , , , , , shape made with pointsvisible and possible parts visible but non possible partsvocabulary elements(points) Figure 4.
Shape made with points (top/left), and the vocabulary elements it is composed of(top/right). The possible parts of the shape are those that can be formed by combining one ormore points of the shape (bottom/left). There are visible parts of the shape that are notpossible to be formed by combinations of points (bottom/right).
Prepared using sagej.cls lexandros Haridis (By reflexivity, p is in U p ). Let U = { U p } p ∈ S be the collection of all minimal openparts containing each of the points of S . Then U , augmented with the empty shape ( ),constitutes a basis that generates a finite topology T on S . We call this topology the ordertopology on S . The collection U is a unique minimal basis for T .In the opposite direction, suppose S is equipped with a topology T . For each point p of S , the minimal open part U p that contains it is equal to the product of all open parts in T that contain p .Now, let p and q be any two points of S and suppose U p and U q are the minimal openparts that contain them. Define a preorder (cid:22) on S by, q (cid:22) p if q ∈ U p . (2)A working example, demonstrating the use of (1) and (2), is given at the end of this paper.As discussed in Haridis (2020), many classical concepts for topological spaces willnot always apply to shape topologies in a natural manner. Many concepts may not berelevant in general, especially those expressed in terms of points. This is not, however,the case with topology for shapes made with points. Classical topological concepts,including separability, connectedness, continuity, homotopy type and others that havebeen worked out for finite topological spaces can be adapted for shapes made withpoints from U , without significant alterations. The relevant topological toolkit can befound in the canonical literature on finite algebraic and combinatorial topology ( e.g. Aleksandrov (1956); Stong (1966); Sharp (1966); Kozlov (2008); Barmak (2011)). A fewbasic examples follow for purposes of demonstration only; these are not much differentfrom the equivalent constructions for finite spaces.An order topology T for S can be said to satisfy the T separation axiom if, and onlyif, the preorder relation implied by T is a partial order (that is to say, if the preorder isalso antisymmetric). To decide if a given order topology T for S is indeed T , one mayuse the following well-known criterion (adapted from Sharp (1966)): T is T if for anytwo points p and q of S , U p = U q implies p = q (see also the Example for an illustration.)An order topology T is said to be discrete if, and only if, all points of S are open andclosed at the same time in T , in which case T can also be said to satisfy the T separationaxiom. Note the notion of discreteness here has the same meaning as in topologies forsets. To decide if an order topology is discrete, one may use the following criterion: T Prepared using sagej.cls
Environment and Planning B: Urban Analytics and City Science XX(X) is discrete if, and only if, for all points p of S it holds that p = U p (the point itself is thesmallest open part containing it).When the points of a shape are known and given in advance, to define a topology oneneed only “interrelate” these points in a certain special way. For design applications, thefollowing two ways of defining topologies for shapes made with points appear relevant.Using a set grammar (this special concept of grammar I am referring to is describedin Stiny (1982)), define as points the distinct compositional parts (with or without labels)that participate in spatial relations given in the rules of the grammar. A design generatedby a set grammar is essentially a shape made with points. Induce an order topology on adesign in the following manner: define a preorder relation over the points of the designbased on the sequence of rule applications followed to generate it ( e.g. say that p (cid:22) q if the point q is generated by a rule applied to p .); then, the topology follows from theconstruction described in (1).Another way is to use graph representations of shapes. After representing a shapeas a transitively oriented directed graph (Evans et al (1967)), an order topology can begenerated on the shape based on this graph, by establishing a preorder relation as follows: q (cid:22) p , if there is a directed edge ( p, q ) in the graph of the shape, going from vertex p tovertex q (i.e. if q is adjacent to p ). Thereafter, the minimal open parts for all points of S can be calculated using (1). Similar methods of constructing order topologies basedon graphs are described, for example, in Merrifield and Simmons (1980) for modelingthe structure of protein molecules, and in Kong and Rosenfeld (1989) for the design oflow-level image processing algorithms.What happens with order topology when one does not work with points to begin with?Put differently, is it possible to somehow define an order topology on unanalyzed shapes(when i > )? A straightforward way to do this is to convert a pre-existing topologyfor the shape into an order topology. This conversion amounts to constructing a certainkind of mapping (a many-to-one transformation), which is sketched in the followingparagraph.Suppose that C is a shape in an algebra U i , for i > , and is induced with a shapetopology T C . Further, let B = { b ,..., b k } be the reduced (minimal) basis that generates T C , for some natural number k . Then, shape C can be cast into a set of points relative to T C by considering as “points” the basis elements of B (Haridis (2020)).Let M be the relation matrix corresponding to T C . A preorder relation (cid:22) can bedefined on the points of C based on M by, Prepared using sagej.cls lexandros Haridis b i (cid:22) b j if m ij = 1 (3)where m ij is the value of matrix M at index ( i, j ). Using the preorder obtained by (3),define the minimal open parts containing each of the points of C using (1), and fromthere, generate an order topology T for C in a manner already described.This procedure maps any shape topology T C , with k number of basis elements, intoa corresponding order topology T , with k number of points. It is not so hard to seethat the order topology obtained from this procedure must be at least T . Every relationmatrix for a shape topology corresponds to a partial order (immediate from the fact thatfinite topologies for shapes when i > are essentially partially-ordered sets). Thus, thepreorder (cid:22) obtained by (3) is also a partial order by construction, so that T must be a T topology. T may additionally satisfy the T separation axiom if, and only if, the originalshape topology T C is totally disconnected.Two differences between shape topologies for unanalyzed shapes and order topologiesfor shapes made with points are immediate. For a shape made with n number of points,there is both a smallest and a largest finite topology—the indiscrete topology, with onlytwo open parts, and the discrete topology with 2 n open parts. For an unanalyzed shape,there is a smallest finite topology (namely, the indiscrete one) but there is no largest finitetopology. Moreover, a shape made with points has a fixed and absolute underlying space,namely, the set of its points. This space will not change under different topologies—a newtopology will only change how points are structured into subsets, but never the pointsthemselves. On the other hand, there is no fixed underlying space for a shape withoutpoints; such a space can be defined only relative to a shape topology and it changesunder different shape topologies.There is another noticeable gulf between doing topology on shapes without points ( i > ) and doing topology on shapes made with points ( i = 0 ). The former are determined inan open ended interpretative process, where one has to choose which parts of a shape torecognize in order to define a topological structure—there are indefinitely many ways todo this, since the appearance of a shape without points can be interpreted in indefinitelymany different ways. The latter are determined by different ways of structuring the samefinite set of points, which is given in advance—there are only finitely many ways to dothis, since there can be finitely many different topologies on the same set of points (e.g.see Erne and Steger (1991)). Prepared using sagej.cls Environment and Planning B: Urban Analytics and City Science XX(X)
In conclusion, this paper along with the accompanying study in Haridis (2020) shouldgive a unified portraiture of finite topology for shapes formed with basic elements of thealgebras U i , for i ≥ . When all that matters in a particular investigation is how a setof fixed and indivisible points of a shape are interrelated with one another, then ordertopology becomes a usual pick. On the other hand, if an investigation is concerned withthe more open-ended interface between the structure and appearance of shapes, thenshape topologies are the more appropriate choice. p4p2p6p5 p3p1 a.b. d. c. p1 p1p2p6 p3 p4p5 p1p2p6 p3 p4p5p1p6 p3p4p5p5 S = = p4 p4p3p3 p3p4 p3p4p4p6 p4p6p4p6 p4p6p5 p3p3p4p6 p5 p4p6p4p6 p4p1 p3 p4p1 U p1 = { } U p2 = { } U p3 = { } U p4 = { } U p5 = { } U p6 = { } Figure 5. (a) Shape made with points ( S ) and empty shape ( ). (b) An order topology T onthe shape. (c) A lattice diagram describing the order relations between the points of the shapebased on T . (d) Minimal open parts for each of the points of the shape. Prepared using sagej.cls lexandros Haridis EXAMPLE Let S = { p , p ,..., p } be the shape in Figure 5a formed with the pointsgiven in Figure 4; the labelling of the points of S is arbitrary. (Note a labelled and a non-labelled shape made with points have no actual mathematical differences. The labellinghere is only for convenience, so that it is easier to reference the points of the shape.)To specify a topology on S , build a collection T of recognized parts of S , combiningpoints of S , such that the resulting collection satisfies the conditions for a topology—thisis shown in Figure 5b. In addition to the parts shown in Figure 5b, T must also contain S itself and the empty shape (shown in Figure 5a). The minimal open parts containingeach point pi of S are shown in Figure 5d; for each point, the minimal open part is theproduct of open parts in T that contain it. Thereafter, the preorder relation implied on S based on T can be defined using the statement in (2): p4 (cid:22) p6, p4 (cid:22) p3, p6 (cid:22) p5, p3 (cid:22) p1, p1 (cid:22) p2, and p5 (cid:22) p2. These are summarized with the lattice diagram shown inFigure 5c. If (pi, pj) is an edge of the diagram, then pi (cid:22) pj (this means pj “covers” pi)and U pi ⊂ U pj . One may also want to confirm the minimal open parts shown in Figure5d using the statement in (1). Topology T satisfies the T separation axiom: U pi = U pj implies pi = pj, for any two points pi and pj of S . Thus, the preorder obtained from T onthe points of S is a partial order. Declaration of conflicting interests
The Author declares that there is no conflict of interests.
Funding
This research received no specific grant from any funding agency in the public, commercial, ornot-for-profit sectors.
Acknowledgements
This is a preprint of an article published in
Environment and Planning B: Urban Analytics and CityScience (SAGE), available at: https://doi.org/10.1177/2399808319827015 . References
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