From mathematical visualization
Simon Fraser University
Part I of this paper outlines a visual formalism developed for the purpose of defining abstract visual notation (for rational numbers mod 1). It draws on the methodologies of mathematics, physics, and the visual and performing arts. Part II presents a body of visual work showing the extension of the formalism and practice into real time 3D immersive environments.
Part I: Background to the mathematical visualization
My hypothesis was that visual notation could be defined
for mathematics, and I set out to construct a space without textual symbols
and words. I began with the abstract problem of notating individual rational
Along the way, I developed a visual formalism of sorts, which drew on the methodologies of mathematics, physics, and the visual and performing arts. Questions of perception, space, time, movement, color, configuration, superposition, frame rate, and recognition of visual object came to the fore, not separately, but as interrelated components of an abstract visual representation.
It was suggested I look at dance rather than theatre; my visual objects moved, but did not speak. I did not have to look very far before I found an example of visual abstraction.
In 1913, the première of the Ballet Russes Le Sacre du Printemps provoked a riot.
La musique polyrythmique de Stravinski avec, ses dissonances et ses changements continuels de mesure, présentait des difficultés énormes, mais la vraie révolte se fit autour de la choréographie; ce fut tout un drame pour faire accepter à ces danseurs de formation traditionnelle les pas primitifs de Nijinski. Pourtant, les quelques descriptions du ballet faites à lépoque reconnaissent chez celui-ci un remarquable sens visuel et dramatique, un don de faire exploser lénergie interne des rythmes, des couleurs et de la ligne. Nous navons quà nous imaginer la fin de la 1ere partie du ballet où des femmes en robes éclarlates, leurs tresses noires volant derrière elles courent dans une ronde immense autour de la scène, leurs pas sauvages poursuivant les notes du thème principal. A lintérieur et contre leur ronde de feu des hommes drapés de blanc se divisent sans cesse en petits groupes. Les personnages dansent passionnément sur la terre, dit largument, la sanctifiant et fusionnant avec elle. Même un public daujourdhui serait bouleversé par ces tourbillons de rouge et de blanc contre le vert éclatant du décor, par ces danseurs plongeant au fond des polyrythmes pour libérer une force extraordinaire. 
The extension of the roles of music and choreography each into the domain of the other confronted the established language and conventions of dance at that time. Not understanding the inter-dependence of these roles, the opening night audience considered the music and choreography to have separately failed. Abstractly, it was a simple map. Perceptually, it would take some getting used to. As will mathematical visual abstraction.
To set up an abstract visual space for my visual rational numbers involved the questioning of many (unwritten) conventions. Mathematics as a text-based form inherits conventions from natural language ( Nordon 1994). As a spatial form it will inherit conventions from film, theatre, dance, photography and animation. What are the devices used to punctuate these spaces, and why do they work? ( McAuley 1987,   Herbison-Evans 1988,   Barthes 1964, 1970, 1973,  Svoboda 1993.)
In text, mathematical symbols are separated by brackets, commas, dots, operation signs, arrows etc. In an abstract visual space, visual objects are frequently interwoven. In text, the convention is to proceed from left to right, reading the mathematical symbols, and punctuation marks, according to certain established conventions. In an abstract visual space, at the beginning, the preferred viewpoints and motions do not yet exist. They are established experimentally, and they are, at least to some extent, dependent on the type of mathematics notated, and the type of notation chosen to notate it. In other words, initially the stage was very empty. All of the visual objects were built from scratch.
The "empty stage problem", can be thought of as the problem of partitioning, or punctuating of the space in such a way as to encode certain relationships, in this case mathematical. It is a shift of notation types from linear text-based, to three dimensional spatial. But, this change of medium involves giving up almost all of the conventions established in the former space and starting from scratch in the latter.
This is not just a case of defining an experimental space for visual investigation, the results of which are written up in traditional style. The fluidity of thought offered by multiple juxtaposition in this new virtual visual environment is not yet codified. In short, such work could not exist if it was confined by linear language forms.
The operational conventions for the space were initially thought of in terms of theatre, dance, film, and animation, depending on which was the most appropriate form at the time. In two-dimensions, the concerns of the early abstract filmmakers, frequently arose in abstract visual space: frame rate, texture, shapes as parts of movement, film loops, camera angle in movement through the space.
In three-dimensions, abstract dance, and the way it chose to integrate, or not, image, sound, and movement, was the closest to a non-text based abstract visual form on a stage. Abstract dance frequently used superposition (of human form), to create composite abstract visual objects in time. However, neither of these traditional frameworks was entirely appropriate; the mutability of the number of dimensions of space and time was, (and continues to be), a conceptual challenge.
Thus the process of representation was approached as the choreography of abstract elements into an environment whose spacetime dimensions were not fixed. The results produced could be sometimes like film, sometimes like theatre, and sometimes like dance. The viewing of this multiplicity of forms was seen as analogous to the cubists presenting all faces of the object on one flat canvas. In this context, reference material concerning image, diagram, film, theatre, dance, animation, visual art, and musical notation contributed considerable guidance in the initial design and creation of the work.
A formalism for visual abstraction
Define an abstract visual space composed of visual objects and the relationships between them. The visual objects are thought of as being coordinate free. A triangle is an abstract triangle. A torus is an abstract torus. The act of programming them into the space is analogous to drawing a triangle on a blackboard, a whiteboard, or a piece of paper for the purpose of theoretical discussion.
Initially abstract visual space is empty. It is necessary to start with primitive visual objects. The type of primitive visual objects chosen depends on the mathematical structures(s) to be "notated" in the space. I chose to start with two visual primitives: one defined by color, the other by configuration.
At the pixel level, the visual objects are composed of colored dots. However, these dots are usually grouped into nested levels of previously defined objects. Often the structure of these nested levels is explicitly forgotten in favor of the composite visual object.
Only a finite number of colors is defined. But truncating at a given integer value (n) is a practical constraint. I operate with this map as if it were countable, not finite, with the truncation no different in principle to the practice of listing the first few terms of an infinite sequence: 1, 3, 5, 7, 9, and so on.
When working in the space, it is routine to remember previous visual objects. It is assumed that once a visual object is presented, it becomes a known element of the abstract visual space. It can be recalled at any time and put into visual juxtaposition with other visual objects.
Looking at the particular becomes more important than looking at the general, at least for the time that it takes to become perceptually accustomed to the new visual environment. It then becomes possible to find alternative ways to generalize or classify observed patterns. This involved temporarily giving up even the most basic mathematics, and instead exploring a new kind of space in which the algebra of visual primitives replaced the algebra of symbols.
The exploration of long sequences of diagrams composed only of colored lines, or colored dots, or where animation is completely without symbols, is (for the most part) outside maintstream mathematics. However, one interesting exception was found in the diagrammatic arguments of Category Theory, which require the recognition of diagrammatic representation, where each diagram is composed of symbols and arrows.
Similarly, in abstract visual space, the abstraction involves the recognition of visual object as composed of mappings of other visual objects. Hierarchical layering of increasingly complex visual objects is the visual equivalent of the algebraic expression of functions of functions of functions, and so on.
What you are doing is layering abstraction on concrete example; you are preserving the context in a way that arrow diagrams specifically eschew, by taking advantage of visual / cognitive backgrounding. In essence you let parts of your objects become merely iconic ('auto icons') for themselves while focusing on other parts. It's the 'freezing subexpressions' notion from symbolic maths software raised to the level of presentation. 
It is both unnecessary and counterproductive to label all of the visual arrows between these visual objects with symbols, and then to keep track of which one is in use at any particular moment. It is possible to navigate the space, changing (algebraic) viewpoint whenever such a shift assists in understanding the structure.
This is not to say, however, that this shift in perspective is trivial. The visual notation itself encodes algebraic properties of the mathematical structure. In this sense it is less arbitrary than a text-based definition. The term "elegant", which has frequently been used in mathematics to describe an insightful and succinct algebraic proof, can be applied in exactly the same sense to describe particular visual objects in an abstract visual space.
In effect, transforming, concatenating, juxtaposing visual objects, and moving through them in a well defined path in the space, creates the visual arrows of abstract visual space. When two or more visual (sub)objects are mapped together into the same composite visual object, the process is defined as superposition. When two or more visual subobjects are shown separately to occupy the same place in a visual object, in a specified sense, it is defined as substitution. These visual (sub)objects can then be interchanged, one for the other in a visual object of which they constitute a component. This process is analogous to algebraic substitution using symbols.
Theoretically, these forms of visual juxtaposition are the same. However, at the practical level, using superposition usually results in the creation of a new visual object, whereas using substitution adds or removes detail, or changes the visual and mathematical focus altogether.
The transformation, concatenation and juxtaposition of visual objects can also take place as motion in time.
Visual abstraction in time
Recall that the visual primitives encode mathematical structure. Visual objects built from them satisfy (the mathematical structures encoded in) these primitives in the same way that statements made about traditional mathematical spaces must satisfy the axioms of those spaces. What is conceptually more interesting is that "motions" defined on and between these visual objects are also consistent with the primitives of the abstract visual space. In the case that the motion is simple and elegant, that generic motion becomes a visual notation for that structure.
By construction, such visual subobjects, defined mathematically and sequenced in time, have an a priori "regular" structure, but the perception of this structure is a non-trivial question of choreography. They are abstract films in which the content is more often rhythm than narrative, as is the case in Ferdinand Legers Ballet Mécanique:
"It is also the first explanation of another form of rhythm that constructed from the change of the image itself. Lengths of film are used deliberately in a rhythmic beat structure. Though this structure may relate to movement or rhythm within the frame, it is essentially of a more basic order, analogous to the relationship of drumbeat to melody in music." 
When a pattern is perceived in a motion it can be analyzed and backwards decomposed to the original visual primitives. The analysis of patterns as visual subobjects formalizes them as known elements of the space. In the initial abstract visual space, subsequences of rational numbers appeared as rhythms, textures or interweavings, visible only in animation. (In the extension of the work into 3D immersive environments my focus has shifted almost entirely to perception of visual rhythmic environments.)
Many musical structures are perceived as a result of the physical relationships between the individual components. In both multiple aural juxtaposition (polyphony), and multiple visual juxtaposition (visual objects defined in time), patterns can be perceived independently of the musical or algebraic structures which determine those patterns.
Are there conventions which have been developed for the recognition of dynamic structures in musical notation which can be applied, or modified, in abstract visual space, for in both cases the dynamic abstraction is explicitly defined in time, and then perceived in time, without language?
"Throughout the twelfth, thirteenth, and fourteenth centuries the mechanics of notation were in a state of continuous flux and rapid change, produced and paralleled by an evolution in musical style the progress of which lies mainly in the field of rhythm." 
Aural abstraction in time
Rhythm can be thought of as partitioning the abstract notation space.
The intricate rhythmic structures of the School of Notre Dame of the 14th century (music composed by the calculated partitioning of its time dimension) could not have been understood, or even imagined by the monks of the 11th century (for whom music was associated to the linguistic structure of the phrase sung to God).
It took 300 years from the writings of Guido dArezzo (1025),
"[...] The sounds, then, are so arranged that each sound, however often it may be repeated in a melody, is found always in its own row. And in order that you may better distinguish these rows, lines are drawn close together, and some rows of sound occur on the lines themselves, others in the intervening intervals or spaces. Then the sounds on one line, or in one space all sound alike. And in order that you may also understand to which lines or spaces each sound belongs, certain letters of the monochord are written at the beginning of the lines or spaces and the lines are also gone over in colors, thereby indicating that in the whole antiphoner and in every melody those lines or spaces which have one and the same letter or colour, however many they may be, sound alike throughout, as though all were on one line. For just as the line indicates complete identity of sounds, so the letter or colour indicates complete identity of lines and hence of sounds also." 5
Guido from Prologus antiphonarii sui [ca. 1025] 
to the writings of Jean des Muris (1319),
At the end of this little work be it observed that music may combine perfect notes in imperfect time (for example, notes equal in value to three bréviores) with imperfect notes in perfect time (for example, notes equal in value to two breves), for three binary values and two ternary ones are made equal in multiples of six. Thus three perfect binary values in imperfect time are as two imperfect ternary ones in perfect, and alternating one with another they are finally made equal by equal proportion. And music is sung with perfect notes in perfect time, or with imperfect ones in imperfect, whichever is fitting. 
During this time, development of a notation for rhythm was hampered by the assumption that one could not sing multiples of two to God; only multiples of three. In 1319, Jean des Muris (above), mathematican, University of Paris explained that two times three is equal to six. A Papal Bull banning polyphony was issued in 1324-25.
A paper by Dufourt, Musique, mathesis, et crises de l'antiquité à l'age classique, in Mathématiques et Art led me into the history of polyphony, in the context of its spatial notation. Two parallels were of particular interest; the defining influence of the notation on the direction of western music:
"La notation diastématique est à la fois un procès organisateur, tectonique et un procès analytique, déterminant. Elle assigne une place fonctionnelle à des termes qui vont se déterminer lun à lautre. La portée de Guy dArezzo, cest à la fois un médiat dopérations transformatrices et la distinction du tout et de la partie. La démarche nest pas seulement abstractive, elle est instaurative. Elle implique lintelligence des rapports, le deploiement dun système de relations, lobservation des règles dappropriation objective.
Elle announce une histoire en la rendant possible." 
and the indirect profound transformation it had on western thought:
"Primat de loralité, de la linéarité, du déploiement univoque dun ordre syntaxique, référence absolue de la musique au code phonétique: telle est linjonction de Jean XXII.
La fameuse bulle qui condamne les tendances modernistes dans la musique est lun des documents les plus significatifs de lépoque, témoignant non seulement dun conflit entre deux poétiques ou deux styles antagonistes, mais surtout dun désaccord entre deux façons de concevoir la musique. Lune la rapporte au langage, lautre à lespace.
[ ]Lécriture musicale remplit ainsi une fonction dobjectivation qui se distingue de la fonction de communication du langage naturel et suscite, avec lintroduction de ses nouvelles syntaxes, des transformations catégorielles profondes dans lappréhension de la réalité." 
It is my personal opinion that in mastery of the new visual notation space offered by the graphic computer interface we are closer in time to Guido dArezzo than Jean des Muris.
Part II: Immersive abstraction
The aforementioned journey provides me with a visual formalism, which I no longer confine to mathematics alone. My work has become immersive abstraction. I do not see colored dots, but intricate interwoven hierarchical structures flowing in time.
The elements of these spaces are not single points (which as often as not have finite periodic orbits), but alignments which coalesce and disperse; they come into and out of existence like elementary particles. Their components are each defined across the entire space and time and driven by a single deterministic flow, yet is it their local proximities in space and time that together create the visual rhythms in the space.
I am interested in them as a means of perceiving mathematical structure as motion, as a means of visualizing non-local relationships between elements, as motifs or brushes in immersive visual environments and as recognizable visual linguistic elements. My understanding of these objects is interrelated across the interdisciplinary objectives outlined; each feeds into the exploration of the other. This is reflected in collaboration with mathematicians, with visual artists / performers, and with those working in data perception / information visualization.
Flowing rational lattices a study in visual rhythms
A rational lattice flowing in a unit cube.
Create a 3D rational lattice by scaling integer lattices to lattices in the unit cube. Place the lattices together in superposition. Rescale the x, y, z coordinates of each point by integer frequencies A, B, C. Multiply by a flow time parameter which starts at zero, mapping the entire visual object to a single point at time zero. A 2D side view is shown:
Now start the flow, but take a quotient of three dimensional space by an integer lattice. Points leaving the cube at one face will appear immediately on the opposite face.
Observe that the visual object is flowing with three frequencies in orthogonal directions. Rational alignments can occur in one, two or three dimensions, at different times and locations in the cube. Three Fakespace screen shots are shown on the following page:
A rational lattice flowing on a torus.
Reduce to a 2D rational lattice in the unit square. Rescale the x, y coordinates by two integer frequencies A, B. Start the flow, take a quotient , of two dimensional space by an integer lattice, and this time view the result on a torus, rather than on a unit square.
Effectively the rational lattice endpoint (1,1) flows along the torus knot defined by the (now) longitudinal and meridian frequencies A, B of the scaling.
The dominant visual structure is the local proximity (in time) of simple rational alignments in the flow. Lattice points converge to, and diverge from, flow time events.
However, at an exact rational alignment in the flow (above center), many points are mapped simultaneously to the same point, and much of the visual information in the object is lost momentarily.
Spiralling the flowing rational lattice into a volume visual artifacts
By staggering the longitudinal and meridian radii of the torus as functions of the flow time, visual information at exact rational alignments remains visible. This map also introduces interesting visual artifacts that will be harnessed in later examples.
Three laptop screen shots are shown on the following page:
Visual artifacts as motifs or brushes
As the flow time increases the staggering of the longitudinal and meridian radii result in the appearance of spiral tendrils whenever a rational alignment approaches. This visual artifact is interesting in its own right.
It can be extracted by retaining information about the flow time of its alignment. As a motif or brush, it can be repeated to form a new visual object. Below is a screen shot of a breathing object made of spiral tendrils.
Introduce a membrane around the spiraling tendrils using a lattice flowing on a breathing sphere. Screen shots from Starfish Ocean JK16 (Rotterdam July 2002).
Visual artifacts as linguistic elements
The visual artifacts of these spaces are not single points but alignments which coalesce and disperse. Teased apart and viewed in visual juxtaposition, they have cycles and behaviors that can be recognized, and possibly learnt.
In a sense, they are dynamic squiggles. Human writing started from squiggles.
So did musical notation.
Visual formalism, historical conceptual shifts in the arts, the evolution of abstract notation systems, the play of mutable modalities in new media and technologies, the problem of partitioning or punctuating visual time-based immersive spaces, the challenge of defining recognizable visual linguistic elements, and the possibility of mathematical visual abstraction are all intimately related. Encoding these interrelationships directly or indirectly into computer based environments is non-trivial, but already happening. As often as not, conceptual and representational shifts in notation and thought came out of the arts. In this context, the role of computer art would seem central.
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About the history of western musical notation:
63 Apel, Will (1970) Harvard Dictionnary of Music. (1973) London: Heinemann Educational Books Ltd
64 Dom Anselm Hughes (1954) The Birth of Polyphony (pp270-286), Music in the Twelfth Century (pp287-310), Music in Fixed Rhythm (pp311-352), The Motet and Allied Forms (pp353-417) in The New Oxford History of Music II, Early Medieval Music up to 1300, First Edition 1954, Revised in 1955, reprinted in 1961 and 1967.
65 Domling Wolfgang (1985) Musique du Moyen Age translation of Musik der Gotik, notes accompanying CD Music of the Gothic Era, The Early Music Consort of London (1976)
66 Dufourt, Hugues (1995) Musique, mathesis, et crises de l'antiquité à l'age classique, in Mathématiques et Art, Colloque 2-9 Sept 1991. Paris: Hermann
67 Grout, Donald Jay (1970) A History of Western Music. (1960) Revised Edition London: WW Norton and Company Inc DENT
68 Guido dArezzo (ca.1025) Extract from Prologus antiphonarii sui in Strunk, Oliver (1965) Source readings in music history: Antiqity and the middle ages, Selected and annotated by Oliver Strunk New York: WW Norton & company
69 Harman, Alec (1962) Man and His Music, The Story of Musical Experience in the West. London: Barrie and Jenkins 1973
70 Jean des Muris (1319) Extract from Ars novae musicae in Strunk, Oliver (1965) Source readings in music history: Antiqity and the middle ages, Selected and annotated by Oliver Strunk New York: WW Norton & company
71 Lang, Paul Henry (1941) Music in Western Civilisation. (1978) Revised Edition London: WW Norton and Company Inc DENT
72 Miller, Hugh H (1972) History of Music. 1947, 1953, 1960, 1972 New York
73 Munrow David (1976) Music of the Gothic Era, notes accompanying CD Music of the Gothic Era, The Early Music Consort of London (1976)
74 Pérès Marcel (1984) , notes accompanying CD Polyphonie Aquitaine du XIIe Siècle Saint Martial de Limoges, Ensemble Organum
75 Reaney, Gilbert (1960), Ars Nova in France in The New Oxford History of Music III (pp1-30), Ars Nova and the Renaissance 1300-1540, Oxford University Press 1960, First published 1960, reprinted 1964.
76 Risset, Jean-Claude (1991) Aujourdhui, le son musicale se calcule in Loi, Maurice (Ed) Mathématiques et Art, Colloque 2-9 Sept 1991. Paris: Hermann (1995)
77 Seay, Albert (1975) Music in the Medieval World. 2nd Ed Englewood Cliffs New Jersey : Prenctice Hall 1975
78 Strunk, Oliver (1965) Source readings in music history: Antiqity and the middle ages, Selected and annotated by Oliver Strunk New York: WW Norton & company
79 Stephen P Spackman, (2000) discussions April-May 2000 <firstname.lastname@example.org>
80 Apel,The notation of Polyphonic Music (Cambridge, Mass., 1942), pp.199-200
81 from Antiphonarium E-2. Gesange fur den vierten und funften Wochentag der Karwoche. Schottenstift, Wien.12 JH. 1200. p52 in Musik in Oesterreich, eine chronik in daten, dokumenten, essays und bildern. (1989) Publisher Verlag Christian Brandstatter. Christian B Verlag Editions, Wien.
82 Tolmie, J.A. (2000) Visualisation,
Navigation and Mathematical Perception: A Visual Notation for Rational
Numbers Mod 1, [Ph.D. Dissertation], Canberra: Australian National University,
4CDs, 300 animations.
83Tolmie, J.A. (2002) Starfish Ocean JK16, rational particle system work in progress, residency July 2002, V2 the Institute for the Unstable Media, Rotterdam.