Originally published in Leonardo, Vol. 22, Nos. 3/4, 1989, pp. 397-402.

Holopoetry and fractal holopoetry: Digital holography as an art medium

Eduardo Kac

A holographic poem, or holopoem, is a poem conceived, made and displayed holographically. This means, first of all, that such a poem is organized non-linearly in an immaterial three-dimensional space and that even as the reader or viewer observes it, it changes and gives rise to new meanings. Thus as the viewer reads the poem in space — that is, moves around the hologram—he or she constantly modifies the structure of the text.
A holopoem is not a poem composed in lines of verse and made into a hologram, nor is it a concrete or visual poem adapted to holography. The sequential structure of a line of verse corresponds to linear thinking, whereas the simultaneous structure of a concrete or visual poem corresponds to ideographic thinking. The poem written in lines, printed on paper, reinforces the linearity of poetic discourse, whereas the visual poem sets words free on the page. Like poetry in lines, visual poetry has a long ancestry, which runs from Simias of Rhodes, through the Baroque poets, to the Modernists Marinetti, Tzara, Cummings and Apollinaire, and most recently to the experimental poets of the 1960s and 1970s.
Following in this tradition, while at the same time attempting to open up a new path, holopoetry began in 1983 by freeing words from the page, using a system that allows duplication and mass production. As distinguished from visual poetry, it seeks to express the discontinuity of thought; in other words, the perception of a holopoem takes place neither linearly nor simultaneously but rather through fragments seen at random by the observer, depending on the observer’s position relative to the poem. Perception in space of colors, volumes, degrees of transparency, changes in form, relative positions of letters and words, and the appearance and disappearance of forms is inseparable from the syntactic and semantic perception of the text. Color is not simply color; it has a poetic function as well. A letter is not just a letter but also a pictorial shape.
If we compare the elements of language with the basic concepts of Euclidean geometry, as Bense has done in the analysis of visual texts [1], we may think of letters as points, words and sentences as lines, and visual texts as planes. Thus, letters would have dimension 0; sentences, dimension 1; and visual texts, dimension 2. By extension, holopoems, which free the text from the page and project it into space, would have dimension 3.
But a hologram need not necessarily be three-dimensional for fractal geometry tells us that there are dimensions in between those numbered with whole numbers, and we have software tools for creating images with fractional dimensions. Fractals teach us to accept the fraction, the passage from one dimension to the next, as a new value in its own right. Euclidean geometry then becomes a part of fractal geometry, since dimension 2 is in between dimensions 1.9 and 2.1, for instance. To work with holographic fractals is to generate holographic images with dimensions other than 3.
In mathematics, being a fractal means roughly being between a given dimension and the next higher or lower one. In art, being a fractal may mean, by analogy, being between the verbal and the visual dimension of the sign [2]. Taking Bense’s analogy a step further, we might try to conceive a language — moving and changing in space-time — that would consist of this passage from the verbal code (the word) to the visual code (the image). Perhaps the aesthetic experience generally, or the poetic experience specifically, will be enriched if the viewer or reader sees a work that is alternately a text and an image.
Neither the present essay nor the fractal holopoem Quando? (When?), which I shall discuss here, purports to offer definitive solutions for the problems that arise on this new path. The object is rather to explore experimentally the limits, and the prospects, of holopoetry.


Poetry is an art that uses words as its raw material. Visual poetry enriched the word, giving it physicality on the surface of the paper and extending this physicality to other materials, as in the case of poems made from wood, plexiglass, glass and metal. The Brazilian neoconcrete poets made such experiments in the 1950s and 1960s [3].
Holopoetry belongs to the tradition of experimental poetry, but it treats the word as an immaterial form, that is, as a sign that can change or dissolve into thin air, breaking its formal stiffness. Freed from the page and freed from other palpable materials, the word invades the reader’s space and forces him or her to read it in a dynamic way; the reader must move around the text and find the meanings and the relation that the words establish with each other in empty space. Thus, a holopoem must be read in a broken fashion, in an irregular and discontinuous movement, and it will change as it is viewed from different perspectives.
When one reads a conventional text or looks at the world around one, slightly different images are perceived by each eye. But in the reading of a book, newspaper or printed poem, this perceptual process is not evident, nor does it affect what is being read in any fundamental way: what the left eye sees is virtually the same as what the right eye sees. In the case of a holopoem, however, the reading is a synthesis of the two different inputs received by the eyes and is therefore something more complex and intense. This is where the concept of ‘binocular reading’ comes in: we are constantly changing the way we mentally ‘edit’ the text, based on the different inputs taken in during the different fixations of each eye on the letters in space.
The linguistic relation that produces meaning — syntax — is constantly changing because of the reader’s perceptual activity. The holopoem’s ‘perceptual syntax’ is conceived so as to give it a mobile structure and thus extend its expressive power to encompass time, since the words are not fixed upon a surface but rather float in space. For instance, in Quando? the viewer will read, depending on his or her perspective at the moment, the adverb lentamente (slowly) or see it change into the noun mente (mind) and the adjective lenta (slow). From a third point of view, one can read mente as a verb preceded by a luz: a luz mente (‘light lies’, in the sense of ‘tells lies’).
Holopoetry started out as a research project on the possibilities of holography in poetry and visual art, and each of my holopoems made between 1983 and 1988 explores some of these possibilities. Current emphasis is on experiments with the interval or passage from the poetic to the pictorial and vice-versa.


Holo/Olho (Holo/Eye), my first holopoem, made with Fernando Catta-Preta (1983), is a combination of anagrams in which the word holo mirrors olho and vice-versa. The mirroring effect, however, was conceived so that fragments of the poem would contain enough letters to form both holo and eye. The arrangement of letters in space was holographed five times; each hologram was fragmented and the five holograms were reassembled in a new visual unit. This holopoem was an attempt to recreate, in its own syntax, a structure that would correspond to the holographic model, according to which the information of the whole is contained in the part and vice-versa.
Then came Abracadabra, a holopoem created with Catta-Preta between 1984 and 1985. This is the work that best illustrates the concept of discontinuous space, because the use of three reference beams (laser beams aimed at the holographic film rather than at the object, when the image is produced) allowed us to predetermine the region in space where each letter was to be placed, as well as the specific angles at which they would become perceptible. Thus, at no time can the reader simultaneously perceive the complete set of letters that make up the word: one is forced to read discontinuously, in broken fashion. In this holopoem, the letter A, which symmetrically structures the word AbrAcAdAbrA, was image-planed (with part of the image in front of and part behind the plate) in the center of the visual field, while the consonants were placed around it (B and C as real images; D and R as virtual images) as if the vowel were an atomic nucleus and the consonants were the particles orbiting around it.
Also with Catta-Preta, I created the holopoems Oco and Zyx in 1985. Oco employs two holograms, one with the letter I and the other with the word OCO. The first is displayed in front of the second, multiplying reading possibilities. In Zyx I used the three letters that name the axes of three-dimensional space to form new, nonexistent, bizarre-sounding words. The actual work is a set of fragments against a reflecting background that duplicates the reader’s face inside the hologram and presents the letters X, Y and Z in discontinuous fashion. In this holopoem, the volume of each letter dissolves into colors.
In 1986 I made three new pieces [4]. The holopoem Chaos combines neon and holography. The letters C, H and A are chaotically distributed in pseudoscopic space (space where the image is inverted, inside out — the opposite of orthoscopic space), so that they move in space in a direction opposite to that of the reader’s movement. This work opens the possibility of a letter changing into an abstract color image and vice-versa, for pseudoscopic space does not respect optical conventions regarding the proportion and conservation of forms. The letters S and O complete the reading in neon, flashing on and off and eliciting SOS from the word CHAOS.
Also in 1986 [5], I made the holopoems Wordsl No. 1 and Wordsl No. 2. The first is an experiment in optical anamorphosis: the letters of the words World and Words were holographically combined into a new word, WORDSL, and placed in a 180° arc around my head. This information was transferred to a 90° hologram, through a process of contraction in virtual space (space within the hologram) that changed the forms of the letters; some of the letters, however, seem to go around and behind the hologram, reappearing in their proper proportions in real space (space in front of the hologram). The curvature itself of the integral hologram (so called because it integrates motion pictures and holography and because it recreates the integral movement of a scene) is the cause of this phenomenon. This relates to the topic of visual deformation in variously curved spaces, which was investigated by Georg Riemann in 1854 in his non-Euclidean geometry and which greatly interested avant-garde artists early in this century.
Wordsl Nº 2 displays the same structure, only this time in a space that is both real and pseudoscopic. This piece, which is a study of an unorthodox way of visualizing a hologram, proposes a reading in a succession of vertically oriented strips (from the bottom up and vice-versa), a sort of scanning instead of a global sighting of the scene or object.
In 1987, I worked with computer artist Ormeo Botelho to create the holopoem Quando?, which was digitally synthesized using fractal software. The concept of fractal dimensions is of the utmost importance for this work and thus calls for closer analysis.

Fractal dimensions

The interaction of art with other fields of knowledge, such as physics and mathematics has been important to the pictorial and poetic vocabularies over the last century. By the next century, this interaction will be of fundamental importance, and one of the spotlights that will illuminate this interdisciplinary dialogue is the concept of fractal dimensions developed by Benoit Mandelbrot in his fractal geometry [6].
Classical mathematics, employed in physics, derives from Euclidean geometry and is based on the mechanics of Newton. Echoing Galileo, Cézanne was inspired by the mathematics of regulal shapes when he advocated the method of “see[ing] in nature the cylinder, the sphere, the cone” [7].
Around the turn of the twentieth century, however, some researchers broke radically with Euclidean and Newtonian notions, stimulated by the discovery of new mathematical objects, for example, Koch’s snowflake, Cantor’s sets, Peano’s curves and Hausdorff’s dimensional continuum. Contemporaries of Cubism and atonal music, these and other mathematicians showed that pure mathematics contained a wealth of possibilities that went far beyond the simple models applied to natural forms. Like twentieth-century art, modern mathematics developed by overstepping the limitations imposed by its close links with nature. Today the influence of non-Euclidean concepts — the fourth dimension, n-dimensional geometry and relativity theory — in avant-garde art of the first 30 years of this century is well known, especially in the works of artists such as Malevich, van Doesburg, Larionov, Picasso, Weber and Duchamp [8].
Mandelbrot has provided a new approach to the study of natural forms, and particularly of their irregular character, which deviates from Euclid’s regular geometric forms. For over 10 years he has been developing a new geometry known as ‘fractal’, from the Latin fractus (irregular, fragmented). Fractal forms have dimensions between those defined by whole numbers. Fractal geometry draws from the work of such men as Koch, Cantor, Peano and Hausdorff and analyzes the irregular shapes of nature in an unprecedented way. It is not just a branch of twentieth-century mathematics; rather, it is a new mathematical, philosophical and cultural synthesis that brings together pure mathematics, the natural sciences and computer graphics. Its influence extends to physics, chemistry, biology, geology and meteorology as well as to filmmaking, graphic design and art.
In order to illustrate the concept of fractal dimensions, Mandelbrot states that “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” [9]. On the contrary, various forms in nature are so irregular and fragmented that, compared with the Euclidean vision, they present a higher degree and a different level of complexity: fractal curves have a dimension between 1 and 2, fractal surfaces have a dimension between 2 and 3, and “it is possible to generate fractals with a topological dimension 3 whose scaling irregularities raise the fractal dimension to 3 < D < 4” [10].

Self-similarity and holism

One of the most important basic concepts of fractal geometry is that of ‘self-similarity’. This is easy to understand because it is close to our intuitive notion of ‘dimension’, but it is not necessarily expressed by a whole number. A straightline (D=1) is similar to any part of itself, just as a square cut from a larger square (D=2) is similar to the larger square and a cube cut from another cube (D= 3) mirrors the larger cube. The change in scale does not affect the characteristics of the objects, which for this reason are called ‘self-similar’. A fragment of such a mathematical object can be described in the same way as the main figure.
Fractal objects are also self-similar, but, unlike the simple forms described above, they are generated by complex, nonlinear dynamic systems. Fractals are structured through feedback processes in which there is a “nonlinear relation between input and output” [11]. They are thus placed between mathematical order and the chaos of forms; they are the products of the unprecedented relation between the two, and they illustrate the transition from one to the other, revealing just how complex the region of transitoriness is. Realistic, irregular computer-generated images, such as mountain simulations, are visual expressions of mathematical calculations; on the other hand, abstract, irregular computer-generated images reflect the fractal forms found in nature.
This fascinating new frontier for artistic exploration can be investigated with a computer through random numbers, so that in the works resulting from this process there is a complex mathematical structure that can generate a new type of holistic space, where the fragment mirrors the whole and enlargement implies increased resolution.
Although common sense assigns an effective dimension of 3 to the holographic word-image, I believe that its dimensional complexity can also be understood through the concept of ‘statistical self-similarity’ [12] developed in fractal geometry. According to this concept, the structure of certain objects does not change statistically on any scale, and at the same time it proves to be different in detail on different scales. Statistical self-similarity may be defined mathematically by a dimension whose numerical expression is a fraction.
Just as in a fractal the degree of irregularity and/or fragmentation is similar on all scales, a hologram of an object contains in any one of its fragments the entire information about the object, but only as seen from those view points allowed by the scale of the fragment. It is as if the structure of the hologram itself were statistically self-similar. This conclusion allows holographic space to be approached from the point of view of fractal geometry as well and suggests the creation of holopoems in which the entire linguistic spatial modeling is digitally produced, so that the work quite literally gains a new dimension. In connection with holography, the generation of fractals in computer graphics with an output in space is one of the most fertile possibilities for the art of the future.


The fractal holopoem Quando?, created with Botelho, is my first experiment of this kind. In our first meetings, we tried to define three elements: the text, the fractal, and the relationship between them. After months spent discussing and conceptually formulating the problem through wireframe tests with solid rendering and animation, we decided to use a fractal software program generally employed in mountain simulation to create a monolithic abstract shape rotating around its own axis, alternately disclosing and concealing the words of the text as it spins.
I also wanted to create a 360° hologram, but not a 360° image that could be seen as one sees a sculpture or an ordinary object. It was then that I realized that the fractal might rotate so as to accomplish almost two full turns inside the hologram and thus widen the 360° space to nearly 720°. This gives rise to a perceptual paradox only made possible by holography: although one sees a 360° Plexiglass cylinder inside which there is a 360° holographic film, the fractal turns and multiplies the holographic space.
The text was conceived so that it could be read at any angle, but there is a basic structure that allows it to be read either clockwise or counterclockwise. Counterclockwise the viewer reads A LUZ / ILUDE / A LENTE / LENTA / MENTE (the light/deceives/the lens/slow/ly); clockwise the text is A LENTE / ILUDE / A LUZ / MENTE / LENTA (the lens/deceives/ the light/slow/mind). Other readings, just as valid as these, may arise, for instance, A LUZ/ MENTE / LENTA / A LENTE / ILUDE (the light/lies [i.e. tells lies]/slow/the lens/deceives). In Portuguese, the adverb lentamente (slowly) is made up of the adjective lenta (slow) and the adverbial suffix -mente (-ly), which as an autonomous word may mean either ‘mind’ (noun) or ‘lies’ ( ‘tells lies’).
These words never appear all at the same time; they become visible as the fractal turns inside the hologram and restructures its space. The words float before the fractal, and every time it turns, a new one appears. It is the fractal that causes the passage from one word to the next. As the fractal turns and passes from one word to the next, the words, which are legible when viewed frontally, are seen sideways, thus becoming illegible and being seen as abstract forms. In this case, the text loses its meaning and the entire set changes into a nonverbal form; thus the revolving fractal makes the viewer see a text in a reversible process. As the fractal turns, the boundary between word and image is assigned to time.


The expressive possibilities of holographic fractals or fractal holograms as an art form are much broader than this first work can suggest. For computer graphics, holography is the tool that makes it possible to draw the image out of the monitor and set it free in space. For holography, it is computer graphics that makes it possible to create mathematical shapes of great complexity such as fractals. Finally, for art, the integration of both techniques allows their complementary use, so that one helps to overcome the aesthetic limitations of the other.
Holography challenges us to accept the existence of three-dimensional objects with no palpable existence. Fractal holography challenges the viewer to see in space holographic images of a dimension other than 3 — a visual paradox that dramatizes the immateriality of the holographic image.
With holographic fractals, the viewer’s consciousness is invited to go beyond Euclidean geometry and penetrate a new territory as rich and irregular as the boundaries between order and chaos.

References and notes

1. M. Bense, “Textos Visuais”, in M. Bense, Pequena estética (São Paulo: Perspectiva, 1975) pp. 176-177.
2. Sign, here, is used as in semiology; a word is a symbolic sign, and a photograph an iconic sign.
3. From 1959 to 1962 Suplemento Dominical do Jornal do Brasil (Sunday supplement of the Brazilian newspaper Jornal do Brasil) published manifestos by the neoconcrete poets and reviews of their shows and publications. The neoconcrete poets were Ferreira Gullar, Reynaldo Jardim, Osmar Dillon, Theon Spanudis, and Albertus Marques. I have recorded an interview with Reynaldo Jardim which, to this date, remains unpublished. For more information on Ferreira Gullar’s neo-concrete poetry, see: Luzia Navas-Toríbio, Gullar’s Pre Concretismo Neo, Polikron, São Luís, Maranhão, 1991.
4. During this period I was working as artist-in-residence at the laboratory of New York’s Museum of Holography.
5. This piece was imaged by Larry Lieberman in his Ohio laboratory, through the Jason Sapan Holographic Studios in New York.
6. B. Mandelbrot, The Fractal Geometry of Nature (New York: Freeman, 1983).
7. F. Elgar, Cézanne (London: Thames and Hudson, 1974) p. 104.
8. L.. Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton, NJ: Princeton University Press, 1983).
9. Mandelbrot [6] p. 1.
10. R. Voss, ‘’Random Fractal Forgeries”, in Fundamental Algorithms for Computer Graphics, Rae A. Earnshaw, ed. (Berlin: Springer-Verlag, 1985) p. 811.
11. H. O. Peitgen and P. H. Richter, The Beauty of Fractals (Berlin: Springer-Verlag, 1986) p. 5.
12. Voss [10] p. 808.

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