evolving an aesthetic of surface economy in sculpture

Leonardo

Evolving an Aesthetic of Surface Economy in SculptureAuthor(s): Brent CollinsSource: Leonardo, Vol. 30, No. 2 (1997), pp. 85-88Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576416 .

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ARTIST'S NOTE

Evolving an Aesthetic of Surface

Economy in Sculpture

Brent Collins ABSTRACT

Th,o i,.,h,r tlicrl C I I1C dULIIUI UIO)UO;

development of his scull over a period of several

rom my vantage point of the present, it is ap- opportunity to view computer as an intuitiveart of viS ematics. He describes ii

parent to me that in finding my voice as a sculptor I have cre- graphics of these surfaces) I dis- recent motif cycle in wh ated an art of visual mathematics. Being a nonmathematician covered that the hyperbolic mod- minimal surfaces are de I have done this through an intuitive denouement of pure in- ules in these sculptures were toroids. Also discussed vention whose mathematical nature I only gradually came to actually truncations from the artist's collaboration wit

puter scientist Carlo Sit realize. In the beginning I had no real concept of visual math- Scherk surfaces. It was a physicist pwhich has le to Sirt ematics as a possible art form. But I had a drive to resolve who first told me I was resolving prototyping program de' sculpture to essentials of holistic resonance which were ca- the surface area of my sculptures by Sequin for this motif

thartically significant for me. And I aspired to a sculptural in- to an approximation of soap film This computer program

telligence of subtle analytic rigor. Now I understand that pur- economy, though at some level a the evolution of the artis of work beyond what wc

suing this vision was also to practice mathematics as the sense of nature's aesthetic tem- ously have been possibl science of patterns. It therefore seems almost inevitable that per had always been an inspira- patterns would eventually emerge in my work. All the pat- tion to me. terns have, however, originated in visual intuitions that pre- date my knowledge of the corollaries that these patterns may have in mathematics or nature, which have so often subse- quently come to my attention.

By the early 1980s I was tending intuitively to minimize the surface area between edge constraints of my sculptures [1 ]. I was in effect approximating soap film economy without know-

ing it. In 1989, saddle curvatures (truncations from what are known mathematically as Scherk's minimal surfaces [2]) began emerging in my work as though under their own self-or- ganizing impetus. The first to appear were those from Scherk's second minimal surface, and they resemble the fa- miliar equestrian saddle. (Computer graphics of this surface, by Carlo Sequin, can be seen in Fig. 2 in his companion article.) But I soon progressed to works that either had third- order "monkey saddles," which provide for tails (Fig. 1), or fourth order saddles, which accommodate quadrupeds.

All these early saddle curvature sculptures were serially constructed and hemispherically capped to give them an el- lipsoidal outline. One of them has a 90? helical deformation along the longitudinal axis of its six monkey saddle stories (Fig. 2). With this single exception, these sculptures were never merely truncations of such multi-storied segments, but instead interweave modular truncations of a single Scherk story with other architectonic features into seamless wholes. Excising the hemispheric caps of one, for instance, creates the bracketing ends of a genus-two Costa surface homeomorph whose modular monkey-saddle story has become syn- : !:' onymous with the hyperbolic curvatures at the Costa surface's center of symmetry [3].

This sculpture's underlying connection with the genus-two % Costa surface only became apparent to me several years after its creation, when I was viewing computer graphics of Costa surfaces. Several years later (once again through having an Fig. 1. Wood sculpture, 30 x 7 x 7 in, 1989. Orientable surface

in which third-order saddles are modularly integrated with Brent Collins, 90 Railroad Avenue, Gower, MO 64454, U.S.A. other rchitectonic features. other architectonic features.

ses the pture, decades, ual math- n detail a ich locally ployed in is the h com- quin,

veloped cycle. furthers st's body )uld previ- e.

LEONARDO, Vol. 30, No. 2, pp. 85-88, 1997 85

-

? 1997 ISAST



Fig. 2. Wood sculpture, 36 x 7 x 7 in, 1990. Orientable surface in which six stories of third-order saddles are helically deformed 90? around their longitudinal axis.

In late 1994 I decided to bring Scherk truncations of second-order saddles into toroidal closure. I felt certain that the preservation of their hyperbolic economy throughout this systematic deformation process would be aestheti-

cally engaging. Visualizing these surfaces was difficult, and I realized I would have to work from their inner- most logic outwards. To accomplish this I articulated their linear cores in armatures made of ring sections cut from PVC pipe (Fig. 3). I then extended thin wax sheets outward from these armatures, stretching them tautly at the critical saddle nodes like skin between the digits of a bat's wing. I then ana-

lyzed the resulting wax models of the surfaces in order to mark cut-out patterns in the boards I laminate and carve to create the final sculptures.

The first such sculpture was the Hyper- bolic Hexagon, completed early in 1995

(Fig. 4 and Color Plate B No. 1). I

angled all the holes of its six saddle stories, alternating them at +/-45? with re-

spect to the main toroidal plane, sensing that this would pattern the surface's

symmetry with optimum cogency. (The entry angles of the hexagon's holes cor-

respond to a value of 45? for the azimuth parameter, as defined in Sequin's companion article.) Interestingly, this

hexagonal surface proves to be another unforeseen homeomorph of a genus- two Costa surface when the edge of its central window is constricted to closure in the formation of a single monkey- saddle.

The same relation to the appropriate genus of Costa surface holds true for all Scherk truncations with an even number of saddle stories deployed in a ring formation. Figure 5, for instance, shows a surface of four Scherk stories deployed in a ring. I closed its central window with

hyperbolic curvatures and extended its three remaining borders to create a visu-

ally indistinguishable conformity with the genus-one Costa surface possible at an azimuth setting of 45?.

Though sharing the Costa surface's serene equilibration, the Hyperbolic Hexagon's aesthetic cast is quite different otherwise. Much of the Costa surface's hyperbolic curvature is rather

modestly concealed in its interior

depths, as we see in the quadrilateral Costa homeomorph in Fig. 5. In contrast, these curvatures in the Hyperbolic Hexagon come into greater visibility by virtue of its central window and the re- traction of its other borders to loci nearer the saddle nodes.

COLLABORATION WITH CARLO SEQUIN Shortly after the hexagonal surface was finished, Sequin and I were put in contact by a mutual friend, and our ensuing collaboration led to the virtual prototyping program he describes in this issue of Leonardo. Its parametric analysis el-

egantly systematizes the possibilities of the toroidal phase of the Scherk motif

cycle, and it will extend the evolution of this cycle through the generation of

promising new formulations and optimized solutions for their parameter values (as determined by the collaborative

judgment of artist and scientist). Most of these formulations would never have come to light had the future of the motif cycle been left to my nervous system alone, with its limitations. The search for optimized solutions would have re- mained a matter of sometimes inspired but always fallible intuitive guesswork,

program can now produce will allow me to create sculptures of a complexity I could not formerly manage.

When I told Sequin that the Hyperbolic Hexagon could be viewed as a toroidal

deployment of a Scherk truncation, the

feasibility of a prototyping program was soon apparent to him. By spring of 1995 he proposed a heptagonal sequel to the

hexagon surface in terms of the twist parameter in his program necessary for the toroidal closure of all odd-numbered truncations of Scherk saddle stories. He suggested in particular that I in- troduce the 90? rotation around the toroidal axis of a heptagon needed to effect the orthogonal articulation of all its saddle structures. (Interestingly, the twist necessary to bring odd numbers of Scherk saddle stories into toroidal closure always results in a non-orientable surface, in contrast with the orientability of all even-numbered closures with or without twist.)

At first, Sequin's suggestion left me

feeling unnerved by the complex task of

designing and making a PVC armature

reflecting the core logic of such a heptagonal surface (see Fig. 3). But after a

period of unconscious gestation, I began to have the necessary insights. These led to the armature for a wax model and fi-

nally to the finished Hyperbolic Heptagon in the fall of 1995 (Fig. 6). Several months later Sequin sent me the first batch of graphics from his generator program, and among them were images of hexagonal and heptagonal surfaces that were obviously homeomorphic with the sculptures I had just finished.

The most recent sculptures I have cre- ated in the ring phase of the Scherk cycle

Fig. 3. PVC armature embodying the core logic of a non-orientable heptagon of second-order saddles. (Photo: Philip Geller)

rather than becoming a visibly mutable

spectrum of parameter choices. What is more, the cross-sectional blueprints his

86 Collins, Evolving an Aesthetic ... Sculpture



are two variants (Figs 7 and 8) of a three- hole formulation. (The number of holes or saddle stories always equate as the same

parameter, and three is the minimum number of them needed to preserve a

pleasingly recognizable ring.) Scherk truncations with an odd number of holes can come into toroidal closure when

given twists in odd multiples of 90?. Both of my three-hole variants have 270? of twist, which yields an optimal symmetry for this formulation. The parameter value in which Figs 7 and 8 differ is azimuth. The azimuth in Fig. 7 is set at a 90? angle in relation to the main toroidal plane; the azimuth in Fig. 8 is set at a 45? angle. The 45? azimuth value in conjunction with the 270? twist parameter generates perfect front-to-back symmetry with identical

edge patterns on the opposite faces, while the 90? azimuth value breaks this symmetry, bringing the edge patterns on the opposite faces into maximum divergence. Aesthetically these azimuth variants are

comparably successful sculptures whose

contrasting symmetries are apparent when the pieces are seen in juxtaposition.

While I was composing and making the wax model for the variant of the three-hole formulation with a 90? azimuth, Sequin was independently explor- ing the virtual aesthetics of a 45? azi-

Fig. 4. Hyperbolic Hexagon, wood, 20 x 20 x 8 in, 1995. Orientable surface composed of six second-order saddles in a ring formation. (Photo: Phillip Geller)

muth variant with his generator program. Neither of us knew what the other was doing until the first graphics he sent me of his variant crossed in the mail with the outline drawing I sent him of the different edge patterns on the faces of mine. After a brief period of confu-

sion, we soon realized the two surfaces were homeomorphic. There was a delay of several weeks, however, before we came to understand that the conspicu- ous variation in geometry between the two was a reflection of their different azimuth values. An azimuth setting of 45? was the sole precedent either of us had previously used, and my change to a 90? setting in this instance was actually a

design solution to simplify the construc- tion of the PVC armature rather than a conscious choice to break with precedent. When all this fully dawned on me and I informed Sequin, the reason the two surfaces were differently configured became immediately clear to us. Both of these azimuth variants have a graceful trefoil coherency neither of us could have foreseen at their inception, and its concurrent emergence in his virtual

prototypes and in my sculpture was in-

spiring for us. It seemed like a kind of

speciation event: the emergence of a non-orientable Scherk trefoil.

CONCLUDING THOUGHTS These recent sculptures all have an ex-

quisite surface economy. But the variants with an even number of saddle stories and zero twist all have a sense of tautly

Fig. 5. Wood sculpture, 16 x 16 x 8 in, 1996. An orientable surface Fig. 6. Hyperbolic Heptagon, wood, 21 x 21 x 9 in, 1995. Non- in which four second-order saddles form a ring whose borders orientable surface in which seven second-order saddles form a hep- have been extended to close its center window and otherwise cre- tagonal ring that twists 90? around its toroidal axis. (Odd numbers ate the ends of a visually unexceptionable homeomorphism with a of second-order saddle stories require twists in odd multiples of genus-one Costa. (Photo: Phillip Geller) 90? for ring closure.) (Photo: Phillip Geller)

Collins, Evolving an Aesthetic . . . Sculpture 87



Fig. 7. Wood sculpture, 20 x 20 x 8 in, 1996. A non-orientable surface in which three second-order saddles form a ring. Its trefoil

edge pattern emerges only with a twist parameter of 270?.

Fig. 8. Wood sculpture, 21 x 21 x 9 in, 1996. This non-orientable trefoil formulation has an azimuth parameter of 45?, while that of the formulation in Fig. 7 is 90?. The two sculptures are configured somewhat differently as a result, but are otherwise identical in their parameters even though they are of opposite chirality. (Photo: Phillip Geller)

balanced stability, particularly given the

symmetrical equity of the 45? azimuth seen in the Hyperbolic Hexagon (Fig. 4) and the quadrilateral Costa homeo-

morph (Fig. 5), while the variants with an odd number of stories and twists in odd multiples of 90? have a more dy- namic vorticism. Neither the works with an even nor those with an odd number of holes seem significantly marred by the continuum of strong-to-weak perspec- tives I have found so difficult to over- come in sculpture. And while their high coherency as patterns is instantly recognizable, they all have a subtle strangeness that eludes being grasped in a glance.

In discussing the verisimilitude, stereo

depth and interactive motion parallax

that an advanced computer graphics program can give virtual sculptures, Sequin wonders whether the creation of actual sculptures is becoming less neces-

sary. The making of sculpture might best be viewed, however, as creatively fulfill-

ing the potential for plastic expression, which has been so crucial to our evolution. Archaeologically it is first evident, along with cave painting, in the relatively recent paleolithic period of modern Homo sapiens. Like painting, it involves an organic grace of extemporaneous hand-to-eye coordination that will con- tinue to be needed in generations to come, even as our creativity is aug- mented by advancing computer capabili- ties. Moreover, as a species evolved for

tool-making and use, human beings have aesthetic empathy for handmade art ob-

jects and will always need them as revela- tions of our nature.

References

1. See George Francis, "On Knot-Spanning Sur- faces: An Illustrated Essay on Topological Art," in Visual Mathematics, special issue of Leonardo 25, Nos. 3/4, 313-320, 1992; and in Michele Emmer, ed., The Visual Mind: Art and Mathematics (Cambridge, MA: MIT Press, 1993) pp. 57-64.

2. For a glossary of terms, see Carlo Sequin's companion article "Virtual Prototyping of Scherk- Collins Saddle Rings," immediately following in this issue of Leonardo.

3. See Fig. 8 in Francis [1].

Manuscript received 12January 1996.

88 Collins, Evolving an Aesthetic . . . Sculpture



evolving an aesthetic of surface economy in sculpture

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