Sol LeWitt’s Visual Offering: Composition, Art, Music, and a Lit Sphere
Sol LeWitt’s series of 28 photographs, “A sphere lit from the top, four sides, and all combinations” might at first seem like a simple exercise in permutability, lacking much creativity or expressivity. Indeed, the aim of this series of images is not conventional “expressivity” or even originality. It is, in essence, an artistic compositional etude. What intrigues us in this exercise, ultimately elevating this set from the mundane to something we might consider artistic – that which has meaning beyond the skill it demonstrates and the aesthetics it may or may not possess – is the implication it provokes in such a seemingly banal exercise.
Consider such monumental works in music history as Johann Sebastian Bach’s Goldberg Variations, Musical Offering, or Art of the Fugue; Johannes Brahms’ forth movement to this fourth symphony; or Caesar Frank’s Variations symphoniques. Or even consider more recent monumental works, such as The People United Will Never Be Defeated! by American composer Frederic Rzewski. What do all these works have in common? They seek their genius, expressivity, and artistic merit through cleverly unified and imaginative variation. Combinatorial permutation – a mathematical process of acquiring all unique variations of a set of diverse elements – as expressed in LeWitt’s series of photographs, is not only a serialized acquisition of distinct possibilities, but a means by which one may ascertain all potential inherent within an idea. The goal of artists like Rzewski or Bach in creating comprehensive arrays of stylistically and aesthetically diverse variations is an encyclopedic understanding of an idea, systematically permuting their creative germ to harness as much potential inherent in that seed as possible under their specific stylistic, instrumental, and performative limitations.
LeWitt’s series of photographs seek to express this concept in its purest form: the idea, a sphere; the stylistic limitations, the five angles of lighting. What LeWitt ultimately reveals are the variety of aesthetic and expressive possibilities conceivable given such a simple seed. The question this work asks its viewers, or perhaps the challenge it proposes, is how to extract all the potential in any idea, whether artistic or otherwise. So frequently we are presented with ideas, whose potentials are never fully realized, but as artists – purveyors of ideas, concepts, and dreams – we are responsible for seeking and evaluating this potential: challenging our mind to permute its imagination and discriminate the merits, deficiencies, and implications of its creations.
Thus, if LeWitt, in his 28 lit spheres, thrusts upon us the responsibility of fully understanding the potential of our ideas, in what way does he demonstrate our responsibility to also discriminate the merit of each unique permutation? Furthermore, in what way does LeWitt demonstrate the artist’s responsibility to also give voice to these diverse possibilities. It is one feat to create variations; it is anther to unify them into a cohesive whole. All permutations of any idea are not likely to be equally valid or meritorious. Furthermore, it is likely that there will be some permutations that are poor. Some will combine with others more clearly or elegantly than others. From the same diverse parts, both pattern and chaos could emerge given a particular structuring.
Recall again the music of Bach. Let us particularly explore his second fugue in the Well Tempered Clavier Book I. The fugue in C-minor for three voices is commonly classified as a "permutation fugue," a style of fugue where subjects and countersubjects are vertically arranged and rearranged such that the piece explores all possible thematic combinations while maintaining all rules of strict counterpoint and voice-leading. A permutation fugue among three voices with one subject and two countersubjects, as found in the Well-Tempered Clavier’s second fugue, gives six possible re-orderings of the thematic material.
Refer to the formal diagram of the fugue in C-minor given below (Figure 1). The fugue begins with a typical exposition, presenting the subject in all voices before it reappears in any voice (mm. 1-8). By the end of bar 8, Bach presents the subject and both countersubjects and gives us, in bar 7 and 8, the first permutation of all subjects in all voices simultaneously. As the piece progresses through each fugal episode with periodic reentries of the subject, Bach offers a new three-voice configuration of the themes.
A recording with a score animation can be viewed here:
Below in Figure 2 is a diagram listing Bach’s permutations and their corresponding bars, where A is the subject, B is the first countersubject, and C is the second countersubject.
Despite the availability of six possible three-voice subject permutations, Bach only used five: ABC, ACB, BAC, BCA, and CBA. Bach decided to not use the CAB combination. Why is this? Has he distinctly failed in his permutative task? While it appears as such, if one reconsiders the missing combination, one will realize the omission is completely intentional and necessary. Here, Bach exercises his contrapuntal acumen. The counterpoint formed by the CAB permutation actually fails to follow rules of strict 18th century polyphony, including one errant non-functional second inversion triad. Realizing this, Bach chose to exclude this permutation from his fugue, acknowledging its contrapuntal shortcomings.
Through such combinatorial analysis, we can understand how Sol LeWitt demonstrates a similar visually motivated discrimination in the “counterpoint” of his lit spheres. Through simple mathematical investigation of this photographic series, we will discover LeWitt’s own intriguing process of discrimination used to select permutations. Like one might assume all subject permutations are present in Bach’s fugue in C-minor, frequently cited as a permutation fugue, one might also suppose “all combinations” are present in LeWitt’s lit spheres, given the artist’s own title for the work. However, like Bach’s fugal combinations, there are in fact a few spheres missing from LeWitt’s series. Has LeWitt simply forgotten these possibilities, or, like Bach, has he made a conscious and artistically informed decision to exclude them, leaving us “all combinations” that serve the aesthetic goal? Furthermore, what criteria perhaps motivates this selection?
Permutation and inclusion/exclusion are not the whole of this combinatorial artistic endeavor. Ideas – motifs, configurations, transformations, etc. - are essential to any creative endeavors, and using a permutative process to generate such ideas is not only efficient but comprehensive. However, each idea is nothing without the right implementation. The fugue does not exist until the ideas – the subjects and countersubjects - have been put into a particular form. Until then, the composer only possesses abstract melodies with potential, which must be realized in form. How does LeWitt demonstrate formal considerations in this seemingly serialized set of photographs? It is not as mechanical as it might at first appear.
Before delving further into our combinatorial analysis at hand, we will suppose three axioms inherent in this artistic process of inclusion and exclusion.
The Axiom of Invention: one must first endeavor to understand all implications of their idea to fully appreciate the potential of that idea and use it in its most artistically fluent manner.
The Axiom of Exclusion: one must differentiate between those permutations that do and do not serve the particular artistic aim given whatever constraints one has placed on the implementation of their idea.
The Axiom of Form: one must implement their idea and whatever permutations of their idea, whether many, few, or one, in a meaningful and discerning manner as to compliment every part of the whole.
These three axioms, while ultimately arbitrary, presuppose a self-consistent, perceptible, and conscious intent from the artist, allowing us to assume an artistic language which attempts to communicate a limited, non-random, “composed” set of ideas. To better understand the aesthetic goals in LeWitt’s inclusion/exclusion process, we will discuss LeWitt’s use of combinatorics to create a generative theme-and-variation matrix focused through the lens of our three fundamental artistic axioms.
Below is the combinatorial equation expressing all possible unique combinations of a limited set of items (n) given the number of items selected from that set (m):
For example, if you have seven different toppings for a salad, but you may only select at any time four toppings, the above equation will tell you how many possible salad configurations you can produce.
Considering LeWitt’s photographic series, our set “n“ contains five elements: lighting from the top, left, right, front, and back. Furthermore, the set of selected elements may contain at least one element and at most five – one lighting source up to five simultaneous and different lighting sources. For example, if we want to know the number of unique combinations of lighting sources from our set “n” of five sources given that there can only be two sources selected at any time, we would use our equation as such:
Taking multiplication by 1 as negligible and reducing the fraction so that all elements that appear on both top and bottom are self-canceling (i.e. 3/3 = 1), our simplified form becomes
Thus, there are ten unique combinations of three elements from the set of five elements. The ten unique combinations are:
TL TR TB TF LR LB LF RB RF BF
Where T = lit from the top, L = lit from the left, R = lit from the right, B = lit from the back, and F = lit from the front. There are no other “unique” combinations of two elements. For example, though TL is a different “combination” than LT, they are not “unique” combinations, since both result in the same lighting on the sphere. Thus, we consider TL and LT part of one “unique combination,” not redundantly distinguishing the two.
THE AXIOM OF INVENTION:
To find the total number of possible unique combinations, we find the sum of all successive computations for each value of m. This can be mathematically summarized as follows:
In LeWitt’s photographic series, we know 1 <= m <= 5 and n = 5. Note that we are assuming a value of m = 0 to be negligible, being an image without lighting. Thus, the computation is
Thus, there are 31 possible combinations of lighting for a sphere lit from the top, four sides, and all combinations. If we do assume a darkened sphere to be a possibility, where m = 0, then there are 32 unique permutations.
Intriguingly, LeWitt’s series of photographs clearly only contains 28 images. Thus, LeWitt has neglected three possible unique lit combinations. The below table maps the combinations LeWitt has selected.
If one permutes by hand, all the combinations possible with the given five elements (T, L, R, B, F), one will discover these three combinations are missing from LeWitt’s set of “all combinations”: TBF, LBF, TLBF. See Figure 4 below.
THE AXIOM OF EXCLUSION:
The first and most essential assumption we will make is that LeWitt had the mental capacities, given his history of combinatorial art, to know the full number of unique permutations of lit spheres, rather than assume forgetfulness or flawed arithmetic. Thus, while LeWitt claims to have included “all combinations,” he has clearly demonstrated some kind of discernment in deciding to exclude three permutations. Unlike with Bach’s well-defined rules of 18th century counterpoint, we cannot necessarily know with certainly exactly why LeWitt chose to omit these three images. We could make the hypothesis that LeWitt desired a rectangular layout for the images, and thus to have included all 31 lit possibilities would have made any such arrangement impossible. We then might wonder why a rectangular arrangement was so desirable, and again we could hypothesize an aesthetic aim in LeWitt’s work, suggesting perhaps a rectangular organization contrasts nicely with the circular impression of the spheres or simply is consistent with his known penchant for matrix-like designs.
Perhaps LeWitt’s own limitation of “lit” spheres hindered his ability to see the possibility of 32 photographs, with one a dark sphere without any lighting. If he had allowed such a possibility, a rectangular design would have still been possible with a matrix of 8x4 rather than 7x4. Did LeWitt forget this possibility, or did he consciously exclude it in service to some “lit” aesthetic that prohibited darkness as a lack of developmental potential, just as Bach might have prohibited a fugal exposition without a fugal subject?
In any case, LeWitt decided to exclude three permutations, yet claims that “all combinations” are present. In doing this, does LeWitt then subtly imply that all viable combinations are present, given whatever stylistic, formal, or aesthetic restraints, under which he chose to work. Thus, in this omission, LeWitt demonstrates the need for both the full realization of the potential of an idea and the discernment to judge between those permutations that are viable or unviable, graceful or ungraceful, necessary or unnecessary for the whole of the art. Not simply following a calculated process ad infinitium, LeWitt has clearly exercised the power of his artistry, intervening in his machine. Why? To what end?
THE AXIOM OF FORM:
Interestingly, LeWitt’s artistic discernment manifests itself not only in his exclusion of certain configurations; his consideration of form for the included spheres also begs inquiry. By permuting all combinations inherent in our five element set, we can create a distinct combinatorial pattern. This pattern takes consecutive ordered elements and matches them with all following elements in sequence to form unique groups. In the photographs given in Figures 8 and 9, the ordering of the elements is given in the first five images: T, L, R, B, F. If this set is fixed (as it is fixed literally to the wall), and we follow the above concatenation process according to this ordering, a pattern emerges, progressing systematically through each set of size m, where 1 <= m <= 5 , according to the initial ordering. Figure 5 shows the serialized ordering of all sets.
Following this process, the displayed order of the images (removing those elements the artist has excluded (TBF, LBF, and TLBF) would be as in Figure 6 below. Notice however, this common pattern, while highly similar to the order given initially in Figure 3, does not correspond with the mapping of the display at the Harn Art Museum. As seen below in Figure 7, two pairs have been swapped.
Was this exchange LeWitt’s intent? It certainly goes against any perceptible generative process in combinatorics. We must assume this odd ordering true, otherwise we must assume a rather odd mistake by LeWitt or a mistake in the museum’s exhibit, the first of which seems rather unlikely given the clear formalness of the rest of the pattern, and the second of which the museum’s curator nullified with assurances of no such blunder.
Thus, we should probably assume that LeWitt made a conscious aesthetic, non-mechanical, decision in reordering these elements. While permutation patterns give clear lists, an artist like LeWitt might realize that patterned lists do not necessarily reveal the quality of their materials. To use the fugue as comparison: a subject while having a contrapuntally sound countersubject in vertical relation to it should also have a horizontally interesting answer and development should the fugue hope to be successful. Not all answers are equally suitable at any time, nor are all the infinite possibilities of subject developments equally valid; skilled formal discernment is needed. By altering the predictable pattern of images LeWitt has demonstrated this skillful formal consideration.
There is still one hesitation that remains in my mind however. While it perhaps seems beautifully poetic that LeWitt defied the combinatorial predictability of his process, I am still suspicious that this swapped arrangement is a mistake. If the photographs are in fact in their correct order as intended by LeWitt, the supposition of our three axioms is upset: there is an indecipherable element of randomness that lacks limited interpretability. Thus, I do not feel it contradictory or problematic to our analysis to question this ordering further. Though it is certainly possible that LeWitt has opted for a seemingly systematic, calculated art subtly infused with capricious whimsy, this does not seem the likely solution.
If one found a transcription of Bach – a supposedly exact copy of the original composition but in the hand of another person - with an egregious and unexplainable contrapuntal mistake, should we first assume the mistake Bach’s or perhaps question the accuracy of the transmission?
While the curator of the Harn Museum reassured me that these images are in their intended order, and that LeWitt was known to frequently defy the patterned predictability that was inherent in his style, I am still disconcerted. We can assume, and I think rightly so, that LeWitt was an artist with intent to his choices. Thus, either there is some intention which has yet to be determined, or there is a mistake which is likely not LeWitt’s.
Furthermore, and perhaps most important to the justification of this disconcertion, there is a clear difference between displays of this work by two different galleries: The Harn Art Museum of The University of Florida during the Fall of 2015 and the Fraenkel Gallery of San Francisco, California during the Spring of 2011. Images of the two displays are given below in Figures 8 and 9.
Notice that the image in position (1, 1) (bottom left corner) of Figure 9 from the Harn Museum has a distinct shadow to its right side, indicating it is lit from the left and front (LF). However, this same photograph in the Fraenkel Gallery (Figure 8) does not occur in the (1, 1) position, which has lighting from both its left and right sides. The image in the (1, 1) position in the Harn is actually in the (5, 3) position in the Fraenkel. Interestingly, the Fraenkel display follows the prescribed combinatorial pattern as predicted above, while the Harn museum display has two pairs switched. It must be assumed that either one of these museum displays is incorrect or that there are two orderings (or no orderings) given by LeWitt. Both of these displays were done after his death, thus his direct input was/is unavailable. While I was reassured by the curator of the Harn that their display was correct, these images beg further investigation.
In either case, the artistic axioms proposed here hold true in LeWitt’s spheres and are clearly demonstrated by his methods in creating and presenting them. In LeWitt’s “A sphere lit from the top, four sides, and all combinations” we are given a glimpse into the genesis of art in perhaps one of the purest and simplest ways possible. In essence, this piece is an encyclopedic treatment of contrapuntal variation in concrete form. Through this comprehensive development of materials, LeWitt accomplishes a similar end as Bach’s systematic manipulation of fugal form and contrapuntal variation in his Well-Tempered Clavier and Musical Offering respectively, albeit the marked differences between LeWitt’s and Bach’s aesthetic worlds.
LeWitt’s series of photographs is a “visual offering,” proposing to its viewer a simple idea and then teaching them an approach to realize the fullest potential of that idea under a particular aesthetic. While LeWitt’s spheres are perhaps not as complex as Bach’s fugal expositions or thematic variations, the sphere’s virtue lies in their simplicity and transparency to communicate the fundamental artistic axioms inherent in them. For those artists and lovers of art who take time to sit and think upon them, LeWitt’s spheres can teach us much about our craft.
FIGURE 8: Image taken from the Fraenkel Gallery
FIGURE 9a/b: Images taken by Jordan Key at the Harn Museum of Art, Monday, August 24, 2015