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You wouldn’t know it from the monochromatic Plus Equals, but I spend a lot of time exploring the mechanics of color. And so my interest was piqued when Loscil and Lawrence English, two musicians whose ambient works I’ve long admired, released a collaborative album earlier this year called Colours of Air, with each of its songs named for a different hue, such as “Aqua,” “Yellow,” and “Violet.” These eight ethereal compositions are constructed from manipulated tones drawn from a 19th century pipe organ, and while the music isn’t as overtly synesthetic as the titles might suggest, comparing and contrasting the songs’ essential qualities to those of their namesake hues is an exercise that elevates the listening experience.

But before I even saw the song titles or heard the music, it was the album cover (Fig. 1) that immediately caught my attention. Designed by Scott Morgan (aka Loscil) and Craig McCaffrey, its background fades downward from a light pink to a deep magenta, atop which is a dense field of horizontal white lines whose various contortions subtly suggest folds in a piece of paper. I spoke with Scott over email about how he arrived at this image, and while his process was more subconscious than strategic, the result feels entirely representative of the music in some ineffable way.

I stared at it a lot, fascinated by how these simple, mechanical lines not only introduce an uncanny third dimension, but also evoke striations in muscle tissue, geological formations, and topographic maps. Inevitably, I wondered if a system could be extrapolated from it to generate variations, and I soon found a path forward.

One way to simplify what’s happening on the Colours of Air album cover is to think of it as a series of horizontal lines printed on a piece of fabric, where certain areas of the fabric are pinched and other areas are stretched. Approximating that effect in two dimensions begins with our old friend, the grid. If we take a simple grid (Fig. 2) and mutate it by pulling its intersections in various directions (Fig. 3), we can see a structure forming, something akin to the sort of wireframe that often gives papier-mâché objects their shape.

Adjusting it further, on each vertical line segment, we’ll add a uniform number of points that are equidistant from each other. On short segments, these points will be close to each other, and on longer segments, they’ll be further apart (Fig. 4). Once we add more horizontal lines by connecting those dots, we’re almost there (Fig. 5), and when we remove the vertical lines entirely, we have an image similar to the one from Colours of Air (Fig. 6).

This is a pretty good process, and we can make all kinds of interesting variations on this image by pulling the original grid intersections into different orientations. But for someone like me who prefers finite systems, it’s still too open-ended. Even though the grid itself is ostensibly a limitation, there’s currently no limit to the ways it can be mutated. To rein this in, we’ll give the base grid a sub-grid (Fig. 7). Now, we can insist that any manipulation of the base grid must align with the sub-grid (Fig. 8).

For the images I’ll be making, after some experimentation, I settled on a 4×4 base grid with a 32×32 sub-grid (plus an additional margin of six sub-grid units on each side). When the base grid intersections are manipulated, they’re only allowed to be pulled a maximum of two sub-grid units in any direction. This makes the range of movement for each base grid intersection a 4×4 area on the sub-grid, a tidy microcosm of the 4×4 base grid (Fig. 9).

Also, inspired by a video by the artist Leah Mackin, in which she records the movement of a sheet of paper in a body of water (Fig. 10), I decided to convert my jagged lines to curves (Fig. 11). (See Plus Equals #6 for a crash course in Bézier curves.)

Within this system, I describe each manipulated base grid intersection by what I’m calling its *deviation coordinates*—that is, the distance it has moved from its initial position on the *x* and *y* axes, measured in sub-grid units ranging from -2 to 2. Since there are 25 base grid intersections (Fig. 12) and 25 possible pairs of deviation coordinates (Fig. 13), each intersection can be given unique deviation coordinates, with every possible pair of coordinates accounted for. From there, if I determine every possible sequence of those 25 pairs of deviation coordinates, the image variations that result—which I affectionately refer to as *mutations*—will be a complete set, representing the entirety of what this system can produce. (See Plus Equals #2 for information on factorials and how I’ve previously tackled the problem of comprehensive sequencing.)

But I’m not out of the woods yet. While this system may seem to have reasonable constraints, it still produces more than *15.5 septillion mutations*. That’s a number with 26 digits! If each mutation were a square inch, the full set could cover the entire surface of the Earth more than 19 million times. It’s nearly 2 quadrillion mutations for every single person on the planet. If you lived to be 100 years old and looked at one mutation per second, it would take you almost 5 quadrillion lifetimes to see them all. In short, it’s a large enough number to be indistinguishable from infinity.

I tried including several additional systemic constraints to reduce that number. One is a sort of bilateral sudoku, which insists that in each row and column of the base grid’s intersections, deviation coordinates can’t repeat *x* or *y* values (Fig. 14). Alas, it seems to require more than enough calculations to melt my computer.

Another is radial symmetry, which only allows mutations that look the same when rotated 180 degrees (Fig. 15). It narrows things down considerably, but still produces a hefty 479 million results.

Ultimately I wasn’t able to find a way to get the number of results down to a human scale without either making unacceptable aesthetic sacrifices or making the system conspicuously convoluted.

As I said in the very first issue of Plus Equals, what I like about a combinatorial approach to algorithmic art is the creation of a system whose many possible results are *all* interesting, both individually and as a set. This is something of a collectivist ethos, in that the individual results are best appreciated in the context of the full set and the system that produced it. Crucially, it also allows the system to be its own curator. But since I like this particular system so much even though I can’t seem to sufficiently tame it, I decided to compromise and narrow down its cosmic scope by taking the final curation process into my own hands.

So I built an app that generates random mutations one at a time—either asymmetrical or radially symmetrical—and lets me sort them into “yes” and “no” piles. For each sorting session with the app, I tried not to think about it too much, letting instinct decide which variations were selected and which were rejected, continuing until I had 60 variations in the “yes” pile. Over the course of a few days, I did five sessions each with symmetries and asymmetries, yielding a total of 600 selections and 5,148 rejections. I then reduced those 600 selections down to a final 60 symmetries and 60 asymmetries.

When I look at my collected selections and rejections, some patterns emerge. The rejections have a habit of being too dramatic or too subtle, with bulbous curves and sharp angles at one end of the spectrum (Fig. 16), and uniform density and plain contours at the other (Fig. 17). The selections, on the other hand, are to my eyes more elegant and balanced (Fig. 18). They’re a showcase of quiet interplay between inquisitive curves and declarative horizontals, landing in formations that feel both entropic and composed, punctuated by dense areas suggesting shadows cast by an unknowable network of light sources.

It’s not hard to see in them formal similarities to the work of some artists I admire, with Frank Gehry’s unmistakable architecture (Fig. 19) as the most well-known reference point. Some of the symmetries bring to mind the transcendent geometry of Matt Shlian’s paper relief sculptures (Fig. 20), which have fascinated me since I first discovered them in a gallery last year. Meanwhile, some of the more top-heavy asymmetries remind me of Ursula von Rydingsvard’s towering sculptures resembling craggy rock formations (Fig. 21), which I’ve been lucky enough to have easy access to while living in Brooklyn and Philadelphia.

As an artist who mainly thinks and works in two dimensions, these sculptural and architectural reference points are a pleasant surprise. And the patterns of desirable and undesirable attributes seen in the selected and rejected mutations offer clues to how I might further tune the system for higher quality and lower quantity. But for now, my curiosity is satisfied enough to leave that pursuit for another day.

Rob Weychert

rob@robweychert.com

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Products of a system are generally expected to have a certain rigidity. A system is ostensibly a rational enterprise, meant to achieve its objective with greater efficiency than the improvisations of an unsteady human hand. And with that absence of improvisation comes an absence of character and warmth. After all, though the system may have been created by humans, its products are at least one step removed from that humanity: Les Paul’s signature may be on your guitar, but Les Paul himself almost certainly never touched it, and he may never have even entered the factory that produced it.

My Plus Equals explorations so far have mostly embraced the rigidity their combinatorial systems produce (Fig. 1). Like their forebears in the conceptual and minimalist art movements, they’re a counterpoint to the gestural immediacy of something like abstract expressionism, and their mechanical lines and comprehensive seriality constitute their own kind of regulated beauty. That said, the systems I devise don’t have a mandate to make stiff work, and the prospect of finding something akin to gestural immediacy within a rigid structure is a contradiction that interests me. And one fundamental way to explore that contradiction is to look at the relationship between straight lines and curves.

While working for the automaker Citroën in 1959, the French physicist and mathematician Paul de Casteljau developed a method for computationally describing and creating curved lines. A short time later, an engineer named Pierre Bézier independently made the same discovery and applied it to the design of automobile bodies at Renault, another French car company. Unlike de Casteljau, Bézier didn’t hesitate to publish his findings, and so the method, which is still widely used today in the design of everything from fonts to video games to the aforementioned Les Paul guitars, became known as the *Bézier curve*.

Bézier curves come in a few different flavors, and the one best suited to my purpose is the *cubic* Bézier curve. It’s defined by four points on a plane: two end points—let’s call them A and D—and two control points, B and C (Fig. 2). The positions of the end points determine the beginning and end of the curve, and through a process of linear interpolation, their spatial relationships with the positions of the control points determine the shape of the curve itself. To see how it works, first draw straight lines between A and B, B and C, and C and D (Fig. 3). Next, mark the midpoint on each of those lines and draw lines connecting the AB midpoint to the BC midpoint, and the BC midpoint to the CD midpoint (Fig. 4). Finally, mark the midpoints of *those* lines and draw one final line connecting those midpoints. The midpoint of that final line, which we’ll call P, is the midpoint of the curve (Fig. 5). If we keep the original four points in the same positions, but redraw the rest of the apparatus, this time placing the interior points 25% of the way across the connecting lines, P now marks the 25% point on that same curve (Fig. 6). Redraw the apparatus enough times, moving the interior points incrementally each time, and P will eventually shape the entire curve between A and D. (Fig. 7).

A chain of curves produced in this manner can form any shape imaginable (Fig. 8), and to do so by applying a relatively simple algorithm to a limited number of point coordinates is incredibly powerful, so it’s not hard to see why the discoveries of de Casteljau and Bézier have been so influential. (For a deeper dive into Bézier curves, I highly recommend Bartosz Ciechanowski’s interactive explainer, “Curves and Surfaces,” and/or Freya Holmér’s video, “The Beauty of Bézier Curves.”)

Bézier curves have the potential to give me what I’m looking for: Their end points and control points can be plotted according to a strict system, and the resulting curves can still evoke the looseness of human gesture.

The foundation of my strict system is a 3×3 grid. The first part of the plan is to find every possible sequence of points within that grid, where 1) each sequence starts at the same point, 2) each point is a distance of 1×2 or 2×1 grid units away from the point preceding it, and 3) no point is used more than once in a sequence. Each sequence will plot the end points for a chain of Bézier curves.

I begin by choosing point 2,2 as a somewhat central origin point for all sequences. There are four points in the grid that are the specified distance from that point: 0,1, 0,3, 1,0, and 3,0 (Fig. 9). These make the beginnings of four sequences, and each of them is able to branch out to one or more additional points (Fig. 10). Any time a sequence has more than one candidate for its next point, a new sequence is formed for each additional candidate (Fig. 11). A sequence is complete when it reaches a dead end (Fig. 12). This process ultimately produces 562 distinct sequences, with lengths ranging from four to 14 points.

Now for the control points, which will be positioned on the same grid. Using the same point names from our earlier Bézier curve demo, I’ll place each B control point one grid unit away from its corresponding A end point, and likewise, I’ll place each C control point one grid unit away from its corresponding D end point. To give the curves some variety, the control points can be positioned horizontally, vertically, or diagonally, relative to their corresponding end points (Fig. 13).

The control points could simply cycle through those directions: the first end point in the sequence gets a horizontal control point, the second is vertical, the third is diagonal, the fourth is horizontal, the fifth is vertical, and so on. However, many of the end point sequences are identical to each other apart from their last few points, and having every sequence use one directional cycle for their control points would make most of their curves identical too. If I’m going for an effect of spontaneous gesture, that kind of repetition won’t help (Fig. 14).

So I made a set of six different directional cycles (Fig. 15). The first end point sequence’s control points use the first directional cycle, the second sequence uses the second cycle, and so on. This allows similar sequences of end points to produce very different curves (Fig. 16).

With repetition obscured and variety achieved, the final product is a series of 562 scribbles that you might not have guessed were all drawn by an algorithm (Fig. 17).

One of the first things I noticed about this series is that despite the uniqueness of its individual scribbles, they all seem to have a consistent personality. Throughout the series, the system generates some loose motifs, which imbue the scribbles with a shared character, as if they collectively represent one person’s handwriting in the absence of an alphabet. And that person seems to be expressing something, albeit in a subtly regimented manner. It’s not hard to see in these scribbles anger, joy, confusion, or even boredom, and yet these emotional qualities never manage to empower the scribbles to escape their confinement. They all occupy their uniform allotted space with robotic obedience.

As much as I’m enjoying the discoveries that come with venturing into seemingly spontaneous and organic combinatorial forms, I’m ambivalent about the achievement.

Artificial intelligence has made real strides this year, most noticeably in the form of machine learning models that generate digital images from natural language descriptions. These models have been trained to associate images with relevant words by studying hundreds of millions of captioned pictures. When given a descriptive prompt, like, say, “Photo of an astronaut playing the piano, in the style of Dorothea Lange,” they can generate photorealistic images in mere seconds that render that scene with startling accuracy (Fig. 18). As a landmark technological advancement, it’s in line with what we’ve come to expect from Silicon Valley, in that it’s both amazing and unnerving, shaped by deep pockets and fanciful libertarian ideals, and there’s no shortage of reasons to be skeptical that we’re ready for its implications.

The trajectory of the internet over the last 30 years—from its techno-utopian colonization in the 1990s, to its capitalist co-option in the 2000s, to its obliteration of the very notion of objective truth in the 2010s—has made Luddites out of many of us who were once optimistic about its promise. It’s hard to square the undeniably positive effects of computers’ growing ubiquity with their ability to enable harm at a previously unimaginable scale. AI has the potential to cure disease, feed the world, and amplify our imaginations. But can it avoid embodying our worst biases and being weaponized for targeted abuse, the spread of disinformation, and who knows what else? Can it re-establish our social safety net with the same speed it displaces workers? Can it reshape our ideas around authorship and ownership?

The series of scribbles I’ve generated here is obviously nowhere near the level of sophistication of artificial intelligence. But it does live somewhere in the uneasy realm of computers pretending to be human. We’ll all be living in that realm eventually. Here’s hoping that against the odds, it’s a harmonious one.

Rob Weychert

rob@robweychert.com

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Marcel Duchamp once said that “while all artists are not chess players, all chess players are artists.” Whether or not you agree with that, there’s no denying that chess enjoys a distinguished reputation, not only as a game capable of prodding the deepest recesses of the intellect, but also as an egalitarian pastime whose appeal has endured for centuries across cultures and social strata. Easy to learn and hard to master, chess’s simple rules belie a vast ocean of possibility, and after hundreds of years of exploration, we still haven’t gotten to the bottom of it.

To excel at chess is to envision staggeringly complex branching sequences of moves, to live simultaneously in the present and many possible futures. In this light, it might be likened to the Cubist paintings of the early 20th century, which often merge multiple movements, moments, and perspectives into expressive hybrid compositions. And indeed, Duchamp’s fascination with chess spilled into some of his Cubist works (Fig. 1).

So I think it’s fair to say that at the very least, chess *enables* art. And chess’s seemingly boundless framework for intricate strategic and tactical invention—derived from a deceptively simple playing field—is enticing for an artist interested in combinatorics, as I happen to be.

In an influential 1950 paper entitled Programming a Computer for Playing Chess, the mathematician Claude Shannon estimated the number of ways a chess game could possibly be played out: 10^{120}, which is larger than the number of atoms in the observable universe. If anyone ever manages a full accounting of those possibilities, my hat is off to them, but it’s *just* a bit too ambitious for me.

Instead, I skipped ahead about 50 years to a computer game called Battle Chess (Fig. 2), one of the many beneficiaries of Shannon’s pioneering paper. Battle Chess plays like any other computer chess game, but its name comes from its distinctive visuals. When one piece attacks another, the attack is literal: a knight decapitates a bishop, a rook devours a queen, a king blows up a knight, and so on. These animations are amusing, if unsubtle, but I’m less interested in their particulars than their logistics. In all, 36 separate animations (Fig. 3) needed to be created to cover every possible attack, which is much more workable as a starting point for a combinatorial exploration of chess than the mind-melting Shannon number.

A square on a chess board can’t be occupied by more than one piece at a time, and in that sense, those 36 attacks are about territory: one piece displaces the other. But what if the chess board’s territory weren’t so exclusive? A square simultaneously occupied by multiple pieces introduces new avenues of thought about those pieces and their formal, functional, and symbolic traits. What possibilities do merging these pieces suggest?

The idea of merging chess pieces recalls not only the aforementioned Cubist paintings of Duchamp and others, but also 0 through 9, a series of paintings and prints Jasper Johns produced beginning in 1960. Each work in the series superimposes the numerals zero through nine on top of each other (Fig. 4). The process of navigating the images’ visual morass to discern the individual numerals forces the viewer to focus on the numerals’ form, to truly *observe* something that’s usually merely *seen*. As a typography enthusiast, one aspect I find especially interesting is how the numerals interact with each other so chaotically despite coming from the same typeface. If I were to merge chess pieces in a similar way, how might the outcome be affected if the pieces’ designs anticipated their merger?

To find out, I made my own two-dimensional chess set, with the design of its six pieces paying careful attention to how they overlay each other (Fig. 5). The simpler the design, the less chaotic the overlays, so I made a point of establishing some fairly tight constraints, settling on a 50×50 grid as a shared basis for the six drawings. And to keep everything as recognizable as possible, the set is modeled on the Staunton style, the standard for international tournament play, familiar to beginners and grandmasters alike.

With the design established, the combinatorial intention is to find every possible merger of two, three, four, five, and six pieces (Fig. 6).

We first find all the pairs, beginning with the pawn and merging it with each of the other five pieces, in ascending order of value: pawn/bishop, pawn/knight, pawn/rook, pawn/queen, pawn/king. Next, we do the same with the bishop. Since we already have a pawn/bishop pair, we can safely ignore bishop/pawn and skip ahead to bishop/knight, followed by bishop/rook, bishop/queen, and bishop/king. Like the bishop’s redundant first pair, the knight’s first two pairs—knight/pawn and knight/bishop—are also already accounted for, as are the rook’s first three pairs, and the queen’s first four pairs, all of which can be ignored. By the time we get to the king, every pair that could possibly include it has already been produced. In all, the six pieces make 15 unique pairs.

Now the process is repeated for each pair. Pawn/bishop yields the trios pawn/bishop/knight, pawn/bishop/rook, pawn/bishop/queen, and pawn/bishop/king. The next set of trios are based on pawn/knight. Since we already have a pawn/bishop/knight trio, we can ignore the redundant pawn/knight/bishop and skip ahead to pawn/knight/rook, followed by pawn/knight/queen and pawn/knight/king. A clear pattern begins to take shape: when adding a piece to a merger, its value can’t be less than that of the preceding piece, because the merger that would result has already been produced. Once we’ve used this formula to find all the sets of three, we can repeat the process to find all the sets of four, which will in turn yield all the sets of five.

When the dust settles, the six pieces can be merged 57 different ways. Including the original six individual pieces brings the tally to 63, which is one less than the number of squares on a chess board. That final square contains all that remains: the empty set, a merger involving no pieces at all.

Of the many ways to think about these mergers and their formal, functional, and symbolic traits, the recurring themes for me are accord, conflict, and the inconvenient reality that those notions aren’t mutually exclusive.

Functionally, the mergers are force multipliers, since they usually have more mobility than any of their constituent parts would have on their own. A pawn/rook can move as far as it wants horizontally or vertically, *and* it can attack diagonally. Adding a rook to a bishop effectively turns it into a queen. And even the queen, the most powerful piece on the board, benefits from merging with a knight (Fig. 7).

But as much as we might appreciate what this says about the virtues of collaboration, we also see that the more complex mergers noticeably struggle to maintain their formal integrity. Even though the pieces have all been designed to work together, the mergers can’t help but get jumbled as more pieces are thrown into the mix, raising questions about where we draw the line between an effective team and an inefficient bureaucracy (Fig. 8).

Chess’s preexisting symbolism offers the mergers even more specific allegories. Those of us who firmly believe in the separation of church and state might see something disquieting in how fluidly the bishop merges with the king (Fig. 9). Critics of the military industrial complex may find it significant that the knight’s lack of symmetry makes its presence consistently conspicuous (Fig. 10).

I’ve been playing chess for most of my life, and I’m not bothered by the fact that its inherent beauty is something I’ll never be able to adequately articulate. But getting even a tiny step closer with an exploration like this is gratifying. The encodings attached to pawns, bishops, knights, rooks, queens, and kings—the way they look, behave, and reflect our understanding of each other—have a richness that extends far beyond the bounds of the chess board. Those six little chess pieces contain multitudes.

Rob Weychert

rob@robweychert.com

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I made some changes to my look this year. My large signature beard isn’t so large anymore, I’m wearing more colors, and most recently, my new glasses are introducing new angles to my face (Fig. 1). The hexagons now framing my vision are a constant reminder of the isometric cube, whose contours are bounded by a hexagon (Fig. 2). Regular readers may recall that one of the seeds for Plus Equals was planted a few years ago with Incomplete Open Cubes Revisited, my extension of a Sol LeWitt work. I learned a lot about isometric projection from that project, but my affection for the concept didn’t begin there. Whether I’m looking at a Chris Ware illustration (Fig. 3) or an exploded-view technical drawing of a complex machine (Fig. 4), an isometric rendering always stirs something in me.

There may be a nostalgic element to that: The first technical drawing I saw was probably an instruction manual for a Lego kit (Fig. 5); isometric views’ enduring tenure in video game design began in the arcades of my youth (Fig. 6); and M.C. Escher’s mind-bending lithographs were among my most significant adolescent art obsessions (Fig. 7). However, I think there’s something more fundamental at work in the satisfaction I get from these images, something beyond the happenstance of what I was exposed to as a child.

Isometric projection is a form of parallel projection, a means of rendering three-dimensional objects in two dimensions. It differs from the naturalistic perspective you might see in a photograph in that objects maintain their size relative to each other rather than appearing larger or smaller relative to their distance from the viewer, and parallel lines remain parallel rather than converging on a vanishing point, hence the term “*parallel* projection” (Fig. 8).

Of the various forms parallel projection can take, the isometric version interests me most because the relationships between its three coordinate axes are all identical: The angle between any two of them is 120 degrees. In the case of a cube, this allows us to see each of its three visible planes in exactly equal measure. It also means an isometric grid is made of interlocking equilateral triangles, which is an irresistibly tidy means of representing a three-dimensional space (Fig. 9). (Some of the examples I cited earlier have slightly different angles and are therefore not technically isometric, but we won’t hold that against them.)

As perfect as it all seems, though, any two-dimensional rendering of a three-dimensional thing is inherently ambiguous, since there’s only so much we can know about an object or scene when viewing it from one fixed angle. And for as long as people have been making images, they’ve been finding creative ways to exploit that ambiguity. A classic example of this is in an iconic scene from the 1923 film Safety Last!, in which Harold Lloyd appears to be hanging precariously from the side of a building, high above a busy city street. In reality he’s hanging from a fake facade just a few feet above a padded platform on the building’s roof (Fig. 10).

Despite its orderly appearance, isometric projection is fertile ground for this sort of manipulation, particularly with simple geometric objects, since multiple interpretations of those objects and their spatial relationships to each other are often available to the viewer (Fig. 11). It’s also rife with opportunities for creating impossible objects that defy rational understanding (Fig. 12).

And so goes the seduction with isometric projection: Its idealized geometry suggests an accessible codification of objective reality, a shortcut to a structural understanding of an otherwise chaotic universe; and yet its idiosyncratic visual language is just as ambiguous and falsifiable as any other, perhaps even more so. Isometric projection is a beautiful lie, and one for which I can’t bring myself to fault the liar.

After spending some time experimenting with combinatorics in isometric space, I landed on a two-layered exploration, with each layer addressing, respectively, the beauty and the lie of isometric projection: 1) What are the various ways a distinct set of objects can be arranged in three dimensions within the regimented boundaries of a defined space? 2) What are the various ways those arrangements can be scrambled by the ambiguity of isometric projection?

For the first combinatorial layer, a 3×3×3 cube is the defined space—specifically the cube’s three visible surfaces—accounting for a total of 19 available units of space (Fig. 13). The objects meant to be arranged in that space are three blocks: a 1×1×3 block (let’s call it Block A, Fig. 14), a 1×2×2 block (Block B, Fig. 15), and a fusion of two 1×2×2 blocks joined at a 90-degree angle (Block C, Fig. 16). The three blocks collectively account for 13 units of space.

To find all the ways these blocks can be arranged, first assign a unique number to each of the 19 available units of space (Fig. 17). Next, find every possible position each individual block can occupy in that space, and name each position according to the numbered units it occupies. There are 15 possible positions for Block A (Fig. 18), 12 positions for Block B (Fig. 19), and six positions for Block C (Fig. 20).

Finally, start making arrangements of the three blocks, beginning by comparing each of Block A’s positions to each of Block B’s positions (Fig. 21). For each comparison, if the two positions share any numbers, that means they overlap, and the pair is disqualified. If they don’t overlap, the pair is put into a list, which will ultimately contain 111 valid pairs. Once the full list of valid Block A / Block B pairs is assembled, repeat the process again by comparing each pair to each of the Block C positions. As before, any arrangement with no repeated numbers is deemed valid. When all is said and done, 846 comparisons have been made between the three blocks, determining that they can be validly arranged 54 different ways.

At this point there appears to be a problem. The blocks in these arrangements may not overlap in theoretical three-dimensional space, but their two-dimensional isometric projections do. Every arrangement has at least one block partly obscured because another block is positioned between it and the viewer, which is to be expected. But the two-dimensional versions of the blocks are effectively flat decals, and while the system that generated their arrangements knows *where* to place each of them, it doesn’t know their correct stacking order, and that stacking order varies for each arrangement (Fig. 22). I could modify the system to solve this or manually edit all 54 arrangements, but I prefer to see this as an opportunity rather than a problem to solve, which brings us to the second combinatorial layer of this exploration: embracing the ambiguity of isometric projection.

Every arrangement consists of three blocks, which means one is in the back, one is in the middle, and one is in the front. And since there are three blocks, they can be sequenced six different ways: ABC, ACB, BAC, BCA, CAB, CBA. For each arrangement, one of those is the intended stacking order, rendering the scene appropriately within the boundaries of the established three-dimensional space. The other five stacks create alternate realities, disregarding the boundaries and changing the blocks’ spatial relationships (Fig. 23). This kind of scrambling of perception is a big part of what makes isometric space so fascinating to me, so the “incorrect” stacks should be considered every bit as vital to this exploration as the “correct” ones.

Six stacking orders for each of the 54 arrangements brings the total number of generated images to 324, but even though in theory they’re all unique, in practice some stacks look identical to others (Fig. 24). This is because the shapes of the blocks make it possible for them to avoid overlapping each other, making their arrangement look the same regardless of which is on top. Two thirds of the arrangements have two stacks that appear to be duplicates. Removing them brings the final total to 252 images.

The end result, for me, is both lucid and hypnotic, a sprawling portrait of the uncanny atmosphere of isometric space. The same compelling geometry that can bring such clarity to these 54 arrangements can also easily and indifferently distort them many times over. It’s an unreliable narrator who won’t shut up, and I can’t stop listening.

Rob Weychert

rob@robweychert.com

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My partner and I bought a house last year. After decades of renting, we’re excited to finally have a space we can truly make our own, and our eyes are more open than ever before to the finer details of home furnishings and interior decorating. My knowledge of wood species, LED lightbulbs, and every single dish drain on the market has reached dizzying new heights. My dreams are routinely invaded by patio furniture, window treatments, and paint samples.

One aspect of home improvement that hasn’t come up for us yet but has nevertheless been on my mind (perhaps due to a recent visit to Fonthill Castle) is tiling. I’m a sucker for a good creative constraint, and I love the way tiling’s spatial framework is both rigidly composed and rife with opportunity. Whether they’re used to create colorful mosaic grids (Fig. 1) or seamless organic patterns (Fig. 2), the creative potential unique to tiles has long fascinated me.

In my design career, I’ve created my share of tile patterns, especially in the early days when the background of pretty much every website was a neo-Warholian eternity of stars or balloons or skulls or whatever. In patterns, as in any other aesthetic endeavor, beauty is not easily formulated or defined. Music theory and the golden ratio notwithstanding, the right brain guards its secrets well. But I’ve found that patterns that stimulate my *left* brain tend to obey a recognizable principle: the harder the individual tiles are to detect, the better. This often equates to complexity: greater complexity equals greater seamlessness equals greater wonder induced.

Achieving that kind of complexity in tile pattern design has typically evaded me, to say nothing of making it actually look good. But recently I started thinking about how complex, seamless tile patterns could be derived from relatively simple systems, and it wasn’t long before combinatorics entered the fray.

The simplest square tiling pattern is a single shape—say, a circle—that sits in the middle of each tile (Fig. 3). Nothing mysterious about that.

A denser diagonal version of that pattern can be made with two copies of the shape. Cut the first shape in half horizontally and move its top half to the bottom edge of the tile and its bottom half to the top edge of the tile. Cut the second copy in half vertically and move its left half to the right edge of the tile and its right half to the left edge of the tile (Fig. 4).

To make an even fancier pattern, cut a single shape in quarters, move each quarter to its opposing corner of the tile, and add a different shape to the middle for a little spice (Fig. 5).

What these processes reveal is that making a seamless pattern relies on the opposing edges of the tile having a relationship with each other. Whatever happens on the right edge is continued on the left edge, and whatever happens on the top edge is continued on the bottom edge. Understanding this enables us to take things a step further by creating *sets* of tile designs that are all compatible with each other (Fig. 6).

And once there’s a set of things, I want to know how big it can be.

Sharp-eyed observers may have noticed that most of the tiles I’ve made so far have been designed with the help of a 3×3 grid (Fig. 7). I’m going to use that grid as the basis for a system which can produce many more tiles, each one able to seamlessly border any other one on any side. To help obscure the underlying grid, each tile design will consist entirely of diagonal lines, with each line beginning at one of the points on the perimeter of the grid and ending on another. The 3×3 grid gives us 12 perimeter points to work with (Fig. 8), and my first task is to find every unique diagonal line I can make with them.

This is done by naming each perimeter point by its grid coordinates and then listing the points in ascending numeric order. For each point, compare it to every point that comes after it in the list. For each comparison, if the *x* coordinates don’t match and the *y* coordinates don’t match—congratulations!—the two points make a diagonal line (Fig. 9). This process reveals that 38 unique diagonal lines can be made from the 3×3 grid’s 12 perimeter points (Fig. 10).

The next part is where things get tricky. I want to add another combinatorial layer to the diagonal lines I just made by finding every possible set that makes use of all 12 perimeter points and uses each point only once. That means every qualifying set will have six lines, but not every set of six lines will qualify (Fig. 11).

The combinatorics of my two previous Plus Equals explorations were concerned with finding sequence variations, which isn’t really relevant to this task, and my initial ideas for finding all these sets of six lines were tying my brain in knots. So I reached out to someone with a substantially hardier brain: my friend Paul Lopata, who has a PhD in mathematical physics and an abiding affection for puzzles. After we batted the problem back and forth a bit, Paul got us on the right track. His solution involves generating two lists of line combinations, and then using them to arrive at a final list.

We’ll call the first one List A, and it contains combinations of four lines. Importantly, in each combination, all four points in the grid’s zero column are represented. To find every such combination, I’ll use a similar process to the one I used earlier to find the full set of diagonal lines. For each line originating at 0,0, compare it to each line originating at 0,1. For each comparison, if no points appear more than once, combine the lines and repeat the process by comparing the combination to every line originating at 0,2. Once again, for each comparison, if no points are repeated, add the 0,2 line to the combination and compare it to each line originating at 0,3. This process (Fig. 12) ultimately finds a total of 549 unique combinations of four lines, with each combination making use of all four points on the grid’s zero column.

List B contains every pair that can be made out of lines that *don’t* use the grid’s zero column. To find them, list the lines in ascending numeric order. For each line, compare it to every line that comes after it. For each comparison, if no points are repeated, combine the lines and add the pair to List B (Fig. 13). The complete list ultimately contains 53 unique pairs.

The process for building the final list of line combinations should feel pretty familiar by now: For each combination in List A, compare it to each combination in List B. For each comparison, if no points are repeated, merge the combinations (Fig. 14). Of the nearly 30,000 comparisons made between the two lists, less than two percent result in mergers, but it still makes a sizable amount of tiles. The final list contains 425 unique combinations of six diagonal lines, with each combination using all 12 of the grid’s perimeter points. I now have a complete set of tiles, and each one can seamlessly border any other one on any side!

Before I knew how many tiles this system would generate, I imagined adding one more combinatorial layer to see how many different compound patterns the tiles could make, but that number surely stretches closer to infinity than is practical to ponder. Still, this issue’s gallery is arranged to give a sense of the possibilities. In addition to displaying each individual tile in isolation, it demonstrates the pattern the tile makes and how it seamlessly integrates with adjacent tile patterns. The final image’s intricate tangle of lines is a mosaic of all 425 tiles.

I don’t expect to redecorate my bathroom with these tiles, but then again, something tells me they’d inspire some good shower thoughts.

Rob Weychert

rob@robweychert.com

**Visit the shop to buy this issue’s print edition or poster!**

**Plus Equals is also available in print!** Visit the shop to buy subscriptions, individual issues, and posters. Plus Equals will always be free to read online, but if you like my work and want to support it, picking something up at the shop is a great way to do that. Thanks for your support!

The past year was a historic one for many reasons, most of them unpleasant, and I’m using this issue of Plus Equals to mark the moment and create an opportunity for reflection. I’ll get into the specifics shortly, but first, let’s take a step back and talk about how photographs are created and reproduced. We’ll stick to black and white photography to keep it simple.

In traditional photography, a camera is loaded with film, a thin strip of transparent plastic coated with a light-sensitive emulsion. The photographer points the camera at the object whose image they want to capture, and when they press the shutter button, the shutter momentarily opens to expose the film to the light (Fig. 1). Later, the photographer develops the film by putting it in a chemical bath in a darkroom, which makes the exposed images appear on the film in reverse: the light areas are dark, and the dark areas are light. This is called a negative (Fig. 2). Finally, the photographer shines a light through the developed negative to project the image onto photographic paper, which, like the film, is coated with a light-sensitive emulsion (Fig. 3). As a result, the lights and darks captured by the paper are reversed again—back to their natural state—to create a print of the final image (Fig. 4).

Since this process is capable of printing high quality images, you’d be forgiven for assuming printed photographs are always reproduced this way, but that’s not the case. Photo paper is thick, expensive, and generally unsuitable for use in most mass-produced printed materials. In fact, the process used to print newspapers, magazines, and books is essentially a more sophisticated version of a rubber stamp, which can only print a solid ink color, not shades of that color. This works great for text and line drawings, whose color is flat, but what if you want to print a photograph with a rich tonal range? It could contain millions of shades of gray, and printing each of those shades with its own individual ink is unrealistic.

This is where the halftone process comes in. To reproduce those shades of gray (called “continuous tone” in the printing world) with a single ink color, the original photo is rephotographed through a halftone screen, a sheet of glass with a finely ruled grid of black lines (Fig. 5). The film used this time is coated with a high-contrast emulsion that will only record black or white. As the light travels through the screen and onto the film, the image is captured as a vast matrix of black dots. The darker the area of the image, the denser the dots. Taken together, these dots trick the eye into thinking it’s seeing shades of gray when in fact only one color is present: black (Fig. 6). And *voila!* A continuous tone image is rendered in a single color, suitable for printing by a more sophisticated version of a rubber stamp.

Modern photography and print production processes are more digital than chemical, and physical halftone screens are rarely used anymore. But those tiny halftone dots are still at work in virtually every printed photograph we encounter, including full-color photos, which do some extra magic by using four inks instead of one.

I love halftones. Apart from the fact that they’re a brilliant innovation, which has yet to be substantially improved upon after over 150 years, I find them appealing both philosophically and aesthetically.

Halftones remind us that a complex concept is sometimes best understood by placing it on a spectrum between two simpler, more fundamental concepts. “This gray contains an equal amount of black and white.” “This jalapeño margarita is a little sweeter than it is spicy.” “Canada’s firearm-related death rate is three times that of Japan, but just one-sixth the rate of the United States.”

And halftones just *look cool.* When they’re enlarged or exaggerated for graphic effect, they give us a different lens through which to see photographs, abstracting them into decorative patterns, turning soft gradations into hard-edged textures. They’re an anachronism: a precursor to the Industrial Revolution that nevertheless seems computer-generated; a unique aesthetic bridging the organic, the mechanical, and the digital.

But making combinatoric generative art—as is my wont—with halftone imagery doesn’t have an obvious starting point. Combinatorics are concerned with finite systems, but halftone dots, reductive as they are, come in an effectively infinite number of sizes. To rein that in, I’ll incorporate another reductive technique, borrowed this time from digital imaging.

A digital image, displayed on the screen of a computer, mobile device, etc., is a grid of tiny squares called pixels. Like halftone dots, taken together, they trick the eye into thinking it’s seeing continuous tone. Unlike halftone dots, each individual pixel can be a different color (Fig. 7). A large, full-color digital image can be made up of millions of pixels and millions of colors. However, early computers didn’t have enough memory or file storage capacity to handle such data-intensive images. To compensate, images intended for these computers were rendered at low resolutions (fewer pixels) and with a limited palette (fewer colors). An efficient way to manage a limited digital palette is with something called indexed color, a kind of paint-by-numbers system for computers. Indexed color assigns a number to each color in the image’s palette, and each pixel is tagged with the number of its corresponding color (Fig. 8). (One example of an indexed color image file format that’s still in common use today is the irrepressible GIF.)

Since a halftone dot’s size corresponds to the shade of gray it represents—the darker the gray, the bigger the dot—I can limit the number of dot sizes by incorporating indexed color to limit the number of shades of gray (Fig. 9). This will give me a finite system to which I can apply some combinatorics. And within that finite system, I’m going to create variations of an image by rearranging the shades of gray in as many ways as possible.

I was pleasantly surprised to learn that the scope of possible rearrangements is a pretty simple matter of factorials. As Wikipedia explains, “the factorial of a non-negative integer *n*, denoted by *n*!, is the product of all positive integers less than or equal to *n*.” In other words, to find the factorial of *n,* do a series of multiplications beginning with *n* and working backwards through every integer before it, stopping when you reach zero. The result tells you how many distinct sequences are possible for a set of n objects. For example, if I’ve limited my image to four shades of gray, 4! = 4 × 3 × 2 × 1 = 24, which means my four shades of gray can be arranged 24 different ways.

To find out what those specific arrangements are, I can reuse the method I devised in Plus Equals #1 for finding all possible stacking orders for a deck of cards. The difference this time is, since I want each of my image variations to use the entire available palette, I only need the sequences that include the full deck of cards—or in this case, the full set of grays—and I can safely ignore the rest (Fig. 10).

The final process goes like this (Fig. 11): a) Take a grayscale digital image. b) Reduce its resolution and reduce its palette to a limited number of shades of gray. c) Assign each of the palette’s shades a number, and tag each of the image’s pixels with the number corresponding to its shade. d) Replace each of the palette’s grays with a halftone dot of corresponding size. e) Rearrange the palette’s numeric assignments in every possible sequence and render a new halftone image for each.

With the combinatoric system now established, all that’s left to do is settle on a source image and decide how many shades to include in the palette, the factorial of which will determine how many variations of the image are generated. Ideally the source image will have a rich tonal range that makes robust use of the full palette, making the visual distinctions between the generated variations readily apparent. But I also want the image, its subject, and its response to this process to communicate something beyond the particulars of its aesthetics, which brings us back to this issue marking the moment and creating an opportunity for reflection.

I’ve decided to use a six-shade palette to generate 720 variations of George Floyd. To my eyes, there’s a grotesque violence in the ways these variations distort Floyd’s image—a violence that’s resonant with his murder, with its place in the legacy of American slavery, and with the subsequent reckoning’s psychic toll on the nation over the past year. Amid the variations’ ordered chaos, the halftone dots themselves seem to invite conflicting interpretations of our sociopolitical reality. Is their binary black and white emblematic of insurmountable polarization? Is their variety of sizes and positions optimistic that diversity will prevail? Whatever you see in these images, and especially if you, like me, are an unwilling beneficiary of a racist system, I hope you agree that George Floyd and everything he represents are worthy of our prolonged gaze.

Rob Weychert

rob@robweychert.com

**Visit the shop to buy this issue’s print edition or poster!**

**Plus Equals is also available in print!** Visit the shop to buy subscriptions, individual issues, and posters. Plus Equals will always be free to read online, but if you like my work and want to support it, picking something up at the shop is a great way to do that. Thanks for your support!

About eight years ago, I created a simple design concept as a gift for family and friends. It consisted of eight 5″×7″ cards of different colors; two were solid and the other six had geometric shapes cut out of them. Stacking the cards in different configurations could generate seemingly countless patterns, making it a small decoration that could be dramatically altered at a moment’s notice to suit the occasion or just provide a little change of scenery (Fig. 1).

Not long after, I was made aware of Kaleidographs, toys based on the same idea, whose system of 12 two-sided square cards boasts over 500 billion possible unique combinations (Fig. 2). That staggering number made me wonder: How many combinations were possible with my simpler design? I wanted not only to know the amount, but, to the extent possible, to *see* all of the combinations. This seemed like a fairly trivial problem for an algorithm, but my skills in that area were virtually nonexistent. Instead, using Adobe Illustrator, I developed a system through which I could theoretically manually generate the full set of combinations, but the amount of time and labor required proved to be too much and I eventually shelved the project.

Over the next several years, similar works caught my attention, including the self-explanatory poster The 892 Unique Ways to Partition a 3×4 Grid and a clever bit of activism called All the Music, which effectively puts into the public domain every possible melody that hasn’t already been copyrighted. But the one that captivated me most was Incomplete Open Cubes, a 1974 work in which the conceptual artist Sol LeWitt found 122 unique structures utilizing three or more of the edges of an open cube (Fig. 3). As I studied it, I began to realize that many more cube configurations were possible if some of LeWitt’s constraints were removed. Having learned the basics of Python in a workshop led by type designer and programmer Just van Rossum the previous summer, I was able to apply my modest new programming skills to expand LeWitt’s set of cubes from 122 to 4,094. I called the project Incomplete Open Cubes Revisited.

I’m still a poor excuse for a mathematician, but I continue to be fascinated by artistic applications of what I now know to be an area of mathematics called *combinatorics*. Much of the algorithmic art I’ve experienced has been based on feeding various values to an algorithm and cherry-picking the most interesting results. What appeals to me more about a combinatoric approach is the creation of a system whose many possible results are *all* interesting, both individually and as a set.

And so I’ve started Plus Equals as a place to document my exploration of the artistic possibilities of combinatorics. The name comes from an assignment operator common to many programming languages: When *x* += 1, it adds 1 to the value of *x*. Apart from the certainty that the += operator will be used in my work often, I’m drawn to its succinct implication that combination is an act of creation.

For this first issue of Plus Equals, I think it’s appropriate to revisit the eight-card design concept that started me on this path all those years ago. My math skills still aren’t as sophisticated as I’d like them to be, but building on my experience with Incomplete Open Cubes Revisited, I’ve managed to figure out not only how many ways those eight cards can be stacked, but how many unique combinations are possible for any other number of cards.

It begins with a set of two. Each individual card is counted as one stack, as is each card stacked on top of the other, which yields four possible stacks: A, B, AB, and BA (Fig. 4). Adding a third card to the set introduces an additional level of stacking considerations. Card A now yields not only an AB stack, but also an AC stack. AB yields ABC; AC yields ACB. The three-card set ultimately gives us 15 unique stacks (Fig. 5), a four-card set gives us 64 unique stacks (Fig. 6), and just as it’s starting to get cumbersome to count everything this way, a mathematic pattern emerges. Going back to the two-card set, which yields four unique stacks, if we add 1 to the number of unique stacks (4+1), add 1 to the number of cards (2+1), and multiply the two sums, we get 15, which is exactly how many unique stacks are possible with a three-card set. Apply that same formula to the three-card set, and we’ll see another familiar number: (15+1)×(3+1)=64. And so on: 4 → 15 → 64 → 325 → 1,956 → 13,699 → 109,600. By the time we’ve reached 10 cards, nearly 10 million unique stacks are possible!

So, using this method, we know that when up to eight cards are in play, there are 109,600 different ways they can be combined. But in the case of my original eight-card design concept, the number of possible unique *images* they can generate is actually far lower than that. First of all, two of the cards—G and H—are completely opaque. Any combination of cards stacked under one of them will be completely hidden from view, meaning any combination of cards with G or H on top is functionally identical (Fig. 7). Secondly, owing to the arrangement of the shapes cut out of cards A through D, any combination of all four of those cards is also opaque. As with cards G and H, whenever cards A through D are all present, nothing beneath them is visible (Fig. 8).

There’s certainly a way to work around these issues, find the true number of possible unique images, and generate them, but that seems like a fairly convoluted start to this enterprise, and one I’m not really interested in pursuing. Instead, I’ve decided to make things more manageable by paring the deck down to just cards A through D, knowing that every possible combination of them will yield a unique image. I’ve also made them square and replaced their colors with black stripes on white (Fig. 9). The latter decision, born partly of printing budget constraints, has the benefit of introducing added visual interplay between the cards.

As we learned earlier, these four cards can be stacked 64 different ways, a manageable amount I could do manually, but instead I wrote a Python script to do it for me. Run through a vector drawing app called DrawBot and utilizing the combinatoric stacking method described earlier, the script draws each card and generates all 64 unique combinations, which are collected here. Modified versions of this script are likely to power future Plus Equals explorations, and I hope you’ll join me for those. Enjoy!

Rob Weychert

rob@robweychert.com

**Visit the shop to buy this issue’s print edition or poster!**