Plus Equals #4

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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.

Hexagonal eyeglass frames and a geometric drawing of a cube
Fig. 1–2: Hexagono eyeglass frames by Henau (left) and an isometric cube (right)
A New Yorker cover depicting two Thanksgiving scenes and a diagram showing the many parts of a refrigerator
Fig. 3–4: A New Yorker cover illustration by Chris Ware (left) and an exploded-view diagram of an LG refrigerator (right)

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.

Illustration of four steps in the process of building a Lego boat
Fig. 5: Lego instructional diagram
A pixelated video game screen shot of a marble navigating a complex grid environment, and an illustration of a waterfall driving a water wheel, which in turn directs water back to the source of the waterfall, seemingly against the laws of gravity
Fig. 6–7: Marble Madness (1984), an early example of an isometric video game (left), and The Waterfall by M.C. Escher (right)

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).

Two renderings of a pair of cubes, with dashed lines showing the path of their parallel lines. In the first rendering, the lines all originate from one of two points on the horizon and spread outward from there. In the second rendering, the lines all have different origins and different destinations.
Fig. 8: When two objects of the same size are depicted in naturalistic perspective (top), the one closer to the viewer appears larger, and the objects’ parallel lines converge on a vanishing point. In a parallel projection (bottom), the objects remain the same size, and their parallel lines never meet.

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.)

Isometric cube drawing on a dashed-line grid of triangles120º120º120º
Fig. 9: The angle between any two of an isometric projection’s three coordinate axes is 120 degrees. An isometric grid is made of interlocking equilateral triangles.

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).

In the first image, a man hangs from a clock on the side of a building above a busy city street. The second image is the same scene, but zoomed out, showing a movie camera and a padded platform under the man.
Fig. 10: A scene from Safety Last! (1923) (left), and a recreation of how the scene is believed to have been shot (right)

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).

A triangle of three isometric cubes and an isometric triangle whose edges meet in contradictory ways
Fig. 11–12: Are the three cubes on the left resting on the same surface? Is one floating above the others? Are two floating? Several interpretations are available. On the right, a Penrose triangle, an impossible object.

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.

3×3×3 isometric cube, whose units of space are shown with dashed lines
Fig. 13: The three visible surfaces of this 3×3×3 cube are the defined space for this exploration.
Three isometric blocks: 1×1×3, 1×2×2, and a fusion of two 1×2×2 isometric blocks, whose units of space are shown with dashed lines
Fig. 14-16: Block A, Block B, Block C

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).

3×3×3 isometric cube, whose units of space are shown with dashed lines. The units are numbered 1 through 19.010611020712030813030813040914051015111213161714191815
Fig. 17: The 3×3×3 cube’s 19 available units of space are individually numbered.
The various possible positions for Block A are shown within 3×3×3 cubes drawn with dotted lines.01,02,0303,04,0506,07,0808,09,1011,12,1313,14,1514,16,1712,17,1815,18,1911,16,1901,06,1102,07,1203,08,1304,09,1405,10,15
Fig. 18: Block A’s 15 positions
The various possible positions for Block B are shown within 3×3×3 cubes drawn with dotted lines.01,0206,0702,0307,0803,0408,0904,0509,1006,0711,1207,0812,1308,0913,1409,1014,1511,1216,1712,1314,1714,1517,1816,1718,19
Fig. 19: Block B’s 12 positions
The various possible positions for Block C are shown within 3×3×3 cubes drawn with dotted lines.06,07,1112,16,1707,08,1213,14,1708,09,1213,14,1709,10,1415,17,1802,03,0407,08,0907,08,0912,13,14
Fig. 20: Block C’s 6 positions

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.

Two columns of numbers depict the positions of Block A and Block B. For each row, if no numbers are repeated, a checkmark is shown, indicating a valid pair. Otherwise, an X is shown, indicating a disqualified pair.BLOCK ABLOCK BBLOCK ABLOCK B01,02,0301,02,06,0701,02,0302,03,07,0801,02,0303,04,08,0901,02,0304,05,09,1001,02,0306,07,11,1201,02,0307,08,12,1301,02,0308,09,13,1401,02,0309,10,14,1501,02,0311,12,16,1701,02,0312,13,14,1701,02,0314,15,17,1801,02,0316,17,18,1903,04,0501,02,06,0703,04,0502,03,07,0803,04,0503,04,08,0903,04,0504,05,09,1003,04,0506,07,11,1203,04,0507,08,12,1303,04,0508,09,13,1403,04,0509,10,14,1503,04,0511,12,16,1703,04,0512,13,14,1703,04,0514,15,17,1803,04,0516,17,18,19
Fig. 21: Block A’s first two positions are compared to each of Block B’s positions. For each comparison, if the two positions share any numbers, that means they overlap, and the pair is disqualified.

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.

Two arrangements of isometric blocks. The stacking order for the first is BAC; the stacking order for the second is ACB.BACACB
Fig. 22: Two different arrangements with two different intended stacking orders

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 different stacking orders for one arrangement of isometric blocks: ABC, ACB, BAC, BCA, CAB, CBAABCACBBACBCACABCBA
Fig. 23: Six variations of one arrangement. CBA is the intended stacking order, but different stacks change the way the blocks’ spatial relationships are perceived.

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.

Two different stacking orders for one arrangement (CAB, CBA) which nevertheless look the sameCABCBA
Fig. 24: Sometimes two different stacking orders appear to be identical.

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

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