Millennium Park's Cloud Gate is irresistible to the hand and eye. We crouch beneath it and play in front of it. We stroke it. We take photos of our warped, stretched bodies reflected in its curved chrome surface. Or we simply stand and watch the sky change in it.

This past July, however, the focus of passersby was temporarily shifted toward a new curiosity sharing the plaza with the Bean. Mounted atop a tripod was a curved metal grid, which held a large piece of drawing paper, also curved. Projecting from a rectangular metal frame around the grid was a white plaster skullcap. Trevor Oakes, 26, spent hours standing at this tripod with his head in the cap, while his identical twin, Ryan, walked around Cloud Gate with a closed umbrella strapped to his foot.

The typical reaction to this sight was confusion—you could practically see questions being formulated behind the furrowed brows and open mouths. Some people figured they were seeing a performance; others concluded that the two were surveying the plaza using some strange new tool. When someone would finally ask, "What are you doing?" Ryan answered, "We're making a perspective drawing."

If you got right behind Trevor, you could see he was slowly making tiny pen scratches on the paper. Though he faced Cloud Gate, the scratches looked nothing like a drawing of it, as they were all contained within a vertical strip on the paper's righthand edge. If you kept watching, you'd see Trevor slice off that strip with an X-Acto knife and store it in a plastic folder. Then he'd begin making scratches along the new right-hand edge of the paper. Once in a while he'd take out the strips and reassemble them on the grid, and to quiet gasps of excitement and surprise Cloud Gate would appear.

As Trevor drew, Ryan explained. Their tripod apparatus, which they call "the easel," in a sense allowed Trevor to see through the paper he drew on to the object beyond it being drawn. Normally, our two eyes unite what each sees individually into one picture; but the easel is designed to separate what they see. In this case, Trevor's left eye saw the paper, his right eye Cloud Gate and the plaza around it.

Splitting focus this way doesn't come naturally. The brothers liken the effort required to the effort most of us must make to see the three-dimensional picture in a Magic Eye puzzle. "There are no similarities with the way the image is generated," they explained in a recent e-mail from "R + T," "but in terms of focus splitting, it is the same action by the eyes of separating one's sight lines at that paper plane (such that they would be converging at a point behind the paper)."

The effect of this technique at Cloud Gate was that the two images seemed to overlap, as if the paper were transparent and the sculpture could be seen behind it. But this overlap occurred in a narrow band—roughly the distance between Trevor's eyes—which was why he was drawing in two-inch vertical strips. When the paper was whole, Trevor looked past the concave grid that held it, but as it was sliced and narrowed, his right eye looked through the grid. The easel is the product of months of rigorous calculation and physical labor—its bars are precisely designed to all but disappear as Trevor looks beyond them.

Trevor drew only lines. There was no shading, just the lines of buildings, of windows in buildings, of concrete on the sidewalks, of leaves on trees. Far-off lines can become difficult to make out, which is why Ryan attached the umbrella to his foot. It helped guide Trevor's right eye to lines in the pavement that disappeared as they receded in the distance.

One other contemporary artist, Robert Irwin, has explored the splitting of focus. Writing about Irwin in his book Seeing Is Forgetting the Name of the Thing One Sees, Lawrence Weschler quotes him explaining that he trained himself by "placing a dot on a window and gazing both at and beyond it, thus allowing two planes of focus, one for each eye." Irwin played with perception in a famous series of paintings of dots and discs, but the Oakes twins are alone in using split focus to draw perspective. They didn't know this until after they'd done it.

The easel the Oakeses invented to impose image on paper has forerunners that go back centuries. Their device is the first, however, that requires neither mirrors nor lenses.

About 25 years into the 15th century, Dutch painting took a dramatic turn toward realism that can be seen in the works of Robert Campin and Jan Van Eyck. At the same time mirrors began showing up in Dutch paintings—they'd apparently become a part of everyday Dutch life. These were convex mirrors, but a convex mirror turned over is a concave mirror, which could be used by artists to render detail and perspective with vastly greater precision. (Painters belonged to the same guild as mirror makers then, so they would have known what mirrors could do.) According to Charles Falco, professor of optical sciences at the University of Arizona, a concave mirror has the same optical properties as a lens, in that it can concentrate and focus light rays. To demonstrate, Falco has suggested aiming a shaving mirror to reflect the morning sunlight onto a bathroom wall still in shadows: "Move the mirror in and out till things cast there onto the wall come into focus, and what you'll get is a technicolor perfect image of the world outside."

Next came the camera obscura, frequently mentioned in Leonardo da Vinci's notebooks. The artist sits in a darkened chamber viewing his subject, who is outside the chamber in bright light, through a small open window. Light enters the chamber, is collected on a lens mounted near the artist, and projected upside down onto the opposite wall, where paper or canvas has been set up. This process allows the artist to fix essential details, such as the corners of the eyes, the edges of the nose, and the mouth and ears of a face being painted. Once these parameters are sketched, the artist works from eye, fleshing out a more complete portrait. The concept of the camera obscura was written about by Aristotle in Problemata, as early as 350 BC.

Then there's the camera lucida, patented in 1806 by William Hyde Wollaston, an optician, chemist, and physicist. The primary component of the camera lucida is a prism, placed above the paper or canvas, in which an image of the scene to be rendered is captured. The artist looks through the image in the prism to the paper and draws—or traces, in a sense—what he sees. The device, still in use, is difficult to use, but some master it. Oxford art historian Martin Kemp wrote in The Science of Art (1992) that Sir John Herschel, a skilled draftsman, "achieved splendid results over a number of years with his camera lucida." In the 1830s, Herschel and his wife, Margaret, used a camera lucida to create a series of celebrated botanical illustrations that in 1998 were published in the volume Flora Herscheliana.

The Herschels' use of the camera lucida is certain. David Hockney, in his 2002 book Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters, set off a tempest by arguing that work by painters as exalted as Caravaggio, Velazquez, and Ingres also exhibits what "the optical look." This step toward realism crossed Europe so quickly in the 15th century that Hockney believes technological innovation must have been involved. Grounding his theory in the science of Charles Falco, he observed how repeating patterns in certain paintings, such as the intricate carpet in Lorenzo Lotto's Husband and Wife, go in and out of focus. He and Falco identified varying vanishing points in these paintings, and attributed them to the shifting of a lens to refocus on different elements of the subject.

Art historians who think the use of optics is cheating have resisted the Hockney-Falco thesis. But as Hockney notes, "optics do not make marks, only the artist's hand can do that, and it requires great skill. And optics don't make drawing any easier either, far from it. . . . But to an artist six hundred years ago optical projections would have demonstrated a new vivid way of looking at and representing the material world." Before Hockney, Martin Kemp had reasoned that painters of the time were drawn to science and would have considered it a point of pride to be proficient with optical devices.

The camera lucida eliminated the camera obscura's projection of an image onto the picture plane, but it placed a prism between the artists' eyes and their subjects. In the Oakeses' method, nothing but the paper comes between their eyes and what they see; it is the paper itself that is bent into a concave surface, rather than a mirror or lens.

"David Hockney talks about the need for really bright lighting if you're going to use optics," says Ryan, "and that even when the camera obscura became portable, if there wasn't bright enough light you wouldn't see an image at all. But with your human eye looking at the world, we adjust."

"We read Hockney's book in 2006," says Trevor, "and it was immediately very inspiring and fascinating that there had been three other techniques used over the course of history since the 1400s on, and we recognized that this was another optical technique that allowed you to trace the world. If you compare them directly, there are several formal differences, and some of the differences do make it more in common with your eye and with the way that humans see the world. Because it's spherically concave and because light rays approaching your eye are traveling perpendicular to a sphere, it is more in harmony with our eye than using a flat picture plane. A flat plane is not at all related to the shapes you see and the physics of light."

Before reading Hockney, Ryan says, "We weren't focused. We were aware of perspective as a mathematical rule, and that was it. Even how a camera lens worked wasn't something that we put a lot of thought into."

The slow steps the brothers have taken toward their understanding of vision and light began with a series of collaborative projects at Cooper Union in New York, the art school the Colorado natives both graduated from in 2004. They began by building a dome of matchsticks, glued together layer by layer, with the red tips forming the shell. "Building that sculpture was the first time we started to think about other entities in nature that shared that spatial structure, the structure of a bunch of rays perpendicular to a sphere," says Trevor. "And the first things that came to mind were light rays being emitted from a point source, like a lightbulb, which has a spherical burst of rays. And then also, light rays coming into your pupil, condensing back to a point. It was building that structure that really got us to wrap our minds around the science involved, and to start considering the physics of it more specifically."

That same semester in 2002 they also discovered map tacks—specifically some round, reflective ones that were being used to pin drawings to the wall of a Chelsea gallery. "The drawings themselves weren't that interesting," says Ryan, "but the tacks holding the drawings to the wall were fascinating." Trevor adds, "Because they were so small they're kind of like a mathematical point. A point, of course, has no dimensions and can't really manifest, but if you lean very close to the map tacks you could see the profile of yourself and the bright spots of the lights in the room, and you could theoretically see that the entire room was reflected on that surface. You could then infer that the light rays were traveling from every point in the interior of the room to every other point on the surface of the room, that the light was behaving in a much more complex way than what we were actually seeing, that the air was hyperfilled with rays just going in every single direction, and that our eyes were only pinching out a thin slice."

Trevor presented a map tack as a sculpture for class. "It didn't go over very well," he says. "Some said uninteresting, others said incredibly uninteresting. The professor said it was worth a D."

The twins' next project together was a curved sculpture several feet long made of corrugated cardboard. "The corrugation of the cardboard makes it more like a three-inch-thick wall that's curved like the crust of a sphere," Trevor says. "Because all of the corrugation faces forward, toward the viewer, the corrugation is perpendicular to the crust of that sphere, or the eye viewing it." From a distance of about six feet "the entire sculpture becomes transparent, like an open window. You can look straight through it." This sculpture acknowledged that the world is seen through a sphere, the eye, and that light rays are oriented to the crust of that sphere. The easel the brothers now use was built on the same principle. The cardboard sculpture got a B.

Two months later an insight hit them, so to speak, right between the eyes. "A pretty key moment was noticing the profile of your nose," says Trevor. "As you're looking forward, you can see a profile of your nose on either the left or right side of your vision. If you close either eye it becomes more apparent. . . . Too far past the left profile is this area where only the left eye can see, and too far past the right profile is where only the right eye can see. You have these two monocular zones on either side and then one binocular zone in the middle." With a laser pointer and paper tacked to a far wall, the brothers took turns mapping each other's monocular zones.

Finally came the discovery of the ghost image. Holding his sketchbook waist high and standing directly above a leaf on the ground, Trevor found that when he focused on the sketchbook to draw, a double image of the leaf appeared beyond it, and when he focused on the leaf, a double image of the sketchbook appeared. As he looked down and placed the sketchbook in front of his left eye while looking at the leaf with his right, Trevor's visual cortex attempted to merge paper and leaf into one image, creating an overlap that allowed him to essentially see through the paper and trace the leaf in detail. "The ghost image was the pivotal realization for what then turned into the method to capture the world in its proportions, scale, and diminishing perspective, exactly as your eyes see," he says. "This is a way of rendering realism that takes proportion, location, and perspective as a given and automatically gets them right.

"We noticed the double image a few years before we ever realized its potential for tracing. At first we just noticed that the phenomenon occurred, and that it meant that within the three-dimensional space your eyes are looking into, there is only one point at which you can perfectly line up, and that everything else moving away from that point becomes shifted into a double image more and more. Even from that point we didn't see how far it was going to go or that we would be focusing on doing these panoramic drawings."

Once the brothers realized they could exploit the double image to render perspective on paper, the easel wasn't far behind.

"You need to hold your head and the paper very still," Trevor continues, "because if I move my sketchbook just a little bit, lines will start landing in the wrong spot. And when I move my head, things become skewed out. Then we realized that we couldn't use a regular flat sheet of paper because after you stabilize your head, the distance from your eye to the very center of the paper will be shorter than the distance from your eye to the corner of the paper, and that would cause objects in the corner to enlarge in scale and you would get a kind of ballooning distortion and bubbling effect," similar to the distortion caused by lenses.

"So instead, the paper needed to be spherically concave so that the length from your eye to the paper was equidistant for all points on the surface of the paper. We had all these grand ideas of what we could do with it, and the natural course of things led to building our first easel, in May of 2004."

The original easel was made of malleable armature wire and survived only two drawings before it got bent out of shape. Early versions involved a chin rest, which helped keep the head stationary but was uncomfortable. The frame and headpiece came later. It was Ryan's idea to build it out of rigid sheet metal and attach it to a tripod. They enlisted the help of their cousin Willis Bowman, a machinist in Ohio with a barn full of metalworking tools. "It was a pretty labor-intensive process," says Ryan. "We had conceptualized it but we hadn't seen it in action, or seen what it would really look like." They wound up having to carve a dome from a tree stump on a lathe, learn spherical trigonometry, and create a template for cutting sheet metal. "We assembled it on the final day and welded it together, and then all of a sudden this thing was there. We had been planning it in our heads and we had some sketches, but to have it there was kind of surprising.

"As soon as we got it back to New York it took over our lives," Ryan continues. "All of a sudden this thing had shown up and it required us putting a lot of attention into it."

The twins' upstairs neighbor, Tiffany Schauer, immediately commissioned an easel drawing. An environmental lawyer who founded the nonprofit firm Our Children's Earth, Schauer had recently become interested in arts patronage.

"Tiffany just got the ball rolling," says Ryan. "We had three of these concave drawings in our apartment in a flat file [the drawings are exhibited concave but can be stored flat], but we hadn't been getting much mileage out of them and we hadn't had them scanned. The scans are about $200 apiece." Schauer picked up the tab for the scans. "As soon as we had digital files it opened up a whole new world. We were able to start printing them and making reproductions. It allowed us to really distribute those images in a way that we wouldn't have. . . . You know, $600 is not a ton of money, but we wouldn't have taken on that expenditure with our bare-bones income."

Aside from the easel drawings, Ryan and Trevor have individual bodies of work. With a homemade brush, Ryan has created a series of paintings consisting of calligraphic glyphs. Individually, the glyphs suggest the shadows of tiny dancers; collectively they suggest abstract landscapes. Ryan describes them as explorations of what his mind perceives the interior of his head to look like—the feeling of his tongue in his mouth and the space it occupies, the view of the backs of his eyes—intuitive diagrams of his internal self.

Trevor's works are pipe cleaner weavings that, although he didn't realize it when he was making them, follow the principles of hyperbolic geometry—a non-Euclidean geometry of curved, diverging lines. The elaborate weavings resemble beds of coral.

It was these structures that led the brothers to Margaret and Christine Wertheim, also twins and founders of the Institute for Figuring in Los Angeles, an organization (and place) "dedicated to the poetic and aesthetic dimensions of science, mathematics and the technical arts." In 2005 the Wertheims, who're responsible for the Crocheted Coral Reef Project that visited Chicago a year ago and continues to tour the U.S., were giving a lecture on hyperbolic geometry at the Kitchen in New York City. With them was Daina Taimina, a Cornell mathematician who'd invented an algorithmic process for hyperbolic crocheting.

"The dean of the art school at Cooper Union purchased one of the pipe cleaner sculptures from our senior show, a very big deal at the time, and a few months later he forwarded us an invitation to a lecture by the Wertheim sisters," says Trevor. "The dean had recognized the similarity in form between my pipe cleaner weaving and their crochetings on the invitation. I also immediately recognized the connection and decided to attend the lecture with one of my weavings, and to introduce it kind of quietly afterward. . . . But toward the end of the lecture Margaret announced that the Kitchen was going to close promptly when the event was over, so no one was going to be able to hang around. A friend that was with us said, 'Just raise your hand during the question-and-answer session.'"

"I was really on edge," Ryan interjects, "because Trevor's thinking about presenting this sculpture to the whole audience and I'm like, 'That is so obnoxious,' you know. These people are here with their own work, their own hyperbolic stuff, and that's totally uncalled-for if you do that. But our friend was there saying, 'Just do it, just go ahead!' They said, 'OK, we'll take one more question,' and he raises his hand and luckily gets called on and stands up and says, 'I'm a Cooper Union student and I have a similar project with a similar form,' and they say, 'Do you have it with you?' 'As a matter of fact I do.' So he sort of hoists this thing up and it's all glinting orange and green, and oohs and ahhs go through the audience." Ryan laughs. "It was very well received, and they called him down to the stage. They were all intrigued, and people were taking photos. Immediately following, Christine Wertheim invited us to dinner, and on the spot said that they would like to commission a pipe cleaner weaving."

Trevor adds, "Christine also later told us that when I was speaking, she was thinking to herself, 'Oh God, another art student who wants his 15 minutes of fame.' But then after we showed her the project she was inviting us out to dinner with mathematicians."

The Wertheims also invited them to lecture at the Institute for Figuring in June 2007. The trip was funded by Tiffany Schauer.

The Oakes twins, the Wertheims also decided, needed to meet Lawrence Weschler. The Wertheims knew Weschler through David Wilson's Museum of Jurassic Technology in LA, where they'd curated shows. Weschler wrote about the museum in his 1995 best seller Mr. Wilson's Cabinet of Wonder.

They introduced Weschler to Ryan and Trevor in New York last November. He was charmed. "I want to see what you're doing," said Weschler, "but more importantly, Hockney needs to see what you're doing." Hockney is another of Weschler's friends—they met in the early 80s when Hockney read Weschler's book on Robert Irwin and invited the author to his studio to discuss it. The discussion never ended, and next month Knopf will publish his True to Life: Twenty-Five Years of Conversation With David Hockney. It's being packaged with a reprint of the book on Irwin.

"After Thanksgiving," Trevor continues, "Ren [Weschler] came over for a studio visit and we gave him a description of our observations and developments. We were having conversations about perception and our sculptures, about the physics of light and all that was orchestrated and merged together into the perspective system, and he became enthusiastic about it."

Weschler invited them to his Christmas party at New York University, where he heads the New York Institute for the Humanities. "He said to bring a box of goodies to show everybody," says Trevor, "so we did. We brought a few miniaturized panoramic easel drawings and the matchstick sculpture and a bunch of photos, and he kind of just pushed people into our corner all night and had us explain what we were doing."

Shortly after that Weschler set up a meeting for the Oakeses in LA, including the Wertheims, Hockney, Hockney's aide David Graves, LA Museum of Contemporary Art director Jeremy Strick, and actor/author/sleight-of-hand master Ricky Jay. "It was a little forum on perspective," says Ryan. Again, Schauer paid their way. "Amazingly," Trevor adds, "all of the pieces fell into place at the right time."

Weschler is also artistic director of the Chicago Humanities Festival, and in that capacity he commissioned the twins to spend July in Chicago creating several panoramic drawings. They'll be exhibited October 29 through November 30 at the Spertus Institute, where the twins will lecture on November 9 as part of the Humanities Festival. (Irwin will lecture a few hours earlier at the Art Institute.) The Spertus show, organized by Weschler and Spertus director Rhoda Rosen, will also feature a drawing of the facade of the new Spertus building on South Michigan, the easel and its imperfect predecessors, and their early projects from Cooper Union. And a six-foot enlargement of their Cloud Gate drawing will be exhibited on the plaza just east of the sculpture itself.

Since the easel came to be, many things have "fallen into place" for the twins. Alice Gallen, an IT consultant for Accenture was walking past Cloud Gate in July, while the Oakeses were on the plaza. She stopped, thinking the easel was a public viewfinder and wondering why Trevor was hogging it. Upon clearing things up with the twins, she discovered they knew some people she was interested in—Weschler and Irwin.

Gallen invited the twins to dinner at her place in Marina Towers, and Ryan and Trevor found themselves looking down at the Chicago River as it bends at Wacker Drive—the exact scene they had been considering for their next drawing. Gallen invited them to set up shop in her condo. "She has a deck that overlooks the river," said Ryan. "It worked out really well."

The easel has become an ambitious member of the Oakes family, urging them out to see the sights and insisting that they reconsider what they see. "It's shepherding us through the world," say Ryan. "It sort of prescribed in a quick sweeping arc many things that were possible."

Trevor agrees. "Once we had it and stepped out to do a drawing we found ourselves placed in one location for the entire length of the day. Over the course of the day we watched the flow of people on the sidewalk change. We watched bursts during rush hour happen. The same person would leave their office, run an errand, walk back past, and you would see them on both segments of their journey. You just get to observe the world from a single vantage point and pay attention to what goes on in a way that you otherwise wouldn't because you yourself would be in flux. You yourself would be moving. I think in that way it was saying, 'Here's your reason, your motivation, to stand still for five days and watch everything that happens from this point in space.' It fills you in about all sorts of aspects of humanity that you otherwise don't get to see."

"The easel also became a key," Ryan says. "All of a sudden we were able to get access to some remote roof terrace, or all these places in New York that you'd never see. It helps us see in a different way—or it hasn't helped us, it's allowed us. After we'd done the first few cityscape drawings I could feel the volume of that space much more wholly than previously."

That feeling translates to the finished product. Concave easel drawings the twins have done from on high in New York have an almost vertiginous quality, particularly the one from atop the Chrysler Building. Their Marina Tower drawing, the only Chicago scene rendered from a high vantage point, likewise captures the feeling of looking over the edge in a moment of lost balance.

Both twins can use the easel, but Trevor's more comfortable with it than Ryan. "It's given us pause to look at the world," Trevor says, "and just the act of looking really keenly, looking at details, tuning in on what's happening in the distance, sharpens your vision."

"Before," says Ryan, "you sort of perceive the idea that you see flat—you're used to looking at flat pictures and photographs. But having seen the world imaged accurately on a spherical surface allows you, when viewing the world, to think of it in a bigger way, more true to perception, more volumized."

To look through the grid of the easel and feel your eyes fight against their nature is to be reminded of the physicality of seeing. But simply looking at the Oakses' work can do something similar. "It's always really nice," says Trevor, "to hear people respond to it by saying 'I see the world differently because of it,' or 'I feel like I have different eyes because of it.' It's great. Our whole thing was just trying to crack the mysteries of the universe, or something. When we found the double image we were like 'Oh, cool, something we're not supposed to see, yet it's something that happens. Let's draw this.' And then you realize that it's the perfect tool to ultimately capture reality in a way that is so simple it's almost cheating."

Care to comment? Find this story, along with video of the brothers at work and more pictucres, at chicagoreader.com.