Topology is the mathematical study of the properties of geometric objects that are preserved under stretching and twisting but not under cutting and gluing. That is, two objects are topologically the same if one can be continuously deformed (stretched and twisted but not cut or glued) into the other. For instance, a solid sphere (topologists call this a “ball;” “sphere” refers just to the surface of the ball) and a solid cube are topologically the same (you can see this by imagining deforming one into the other (Diagram 1)). On the other hand, a ball and a doughnut (“solid torus”) are distinct. 
The intuition here is that the ball and solid cube have no holes in them but the doughnut does. Mathematicians call the number of holes in an object its genus, so a sphere has genus 0 and a torus has genus 1.
The intuition here is that the ball and solid cube have no holes in them but the doughnut does. Mathematicians call the number of holes in an object its genus, so a sphere has genus 0 and a torus has genus 1. Adam Distenfeld, sculptor at Brooklyn Rockwerks, creates topological sculptures and fountains from large rocks he rescues from construction sites. When a construction crew digs a future basement out of the ground, it unearths many large rocks, residents of that plot of land for millennia, and throws them in a landfill. Distenfeld rescues such rocks from all over Brooklyn and transforms them in his studio into art. He uses industrial drills to remove sections of the rocks, leaving sculptures that present interesting mental challenges, particularly in exploring the dichotomy between their malleable topological identity and their rigid, physical permanence. One way to think about genus with respect to such solid objects is that it represents the number of solid discs that would be needed to fill in the object’s holes to yield a topological ball. 
Another way to think about genus is that it represents the number of cuts that would be needed to cut the object into a topological ball.
Another way to think about genus is that it represents the number of cuts that would be needed to cut the object into a topological ball. Topological theory tells us that any solid, connected, 3-dimensional object can be formed by starting with a ball and drilling holes and adding handles (Diagram 2). ). Note that the act of adding an unknotted handle is the same as removing an unknotted hole (Diagram 3). In particular, this means that any solid (with connected boundary) without any knotting in it is completely characterized by its genus. This, in turn, means that the topology of any of Distenfeld’s sculptures can be described entirely by one whole number, its genus. Since all of the sculptures have genus less than, say, 25, and almost all the sculptures have genus between 0 and, say, 3, there isn’t much topological variation among the sculptures.
If we can’t very well distinguish the sculptures from each other based on their topology, why think about them as topological objects at all? Well, for one, this act itself is an interesting mental challenge, transforming these objects in their solid, physical, permanence into malleable mental objects. Secondly, the variation among the sculptures lies in their appearance. It’s interesting to consider how we react as human beings to the physical sculpture and how these natural reactions relate to the mathematics, the absolute truth in some sense, within the structure. Attempting to reconcile appearance with mathematical identity forces us to confront our perceptual biases and bridge the two hemispheres of our brain in a new way. Try to think of the rock sculptures, in their hard permanence, as topological objects – as objects that we could mold like clay if we wanted to. Within the framework of this mental state, the solid rigidity of the structure and the force required to alter it physically counter the psychological transience of its pure topological identity in a powerful dialectic. Studying the topology of the sculptures guides the mind into this dialectic, balancing the solid geometry of the real world with the topology of the mathematical world.
While purely topological objects are amorphous, genus, the most basic and important topological invariant, is discrete. None of the sculptures have (or could have) a genus of two-and-a-half. In the process of coring the rocks, the topological identity may not be altered at all if the two ends of the removed void were connected on the skin of the rock. (Diagram 5) We can imagine a continuum of potential sculptures, with the voided space moving continually inward. (Diagram 6) At a single “instant” on the continuum, the genus of the imaginary sculpture would be transformed from 0 to 1. In fact, at this exact moment, the mental sculpture would be “singular” – not actually solid but rather infinitesimally thin where the two sides of the void first came together. Similarly, if two voids are removed from the original rock, it will create a genus 2 sculpture if these voids do not touch and a genus 3 sculpture if they do. Again, we can imagine a continuum of potential sculptures with the two voided spaces moving continuously closer. Once more, there will be a single point on this continuum where the mental sculpture is singular. (Diagram 7)
The impossibility of such a singular “in-between sculpture” is interesting in and of itself. Not only would the sculpture be logistically difficult even to approximate, but the brittleness and tensions within the rock would make such an approximation unstable, and the discreteness of size on the atomic level would make an exact construction physically impossible, even in the most theoretical sense. The mental process of trying to imagine creating this impossible singular sculpture is also interesting. It forces us to grapple with the idiosyncrasies of viewing the things around us as purely mathematical objects; the problem of singular objects forces us to abandon our grounding in the physical world when viewing the sculptures as topological objects. That is, of the billions of physical things we encounter on a daily basis, none are truly singular – all have real thickness in all three dimensions. To imagine this transformation is to imagine a physical process that could not happen in our world.
We can also imagine a continuum following the chronological path of the drill through the rock. Again, the topology of the sculpture will be changed at a single point on this continuum, but unlike the last two examples, this singular structure was realized, approximately, at the moment the drill first punctured the skin of the rock from the inside. Somehow, this “impossible” object did exist, in close approximation, in a fleeting moment. Yet, this moment is ultimately uncapturable, for if we were to stop the drill almost precisely at this moment, the torsion within the rock would probably make the sculpture unstable. The difference between this continuum and the first two is that the first two were purely mental spaces, with each point on the continuum representing a potential, finalized, theoretical sculpture that was never actually realized, whereas this is the continuum of time as it actually occurred in the voiding of the sculpture. Still, the mental processes involved are very similar. 
Ultimately, the goal of the sculpture isn’t to struggle in futility to attain these “impossible” objects, or even to explore the impossibility itself. Instead, it accepts this impossibility as a truth and explores the mathematical neighborhoods of the sculptures’ topological archetypes, physically manifested in these imperfect, rigid projections.
Ultimately, the goal of the sculpture isn’t to struggle in futility to attain these “impossible” objects, or even to explore the impossibility itself. Instead, it accepts this impossibility as a truth and explores the mathematical neighborhoods of the sculptures’ topological archetypes, physically manifested in these imperfect, rigid projections. The topology of the voided spaces is also interesting, particularly in its simplicity. Any connected voided space is a topological ball; that is, there is no genus in the voided space. Each sculpture can then be thought of as a topological ball with one or two or several topological balls removed from its interior. Yet, even when only one ball is voided, a sculpture with any genus can be created. How can this be? This variation lies in how the voided ball intersects the skin of the rock. If the void intersects the skin of the rock in only one region, the sculpture will be a topological ball, genus 0. If two regions, the sculpture will have genus 1; three regions genus 2, et-cetera. In general, if a total of k voided balls intersect the skin of the rock a total of n times – if the sculptures of the voids themselves contain n distinct regions of original skin – the voided sculpture will have a genus of n–k (Diagram 8).
In some sense, then, when considering the voided space, we are forced to distinguish between the skin and the interior of the rock. This particular distinction is one of the first we often notice when viewing the sculpture – the contrast in texture and color between the skin and the interior. The skin provides a window into the geological history of the rock, including its sedimentation, splintering, and weathering and its unearthing in a recent building project. Meanwhile, the exposed interior of the rock reveals the contrasting nature of the process of voiding the rock; the stone itself may be a different color, possibly even wearing paint stripped from the skin of the drill head, and its regular texture reveals the precise, industrial violence of the voiding process. And somehow, this most apparent of physical distinctions provides the mathematical machinery to classify the voided space.
We’ve discussed at length the challenge that the rigidity of stone provides in considering the sculpture as a purely topological object. The transience of flowing, dripping, and streaming water provides a different challenge in viewing a fountain as a topological object. One instant a continuous trickle of water will connect the head of the fountain with the basin, and then in an instant that trickle will be cut off, changing the topology of the object that is the space occupied by the water at any given moment. That is, this connecting trickle contributed to the genus of the water, and its sudden truncation reduced by one the genus of the fickle topological object that is the water.
Also interesting to note is that this variation in the streams of the water is a function of the rock’s irregularity. Somehow, the irregular but rigid permanence of the rock directly implies the transience in the water that flows through it. We are drawn back to a profound, underlying dialectic between physical and mental spaces. Watching the dozens of connecting streams come and go, begin and end, “glue” and “cut” constantly, we realize that this topological object entirely lacks permanence, in sharp contrast to the rock structure it surrounds. The transience of the water counters the permanence of the rock almost in a physical metaphor for the deeper dialectic between the physical and the mathematical. The more time we spend with the sculptures, the more we appreciate the layers of these connections, and sometimes even the perceptual and cognitive biases under which we operate on a daily basis. Actually, we need for the surface of the solid to be connected; for the solid to be a filled-in, connected 2-dimensional surface, This will always be true for these sculptures.
Knotting these holes and handles affects the topological identity of the resulting object. We won’t be concerned with this problem, though, since Distenfeld’s drilling process necessarily leaves unknotted voids (Diagram 4).
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