Breaking and Swapping Symmetry II
Breaking and Swapping Symmetry
James P. Buban, Physicist
Pervasive throughout numerous academic fields is the concept of symmetry and its usage:
In architecture, it can be seen in the mirror symmetry of many buildings as well as in the translation symmetry present in arrays of windows, pillars, and even tilings.
In music, symmetry is integral in both song structure–the repetition and alternation of versus and choruses, and meter–the use of time signatures and temporal repetitions in rhythm. Moreover, the breaking of symmetry through variation on harmonic themes is ubiquitous in musical composure.
In mathematics, the field of group theory is devoted to developing language and tools to deal with symmetry, applications of which can be found in solving differential equations and data compression.
In physics, Einstein, with the help of Hibert, exploited the geometrical symmetries of space-time to develop his relativistic theory of gravity, the success of which has led physicists to routinely incorporate various symmetries into most physical theories studied today.
In biology, the symmetry of the daily rotation of the earth and its annual revolution around the sun affect the life cycles of an overwhelming number of the flora and fauna on our planet.
In psychology, Pavlov’s dog was trained to salivate to the sound of a bell using the repetitive symmetry of ringing a bell during each feeding.
In the precise language of mathematics, symmetry is defined as the invariance of a (mathematical) object under certain (mathematical) transformations. A simple example would be the transformation of rotation on a geometric object, such as a square. Assuming the square is unmarked in any way, if an observer viewed a square, then looked away, it would be impossible to determine if the square had been rotated by 90 degrees before the second viewing. To the observer, a square rotated by 90 degrees and a copy of the square that has not rotated are indistinguishable.
From a more colloquial perspective, observing the asymmetry apart from the symmetry helps the observer know when things are out of place in relation to things that stay the same (invariance). It gives us the power to make decisions and even predictions. Thus, a better understanding of symmetry may have been crucial to the evolutionary success of our species. Under such a premise, we could suspect that recognition of symmetries, or pattern recognition, would be deeply intertwined in our primal psyche. Hence, it is not at all surprising that our species is naturally drawn to objects that exhibit symmetry or patterns; we pause to admire the pure symmetry of intricate patterns in art and architecture.
Moreover, we may find that breaking a pattern or symmetry once found true holds a certain power with a magnitude proportional to how fundamental the pattern was perceived to be. This could be as small as the satirical twist of irony or a pun, or as large as the announcement of Copernicus that the earth revolved around the sun.
This is, in a sense, what Haskins’ Skycube has done. By bringing the full vertical dimension of the sky down to the pictorial plane he has broken a perceived symmetry with psychological roots so deep that, undoubtedly, even many animals must understand at some level: the sky is always above. As a result of breaking such a fundamentally perceived symmetry or pattern, the Skycube immediately captures our attention. After the initial disbelief, it brings not only visual delight but also intellectual wonderment that finally causes us to question what other fundamental patterns might be broken, pushing us one step closer to expanding our intellectual and perceptual horizons.
The potential of this kind of shift in thinking cannot be underestimated. As mentioned above, history is replete with examples of individuals whose willingness to question presumed truth often led them to watershed discoveries that catalyzed breakthroughs in physics, mathematics, philosophy, and art. And yet, perhaps what makes the Skycube even more fascinating is that Haskins doesn’t stop at breaking our orientational symmetry. He introduces a new symmetry, scale invariance, dramatically changing our sense of perception. Such a replacement or swapping of symmetry is rarely devised or experienced. Haskins accomplishes this through his use of light, parallax, and fractal geometry.
LIGHT is all our eyes actually see. We visually observe objects surrounding us only by observing the actual light rays that are reflected off the surfaces of these objects or by observing the light transmitted from an object, like the sun, or a light bulb. In the case of the Skycube, the light coming from the front face of the cube surrounding the aperture is reflected light; whereas the light within the aperture is sunlight refracting directly off the atmosphere.
But Haskins has blurred this differentiation by creating a reverse-bevel aperture. By removing the visible thickness that one usually observes while looking through a window or hole in a wall, he has prevented light from reflecting off its interior edges, causing the hole to become imperceivable to the observer. As a result, the reflected light from the wall and the light emitting from the atmospheric changes in the sky appear to both be on the pictorial plane. This begins to interrupt our standard visual cues that aid in determining size and distance between us and the phenomena inside the aperture, giving us our first introduction to scale invariance.
PARALLAX is observed every time we move through space. For example, if a road sign and a tree are both positioned at the same mile mark along a highway, but the sign is 10 feet off the road and the tree is 2,000 feet off the road, then technically when we drive past the sign, we will also be driving past the tree. But perceptually it will appear as if the sign moves quickly past us, while the tree in the distance barely moves at all. This is the effect of parallax, when the amount of apparent change in location of an observed object is proportional to the amount the observer moves as well as the relative distance of the two objects along the line of sight. This spatial relationship allows us to regularly use parallax as a tool to measure distance and negotiate our environment. In fact, parallax is fundamental to our very perception of distance; we are accustomed to seeing objects closer to us “moving” more than objects further from us whenever we move through space.
But parallax appears to operate differently with the Skycube. As we move around the patio, the clouds which are thousands of feet away appear to move faster then the objects around us that are only a few feet away, convincing us the clouds are much closer then they actually are. Furthermore, the sky doesn’t simply seem close; it also appears to possess a kind of skittishness that plays with our normal understanding of movement and scale. Approaching the aperture seems to make the clouds grow smaller, like the sky is backing away from us. And yet when we walk backwards away from the aperture the sky seems to then move in toward us, causing the clouds to grow larger.
Although it might seem like the laws of physics are working backwards, there is a fairly simple explanation. The Skycube's aperture is simply cropping our view of the sky based on our position in the patio in the same way a standard window in a wall crops our perspective of what we see outside in relation to our interior movement. But as explained earlier, the reverse-bevel aperture inhibits us from making this connection. And since what we are looking at is more open, empty, and vast then any space we’ve previously observed outside the walls of a building, we are further led to assume this is nothing like a standard window.
Nevertheless, it is essentially the very same thing. Our left, right, forward, and backward movement determines our perspective. Moving closer to the aperture essentially increases the breadth of the sky we see. Any particular cloud observed is part of a wider composition and therefore seems smaller in relation to the overall frame. When we back away from the aperture we see less of that wider composition because its frame is cropping out the peripheral clouds we saw when we were closer. This makes it seems as if the clouds in view are growing larger in relation with the overall frame.
Despite these truths, the Skycube's unusual construction leaves us perceptually unable to employ parallax as our standard tool for understanding distance and scale in regards to the spatial relationship between us and the objects seen inside the aperture, bringing us another encounter with scale invariance.
To understand FRACTAL GEOMETRY, it is easier to start with the relevant example of the sky. Due to the amorphous nature of clouds and their inconsistent size, it is extremely difficult to make any kind of estimate of the height of a cloud. Perhaps people find it very difficult to distinguish between a large cloud at 30,000 feet, and a smaller cloud at 10,000 feet. Moreover, if the aperture or opening of the Skycube only allows the viewing of a section of a cloud, the chaotic gaseous nature of the cloud also makes determining its size difficult, and, as a result, its distance from the viewer is not easily determined. Effectively, this lack of structure causes the viewer much difficulty with determining any spatial scale, and viewing different objects at different distances can appear virtually the same.
This phenomenon is associated with fractal geometry, which refers to any geometry that has the property that cutting out a piece of any object results in a smaller approximate copy of the original object. Other examples include a graph of the movements of stock markets, where looking at a chart of a decade exhibits the same scale of movements and similar patterns as that of a weekly or even daily chart, and coastlines, where an observer from an airplane will see the same jagged patterns as those apparent in photos from satellites. When objects have a fractal geometry, the smaller pieces exhibit identical behavior to larger pieces, so changing the spatial scale does not change the measurable properties of the objects. We say these unchanging properties are scale invariant.
Thus the reverse bevel aperture, perceptually unusual parallax, and fractal geometry of the clouds make the determination and observation of spatial scale difficult when observing the Skycube. At the extreme, the moments when the sky is completely uniform (either a clear blue sky, or the pure gray or white of unvaried cloud coverage), there is absolutely no reference to discern distance in the direction of observation at all. Without this differentiation, the meaning of distance vanishes. In these moments, and depending on the time of day, the viewer may interpret the light coming in through the aperture as originating flush with the front face of the cube or any number of miles away.
In essence, Haskins’ Skycube breaks the symmetry of orientation, typically taken for granted, and swaps it with the symmetry of scale invariance; direction is lost and distance becomes incomprehensible. We lose our usual tools for visually perceiving space, yet the Skycube is largely a visual experience. With all these normal tools stripped from us, we are forced to become acquainted with a new kind of uncertainty, activating an often unaccessed level of perceptual awareness.
Through the process of breaking and swapping symmetry, Haskins fractures our preconceptions about the interior and exterior phenomena we encounter in our daily lives. We are invited into a process of reorientation by way of disorientation. Perception and observation are detached from assumption, allowing for immediate experiential knowledge. This reordering of our perceptual hierarchy frees us to more effectively relate and engage with our interior and exterior spaces.
James P. Buban received his PhD in Physics from the University of Illinois at Chicago. He was previously Assistant Professor at the University of Tokyo and currently teaches mathematics at the University of California, Davis.
Dr. Buban is also Lead Scientist at the San Francisco startup Predictium and founder and director of MathMavericks.com–connecting students with PhD-level tutors.