Rotation It's All In Your Head I learned about rotational mathematics using my head (after being stricken with VERTIGO) Jeffrey Ventrella, January, 2005 (updated June, 2009)

 Introduction We've all learned about the five celebrated senses: seeing, hearing, tasting, smelling, and touch. But there are a few more which don't get as much attention. One in particular... I woke up one morning with vertigo, and there is one sure way to become acutely aware of this sense - and that is for it to break down, and essentially whack you upside the head and knock you out for days.
 Specifically, I am referring to BPPV, which is short for "benign paroxysmal positional vertigo". http://www.tchain.com/otoneurology/disorders/bppv/bppv.html. http://www-surgery.ucsd.edu/ent/PatientInfo/info_bppv.html.
 While this experience was not fun, there was one rather exciting aspect - I discovered that the human body has a built-in mechanism which corresponds directly to the three axes of rotation that we commonly use in computer graphics and engineering mathematics. These three axes are embodied in the form of three small loops, each oriented roughly orthogonal to each other - they are the semi-circular canals, responsible for giving us the sense of balance - the vestibular system. They care about gravity, and they care about which way your head is rotating - and they are exquisitely good at sending signals to the brain (and eyes) as far as what's going on. The fact that the semi-circular canals correspond to the three axes of rotation is probably not a coincidence. But what then is the nature of this non-coincidence? That is the what I am interested in exploring here. Sensing Space and Time The fact that our inner ears are directly wired to our eyes is another reminder that our sense of space is not just visual, it is vestibular/kinesthetic/neuromuscular. Among the many automatic mechanisms we have for stabilizing our perception of the world is the Vestibulo-ocular reflex which causes the eyes to move in the opposite direction of head rotation. There is a fascinating phenomenon in the eyes called nystagmus (sometimes referred to as "eye-beating") which is caused by stimulation to the inner-ear. This is what doctors look for as clear signs of a vertigo problem. It is also what causes the room to spin when you have a dizzy spell.
 Saccades Our eyes are constantly darting back and forth. If a video were recorded of exactly what our eyes are looking at as we live our lives, it would be more jumpy and disoriented than the wildest MTV video. But our brains are part of a team which includes the eyes, semi-circular canals, and other players, in stitching together these snapshots into a consistent model of the world. Here is a college paper I wrote years ago on this topic: http://www.ventrella.com/Ideas/Four_frames.html

 I Have Rocks in My Head Quite literally, I have rocks in my head. Actually they are small crystals that are hanging around in places they shouldn't be. Most of us have a few of these swimming around in our canals, but they usually don't make a fuss. A recent discovery about these crystals, which are called "otoconia", suggests that they are the usual cause of BPPV when they get dislodged and move to the canals. http://cabinetmagazine.org/issues/31/burnett.php Some doctors specialize in treating this with what is called the Epley Maneuver . For this procedure, the doctor sets the patient's head in various orientations for certain durations, watching the eyes for nystagmus. The order of these positions is arranged specifically to cause these crystals to tumble around and finally come to rest in a place within the labyrinth where they will gradually adhere, and hopefully, they will not get loose and cause more trouble. Who would have thought that such a low-tech medical treatment should have been devised within the last ten years, when we have come to expect that all significant medical advances should be high-tech? Perhaps the discovery of the exact cause of BPPV is what was not possible before the current age of medical visualization and analysis.
 Animal Instincts We evolved from simpler animals (I realize that the current government does not endorse this crazy theory, but I'm sticking with it). Anyway, unlike plants, animals have locomotion - they move through the world (except for sponges, and some people with satellite TV). Animals have a special relationship to the environment as a part of distinguishing self from world - a self that moves throughout the world, thereby continually changing the landscape of stimuli. Animals have evolved internal models of the nature of their motions and actions in the world. Maybe this is the reason plants never evolved brains - not much reason to. The concepts of "up" and "forward" are, I believe, wired deeply in the brains of even the simplest animals. By implication, the concepts of "down" and "backward" are probably hardwired as well. "Left" and "right" are also probably represented deep in the brain, however, the distinction between left and right is not so strong. - and that might be because animals have bilateral symmetry - our left and right sides are mirror-images of each other. In a very common form of dyslexia, left/right distinctions are weak, and often get switched. Is there such a thing as up/down dyslexia? Perhaps in terms of the language of opposites, but, in terms of motor control? I would assume this to be uncommon - but hey, maybe Oliver Sacks knows of someone with this problem.

 The Embodied Language of Orientation In 3D Computer Graphics, these directions are usually denoted with x, y, and z. And they provide a language for specifying movements and positions. They also serve as axes of rotations. I also believe that rotation is a hard-wired primitive concept in the animal brain (how could it be otherwise when such an exquisite mechanism as the vestibular sense has evolved to represent the sense of rotation?). Some axes of rotation, I would imagine, are more salient than others. In humans, turning left or right (yaw, or "heading") is perhaps a more grounded concept than looking down or up (pitch), which is yet more grounded the cocking the head from side to side (roll). The relative importance of these rotations varies from animal to animal, depending on that animal's particular vocation. "Orientation", "Heading", and "Angle" are grounding metaphors - part of the embodied mind of all animals. Whether or not these actual words are used - or whether or not they are even words: the semantic structures are deep in our firmware. According to George Lakoff and Rafael E. Núñez... , these and other grounding metaphors have given rise to the pre-language stuff of our everyday language - it is so pervasive that we don't even realize it's there.

 But, Why Three? Okay, why are there Three axes? I asked Scott Kim this question once. He suggested that perhaps there is some inherent "three-ness" in the universe, and this is why it shows up in our dimensions. For instance, we speak of three-dimensional space, and we (commonly) use three rotational angles to define a full rotation in..."3D". I wished I could have discussed this with Scott more - he is a brilliant geometer and thinker. The question, "why three?" has been around for a while. Lakoff and Nunez would have me conclude that the human mind is a product of the human brain, which evolved in the human body, which has specific needs pertaining to survival in the world. We are a product of a cobbled-together experiment, tuned for eons by our interactions with the environment, and our ways of navigating that environment. What we take for granted as universal truths may just be part of our nature. Could it be that the three axes really have nothing to do with physics per se? This may not have a straightforward answer. All the animals we know about come from Planet Earth, and because of Gravity... , which is ubiquitous, we all have a pretty strong sense of "up". Even plants have this concept (to the extent that plants have concepts). But if we had evolved in outer-space, where there is no gravity, would our brains (and inner ears) have evolved in recognition of "up"? Then again, would there be any reason to have an inner-ear if you lived in outer-space? For whatever reason, the number is THREE - and I have a hard time imagining it being 2 or 4. And that is either because of a limitation of my brain or a property of the universe. Perhaps I can never know.

 Mathematical Artifacts The terms Yaw, Pitch, and Roll, (sometimes, "heading, pitch, and roll") are commonly used in aeronautics to describe the rotations of an aircraft (http://www.allstar.fiu.edu/aerojava/fltmidcont.htm). These terms refer to rotations in the local coordinate system of the plane, and not in the coordinate system of the world. This is used in computer graphics too - when one wants to manipulate the orientation of an object in a simulated scene in terms of its own local frame of reference. I have found that this provides an intuitive way of doing 3D graphics, as compared to hierarchical modeling, especially when the objects are autonomous 3D characters. They are virtual animals, after all, and not machine parts rigidly connected with logical constraints. In computer graphics and engineering, we have various ways of representing the orientations of things. One very common representation is the one I have just been describing, in which three angles are used (Euler angles). Because of the nature of Euler math, the order in which rotations are performed on a virtual object are critical. And if one accidentally or absent-mindedly does rotations in the wrong order, things can get really confusing. This is one reason we have alternative techniques for representing the orientations of objects, such as Quaternions. Here is a paper claiming that quaternions are not only more efficient for modeling saccadic and compensatory eye movements - they are more directly related to the actual physiological variables: http://modelingnts.la.asu.edu/pdf/InvarBK1.pdf. Whenever I encounter techniques that have common problems like the rotation-order issue when using Euler angles, I ask Mother Nature what she thinks about it (not that I always can understand what she is saying, but I do think it's important to at least consult). Does this technique correspond to any grounding metaphors? Does it correspond to anything in physics? Is there some mathematical artifact that is getting in the way of a more natural representation? A free-floating object in the universe of an arbitrary shape does not have an inherent x, y, or z axis. It may be rotating along some arbitrary axis, and moving along some arbitrary linear path. But the imposition of three orthogonal axes onto the object is simply for illustrative, analytical, or constructive purposes. So I would conclude that the problem is not an inherent aspect of physics, but rather an issue of logic. It arises, not in the natural world so much as in the world of machines and architecture. In fact it is an important aspect of Rubik's Cube. It is a pattern in space-time, caused by a series of deliberate, constrained (orthogonal) rotations, probably by some agent, such as an intelligent animal.

 What My Inner-Ear Tells Me This whole yaw/pitch/roll business goes deeper than just being a part of the world of machines and architecture. In fact, it's sitting deep in each of our skulls. Furthermore, the vestibular system may explain why we use the concept of "rotation" (which is either a verb, or a noun with an implied verb), perhaps more often than the concept of "orientation" (which doesn't necessarily imply some action that caused it to end up that way - it just is - "that's just how it's oriented" - or if you are from England, orientated. I think I know now why our mathematical language contains "rotation" as referring to a transformations from some previous state - as a verb. It is because our inner-ears make us think this way. In fact, I would suspect that when you are walking around in a darkened room, there is a reason why you can know roughly where you are, and which way you are facing (up to a point). And it has to do with accelerations in head rotation, and the delicate signals coming from the inner-ear that tell you to re-align your inner compass. Inner ear: "you have just turned left about 45 degrees". You: "OK, I guess I'm now facing the door." As a matter of fact, my experience in having the Epley maneuver has made me even more acutely aware of how much our inner-ears have to do with the way we think about rotation. The Epel Maneuver requires that rotations of the patient's head be done in a very specific order, such that the crystals can end up in a place in the labyrinth that they would not have ended up if a different sequence had been used (even if the final orientation of the head in either case were the same!) It's kind of like a Rubik's Cube.

 The Limits of Our Knowledge According to George Lakoff and Rafael E. Nunez... "the only mathematical ideas that human beings can have are ideas that the human brain allows." ----- "http://perso.unifr.ch/rafael.nunez/welcome.html"... Likewise, if the brain is just a physical organ that reflects the physical realities of our bodies and our relationship to the environment we find ourselves in, perhaps our three-dimensional metaphors are arbitrary, just as arbitrary as the particular way Earth life has come about. Consider the Butterfly Effect: if things had been slightly different when the earth was in formation, the whole chain of events might have taken a different turn, and life could have come out looking entirely different. Likewise, if our life had formed on a different planet, maybe we would be asking questions like, "Why on Za7x are there 5 axes of rotation?" A Final Thought... So I have a final question: Did the brain evolve the grounding metaphors of rotation and orientation as a result of the anatomy of the semi-circular canals in our inner ears? Or..did our semi-circular canals evolve in order to provide us with sensations that correspond to and support our natural language? Chances are, both of these things evolved together, as generations of animals continually refined their inner models of reality so as to navigate the world. These inner models have shown up both as a predisposition to rotational language in the brain, and as two marvelous, intricate pieces of machinery tucked deep within our heads - which can sometimes be a nuisance.

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Jeffrey Ventrella

www.Ventrella.com