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The Euclidian Spatial Imperative That Both Reassures Us – And Misleads Us

(Alan Grant -3/13)

One can propose that fire, the wheel and the plow were the most important inventions of man. I would hark back to a much earlier time to propose the most important event - was the invention of the Plumb Bob (some kind of string attached to a rock). In one dramatic revelation, we understood what was up, and what was down – and as consequence, all we needed to know about gravity. As a derivative of the Plumb Bob, some curious soul began to speculate what would happen if we strayed away from that vertical plane. If we went too far we would end up at the bottom. But if we angled only half way from the top – voila we had created a right angle – A T-Square. Now to couple these two, along came Greek philosopher/mathematician – Euclid. He taught us about parallel lines that would never meet. Or did they?

An Evolutionary Retrospect Eons before primitive apes and then man, there were animate creatures with eyes that pointed laterally (with no overlapping visual fields). Most likely all that these creatures could detect was sudden external motion that would spook them. Then via a long evolutionary process, eyes began to move forward into the skull and our visual fields began to overlap – a genesis of binocular stereo vision. Conjunctive with depth perception came the necessity of Horizon. The Horizon also had a dynamic. The farther away objects were in our forward-looking visual filed, the higher they would appear within that field. Further, objects further away would appear slightly bluer (due to greater scatter of the shorter wave lengths in our visual spectrum). From a physiologic and anatomic we are designed, and are in best homeostasis with an upright and forward-looking posture. We match the vertical plane of the Plumb Bob and we visually lock into that reassuring Horizon.

Another self-correcting mechanism are the semi-circular canals in our inner ears. They are our gyroscopes – creating a righting reflex. No matter which way our heads twist, turn or rotate from that primary default position, our eyes will partially compensate to undo the head deviations. An additional anatomic structure that has evolved uniquely in Homo Sapiens is the configuration of the foveolar retinal area. It presents an elliptical shape with the long (prolate) axis along a horizontal plane. Whereas the short (oblate) axis is vertical –at right angles to the horizontal. While evolutionary changes, though random, also tend to be purposive for survival - It would seem logical that a this prolate foveolar axis would be in best consonance with the visual horizon.

Horizon Disruptions In everyday visual life we are subject to these disruptions a. A computer operator with fixation bouncing back and forth between the screen, the keyboard and any adjacent material that is required to support the task function. b. A violinist, by necessity working with a chronic head-twist, and at the same time have a skewed view of the horizontal plane of the score on the music stand. In addition if this individual has an astigmatism that has been carefully analyzed in examination – and appropriately corrected for, This Rx will be off. If the head is twisted 20 degrees away from the vertical – the eyes (courtesy of the ‘gyros) will rotate about 7 degrees back to the vertical). This implies that what we prescribe – is not always appropriate for the occupational needs of the individual. c. And quite often an unstable horizon on a tossing boat is causality for Mal-de-Mer (seasickness).

Sea Level and Man We are sea level creatures. And when we stray above or below sea level, we impose greater stress upon homeostasis. True that we can venture upward to 10,000 feet above sea-level and despite being in an environment with less partial pressure of oxygen – still be cognizant of our environment. Even within intense clouds, an airplane pilot will be reassured by an artificial horizon displayed on the instrument panel – and whether the plane’s posture jibes with, or is askew from level flight. Fig 1 Artificial Horizon Indicator When we venture below sea level (via scuba diving), our judgment starts to fail rather quickly (below 30-40 feet). We lose that horizon completely. Colors are perceived blue and green. And though when we expire air from our face mask, the bubbles will travel upward toward the safety of the surface. However, the lost horizon can fault our cognizance – and we may turn downward in serious error. There have been any number of divers that have been lost because of inability to mentally assess what was up – as opposed to what was down. It is called ‘The Rapture of the Deep’.

How Does This Relate to Maps? Maps both reassure us – and misguide us. Around mid-sixteenth century, a German mapmaker, Gerhard Kremer devised a flat representation of our world. It illustrated the world that was gridded with flat horizontal lines –and flat vertical lines. Both were a homage to Euclidian plane geometry. His name was Latinized to Gerardus Mercator, and the Mercator Projection has remained the de facto flat map standard for over 500 years.

Fig 2 Mercator Projection But the world is not flat. It is a sphere. And up to our current times, no one has able to take a sphere – and accurately represent both size and distance whilst in a flat projection.

Fig 3 Gore Map Projection One might argue that longitude lines on our spherical world representation are parallel. But that is not so. Only at the precise crossing of a longitudinal line at the equator – might qualify. But as these longitudinal lines proceed either North or South toward the Poles – they curve inward to meet at a point terminus. Euclidian perception has been preferentially supplanted by a curved Riemannian system.

Fig 4 Riemannian Geometry With regard to latitude lines on a spherical surface, there is no parallelism at all. Each successive latitude line above or below the equator represents a smaller circle with a steeper curvature – that immediately departs from the rate of curvature at the equator. What the Mercator projection does is distort land mass sizes. The further away from the equator, Greenland is shown almost equivalent to Africa (which is reality is forty percent larger). And the Antartic continent is grossly exaggerated – to be almost the equivalent of the other major world land masses – combined. Why do we accept this falsity? Because we are more comfortable in a two-dimensional world. Around 1880, a British schoolmaster, Edward Abbot, wrote a novella entitled Flatland. It was a satire on the stratification of Victorian society – divided into three parts.

a. Lineland: Primitive as single line – viewed in plan or end view. b. Flatland: A two-dimensional existence. It was bounded by either horizontal or vertical lines. And if in multiple lines, this reaffirmed Euclidian geometry. c. Spaceland: A special domain, wherein its’ inhabitants had transcended Flatland – and now existed as polygons. Further, as these polygons reproduced, there resulted new polygons with many more sides, And ultimately the polygons became spheres.

We can easily construe Flatland as a metaphor for our daily mundane walk-about existence. Although intellectually we may understand that we live on a sphere, with our feet planted on a curved surface below us we still carry that two-dimensional perspective. So, it turns out that Euclidian plane geometry both comforts us and misleads us – at the very same time.

Spaceland Exceptioners There are some notable individuals that have ascended above the domain of Flatland. Einstein and our particle physicists that speculate on the existence of realms of multiple unseen dimensions. They are a rare bunch. Our space programs have also mandated the teaching of weightlessness to astronauts – thus ignoring gravity and mitigating against a rigid necessity of Horizon.

Denouement Without that visual perceptive bonding to the Euclidian ethos – we Are discomfited by some lost measure of stability. Art always seems to anticipate a future reality. On the advent of cataclysmic World War II (1938) – we were so challenged by a movie – ‘Lost Horizon’.

Euclidean spatial imperative