Sunday, 1 April 2012

Euclidean vs non-Euclidean geometries

I find it rather dismaying that someone like the great Roger Penrose would find the first four postulates of Euclidean geometry a "little strange", especially the fourth: that all right angles are equal. (Roger Penrose, London, UK 2004, p. 29)

I think if Penrose really thought about the origins of Euclidean geometry, he'd realize first that it stems from surveying of land and the laying on of foundations of buildings. In this respect, Euclidean geometry still applies while 'embedded' in three dimensional space, and, as a result, levelling (squaring) of buildings is possible. The natural contours and geodesics of the landscape are "cut" into planes more amenable to "Euclidean" treatment. The levelling of the foundations for the building is first determined, then, the floor-plan is built-up from the boundaries of outerwalls whose angles are determined by the fourth postulate, mind.

In this localized context, the Euclidean plane is perpendicular to the vertical direction of gravity; the rolling of the wheel is still allowed to describe the sine and cosine waves - 90 degrees from each other; two points really generate an indeterminately-long straight line (points and lines being, after all, just idealized entities). Though the number, shape and height of the ground floors may vary from building to building the commonality of all buildings is that abstract plane that allows flush and weight to be balanced, where sea-level would be were it there. A cathedral, no less than an outhouse, is an Euclidean affair. The arches, domes, vaults, buttresses and walls conspire together to illustrate the great as well as the mundane considerations of humanity.

The Pythagorean theorem is given space (and time) to assert itself upon all right-angled triangles therein its realm as in the real world. It is naturally assumed that the Euclidean space is static: true, but that is if one disregards the wheel on top of the plane happily describing sine waves...

Jay

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