A Rational Cosmology: Euclidean Planes and Three-Dimensional Constructs

Essay XVII

This is Essay XVII of Mr. Stolyarov's series, "A Rational Cosmology," which seeks to present objective, absolute, rationally grounded views of terms such as universe, matter, volume, space, time, motion, sound, light, forces, fields, and even the higher-order concepts of life, consciousness, and volition. See the index of all the essays in "A Rational Cosmology" here.

The Euclidean plane, a two-dimensional construct, enables the study of an even vaster and more complex interplay of qualities than does the Euclidean line. Euclidean three-dimensional constructs are capable of describing all of an entity's spatial qualities, though they still omit the quality of matter from the mental model of the entity.

The Plane

The plane is, in effect, a mental model isolating for study all the possible variations that can exist in the combination of any of two of the three linear dimensions. Two-dimensional shapes, curvatures, and patterns may be the results of such variations, which can be found as emergent qualities -- qualities whose existence is based on a certain interplay of more basic qualities -- in entities.

Circles, for example, are a quality possessed by the entity, "cylinder," which, being three-dimensional, can exist in reality. Each of the properties of shape and curve constructs on a Euclidean plane will hold if these shapes and curves are qualities of a given entity; the sum of the angles on the surfaces of a triangular prism will always measure 180 degrees, given that this prism possesses the quality, "triangles."

A three-dimensional projectile will still follow a parabolic path in two of three dimensions (and will not alter its parameters in the third). A cylinder's rim will measure two-pi times the radius of its surface. Thus, we see how the findings of a Euclidean investigation of the isolated interplay of two dimensions can be applied, with perfect accuracy, to actual, three-dimensional entities.