A Rational Cosmology: Euclidean Planes and Three-Dimensional Constructs

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.

Moreover, elementary and ubiquitously accessible empirical observation yields the conclusion that, although entities can never be purely two-dimensional, there is nothing barring the surfaces of entities from being such.

The entity, "cube," for example, is three-dimensional, and, presuming that man possesses a technology precise enough to refine the faces of a real cube so that no ridges, creases, or miscellaneous imperfections may remain on them, the resulting perfectly smooth surface would be two-dimensional.

No matter which point one picks on the side of the cube, it would have the same numerical coordinate in one certain dimension that does not vary on the two-dimensional surface. Rather, such a dimension would constitute the cube's depth, and the measurement of this dimension would be necessary in order to describe those regions of the cube which are beneath its surface.

Whether or not ideal two-dimensional surfaces have yet been observed in nature or obtained via man's technological precision is not the province of cosmology to judge. Cosmology only informs man that such surfaces are conceivable as existing in reality, as parts of real entities. Of course, not all surfaces are two-dimensional. Surfaces may be three-dimensional, as the surface of a sphere, cone, or any other entity with non-planar contours will demonstrate.

Euclidean Three-Dimensional Constructs

Moreover, whenever Euclidean geometry ventures to describe three-dimensional relationships and shapes, it begins to address the entire interplay of linear measurements necessary in comprising an entity.

Spheres, cubes, cylinders, and prisms, for example, are all conceivable as actual entities. Of course, in order to be such, they would also need to be composed of the quality, "matter," which Euclidean geometry does not directly address.

Thus, three-dimensional geometry can express, with perfect accuracy, the entirety of the linear measurements applicable to an entity, and study these qualities in isolation from the remainder of the entity's qualities, such as matter. Though immensely realistic, three-dimensional geometry, like all mathematics, remains a model, not an actual existent.

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