A Preface – As is often the case for myself, I can offer no conclusion… merely commentary with an invitation for scrutiny.
A simple description of the Circle is all points on a plane equal in displacement to a discrete point. Extension then allows definition of radius, diameter, and circumference. This as basis, renders the Circle easy enough to construct. But what of the Circle’s origin. Unintentional as it may be, we tend to forsake the planar nature of the Circle. Ours is a three-dimensional world, and two-dimensional geometric shapes arise as planar sections of three-dimensional objects. Or, do they? As discussed in a previous post, a similar question is posited – is the Circle the parent or child of the conical surface?
Note the use of “the” in conjunction with Circle. As noted in prior posts, the Circle enjoys unique status (included, the Parabola) of existing as a singular entity manipulated only by dilation or scale. Therefore, “the” Circle is differentiated from “a” Circle implying shape variation (e.g., “a” Hyperbola), granted the exception of quantity (i.e., one of several).
First, some framework to aid discussion. A circular conical surface is an open1, curvature in one direction, three-dimensional surface of revolution with azimuthal symmetry. It is formed by a linear generatrix being continuously revolved 360° about a non-parallel axis of coincident plane where the intersection of axis and generatrix defines the three-dimensional conical vertex. This resulting surface constitutes an infinity of generatrices tangent to a Dandelin Sphere (or focal sphere). Here, it is intuitive to view the Dandelin Sphere’s interface with the conical surface analogous to a ping-pong ball dropped into a funnel. Sectioning the conical surface can yield two-dimensional conics of the variety Hyperbola, Parabola, Ellipse, and Circle dependent upon section plane angle. If sectioned by a plane perpendicular to the axis, the resulting conic is the Circle of eccentricity zero.
If the linear generatrix is revolved in a non-continuous manner, with irregular distribution to three or more non-coincident locations, and sectioned as described prior, the result is a polygon. An equally distributed revolution of the generatrix within 360° to three or more discrete positions with sectioning of the same develops a regular polygon. Let’s imagine a Square as our regular 2D polygon as sectioned from a 3D polyhedron or pyramid. Now, instead a conical surface, the pyramid of four sides or lateral faces acts as our funnel. Placed inside, our ping-pong ball rests tangent to these faces, not the pyramid’s corners or vertices which are now synonymous with generatrices. Mentioned prior, and key in this instance, the Dandelin Sphere’s tangency to generatrices. With this in mind, one needs imagine the pyramid being inserted through the ping-pong ball until its vertices rest tangent to the spherical surface. Thus, not an inscribed sphere, instead circumscribed. Quite an important distinction in visualizing how the Circle evolves.
Thus, the faceted sides of the 2D polygon or the lateral faces of the 3D polyhedron represent short-cuts or chords between the vertices or generatrices which lie on the Circle or conical surface. The Circle then develops by the iterative refinement of decreasing chordal deviation from increasing a polygon’s vertices. Expanding the validity of our original definition, the Circle can also be viewed as a feature of ultimate distillation, defined as a conic of infinite polygon vertices whose locus is equidistant from the Dandelin Sphere’s tangency with a sectioning plane perpendicular to the conical axis. While not a definitive declaration of origin, a supplemental perspective nonetheless.
Depicted below are sections of conical surface by eccentricity and a variety of polygons to illustrate the progression of refinement as the number of generatrices/vertices approach infinity.
1) By comparison, a “cone” is a conical surface with closed end forming a primitive solid of finite volume.
