tldr;
In the old days we lived in the trees. Then came granaries, the division of labor, dividers, flexible pieces of wood called splines, and the French Curve for a little while there. Now we have computers that generate algebraic curves. The circle still reigns supreme. But spline-design offers some interesting possibilities.
There’s more to curves than meets the eye. Everything in the universe is curved, except for the few things that aren’t, and a curve is really just that, except for the few curves that aren’t, so there are a million ways a thing could be curved.
From a design perspective, there are three basic kinds of curve:
arcs: segments of a circle defined by a radius and degrees
A circle is also a conic section. From a design perspective, the distinction is about curve-construction. In particular, the historic use of dividers (compasses).algebraic curves: polynomially-defined curves, eg x^2 + 2x + 1, conic sections
Algebraic curves were not practicable until computers.transcendental curves: all the weird ones, logarithmic spirals, sinusoidal curves, the curve of the distribution of the prime numbers.
Transcendental curves can be easily but not arbitrarily or parametrically constructed. A flexed ruler is a transcendental curve.
Transcendental curves, so common in nature, are generally approximated by algebraic curves (via computer-aided Taylor Series) or arcs (by hand and eye).
For a long time, design and layout with dividers was the primary architectural mode.
Before industrialization hung a measuring tape from everyone’s pocket, dividers were commonly used to design and lay out architecture and furniture, not just the curves but the ratios and proportions. Acting like a compass, all kinds of constructions were possible. The golden ratio cannot be exactly located on the number line but the proportion can be exactly constructed with a compass. No straight edge is even necessary.
Designing by divider naturally lends itself to thinking in whole number ratios, which lend themselves to the famously well-proportioned “Classical” design. Greek columns mimicking the curvature of a tree, ogees,
or, say, a flying buttress
In “Systematic calculation of flying buttress parameters by means of geometric regression”, some researchers attempt to reconstruct the parameters that Gothic architects would have used together with dividers to design the curves of their arches, which are not just arcs. In the paper, the researchers worked from best-fit algebraic curves to approximate arcs, but it seems more likely to me that the flying buttresses were transcendental curves approximated by arcs. The “platonically-ideal” arch is not a conic section, but a catenary.
A catenary curve looks like a parabola, but it is not. The parabola is algebraic, the catenary is transcendental. A catenary curve describes the curve of, e.g., a chain hanging between two points. Or the St. Louis arch.
A parabola describes the arc of a ball thrown through the air (but not subject to friction). The Star Trek comm badge and the Nike Swoosh both look parabolic, evoking lightness, motion, shooting stars. If you hit a golf ball just right it will hang there for just a moment in eternity. But these are not things, they are logos.
To return to the flying buttresses, the distinction is maybe immaterial to the researchers, but I must imagine that there is a phenomenal and a noumenal parameterization.
To illustrate: the design software, Sketchup, does not have a circle primitive. Once you create a circle, that thing is just a series of line segments. Now, a more precise CAD will preserve the arc’s parameters (e.g., radius, degrees, center) and generate a view of that circle on the screen, or send it off to the CNC. Sketchup, however, will make your CNC cut curves as line segments because it does not have an internal concept of a circle.
So, the Gothic architects needed to know how to generate a usable construction document but they also needed to have some idea of these Transcendental curves’ “true” parameters.
The Gothic architects probably understood that their arcs were only approximating the ideal curvature, and perhaps the wood frame constructed to construct the arch created a re-hydrated catenary curve by the curvature of the bent wood approximating the parameters of the architect.
Industrial Revolution
Early into the industrial revolution, mathematically-designed and mass-produced French Curves were the way to approximate irregular curves.
All kinds of transcendental curves were hidden in these little guys, e.g., parts of the Euler spiral.
Today, commercially-ubiquitous CAD programs have a few methods for defining curves like arcs and bezier curves. Bezier curves are the standard for “organic” digital curve design. Usually you have two “handles” that you drag around to create two vectors that define a cubic (x^3) polynomial, eg, an algebraic curve. These curves may then be attached end-to-end, to create more complex curves. These “splined” cubic Bezier curves are the most expressive that a computer can easily save and manipulate the parameters for, though there are still some black-boxed outputs (in this case, no easy way to calculate the length of a bezier curve).
Also, CAD tools have some unique capabilities like generating curve offsets.
But at the end of the day most modern architectural designers probably still stick with arcs, which are way more easily parametrized for construction.
Part II
I started thinking about all this when I shaved off a very long piece of a maple from a board. Along 13 feet it went from 1/4” thick to 0. When I bend that piece of wood, what kind of curve is that making?
I still don’t know the answer to that question, but I did go down a short path of investigating Euler’s Elastica Theory, which is basically the math of the curvature of long flexible things, like a ruler, being subjected to compressive forces, like putting it length-wise in a vice and tightening the vice.
One banal but useful implication of these insights might, for example, aid in caul construction:
Cauls are used when gluing smaller boards into one big board; when you clamp them as pairs on top and bottom, it helps align the little boards to each other vertically, before you run the whole thing through the wide-belt sander. Plenty of people just use straight pieces of wood as cauls, but then the pressure on the joints is low in the center. Guides like the one above usually say to give the caul a little curve.
I’m hypothesizing that the ideal curvature for such a caul could be parameterized and constructed by basically cutting a strip of the wood you use for the caul and bending it between the jaws of a clamp, tuning the elevation of the curve, then tracing that curve onto your caul, and cutting it out with a bandsaw. Or maybe that’s just over-engineered bullshit.