Friedrich Schiller: “Art is the daughter of freedom.”
I am not alone in believing his ‘Ode to Joy’ was actually code for ‘Ode to Freedom’ which would be censored in his time. In German, this would scan: ‘Freude’ or ‘Freiheit’. He writes:
Freude (Freiheit) … Joy (Freedom) …
Tochter aus Elysium. Daughter of Elysium.
In this interpretation, Art is the granddaughter of Elysium, that is, Heaven. This seems to equate joy and freedom, rather it sets freedom as prerequisite for both art and joy. Both freedom and art derive from and reflect back to their ancestor, the sacred. Modernity in art, architecture, poetry, etc. conflates freedom with license. Art must respect, create, and celebrate harmonies. License violates them.
So, too, does science find and respect harmony. And so, too, does science need freedom to flourish or even to begin. It must throw off those shackles clapped on by capricious gods. In a world subject to the whims of clay-footed gods created in the image and likeness of us, there is no predictability. No harmony. No laws. No science. Science needs a rational God. Who respects the rational and will not deceive us creatures. A self-limited God who winds up the world and then lets it play out per His built-in harmonies and laws. Free from meddling. Following Schiller then:
Science is the son of freedom.
So, art and science are siblings. Both derive from and reflect up to their grandparent, the sacred. But only a particular sort of sacred. Besides a limited God, we must also be free from one who hides all of reality from us. And free from random antics of multitudinous flawed divinities.
Proportions, ratios, laws obtain in nature too. Physics, in particular, aims to discover those as used by nature’s Artist. For example, the fine structure constant (α) is a dimensionless constant, a constant source of wonder. Being dimensionless implies that both we humans and an intelligent race from another galaxy would arrive at the same value, about 1/137. It is embedded into the very fabric of the universe. Were it to differ by even a few percent from this value, our universe and, therefore, we would not exist. It’s every bit as beautiful to my mind as is a Rembrandt to my eye. Theoretical physics supplies about two dozen other such dimensionless ratios about which we would agree with our extra-galactic counterparts. The same agreement would also hold for these mathematical examples:
● 𝜋 – the ratio of any circle’s circumference to its diameter.
● φ – (golden) ratio of rectangle sides leaving a similar rectangle when cutting out a square
● e – Euler’s number, basis of exponential function which is uniquely its own derivative
Catenary curve derives from Latin catena meaning chain. A hanging chain supported on both ends will approximate a catenary curve. This can be understood by considering that each link in the chain will be pulled by gravity as far down as possible while constrained by its neighboring links. In short, the chain must take the shape which minimizes the entire chain’s gravitational potential (height). It can be shown mathematically that, in order for a suspended chain to meet that condition throughout its length, it must approximate the shape of a catenary curve. Well-known examples are the cables of suspension bridges.
Those cables are just chains with lots more links, each much smaller. In the limit of infinitely many infinitesimal links, it will produce the ideal curve. Then every part of the cable will be pulled only in the direction of its length, – in tension only. At every point, its tension is tangent to the curve. This ideal curve is the hyperbolic cosine which is defined in terms of Euler’s constant (e) mentioned above. This constant raised to a variable power models exponential growth for positive values and exponential decay for negative values. Being its own derivative means that as growth increases, so does the rate of growth. The more it grows, the faster it grows. The hyperbolic cosine is the average of exponential growth and exponential decay. What is the average of growth and decay? Life?
Flip the catenary shape upside down to get the catenary arch. Just as the right-side up catenary cable carries only tension stress tangent to the curve at every point, this arch carries only compression stress tangent to the curve at every point. So, the weight of the arch is spread through the arch only along the arch, that is, there are no forces tending to break the arch. So, the shape is optimum.
That cathedral’s catenary dome reflects the heavenly grandparentage of its art and its science. The shape of a catenary dome is formed by rotating the shape of a catenary arch about a vertical axis. As before, this dome is optimum for spreading its weight to supporting walls. Of course, it can only approximate the optimum because the building materials have extent; they are not infinitesimal and there are not infinitely many of them. We might say a catenary dome is striving toward the infinite while “speaking” to our visual sense as a kind of chained muse, frozen in stone.
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Dean Z. Douthat is a retired engineer residing in a senior living facility in Ann Arbor, Michigan