Visiting my favorite catenary

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I’m on the road this week, visiting my favorite catenary, which is a unique form of natural curve, this one found on a very large man-made structure. Where am I?

The image source gives this away as the Mackinac Bridge (pronounced Mackinaw), which spans the 3.5-mile-wide Straits of Mackinac between Michigan’s upper and lower peninsulas near the northern terminus of Interstate 75. In my earliest childhood years before the bridge was built in 1957, my family would take a ferry between the two peninsulas on the way to visit my grandmother in the Keweenaw, or to visit Mackinac Island, a couple of miles east of the bridge. I particularly remember a summer 1957 trip, the year my younger sister was born, and our last ferry ride before the bridge was completed that November. That yet-uncompleted structure loomed large over the straits.

At the time of its completion the Mackinac Bridge was one of the longest suspension spans in the world, and I have traversed it dozens of times since, traveling back and forth to university, including during winter storms when the roadway disappeared ahead in the blizzard, an unnerving drive to say the least.

What is a catenary?

The cables holding up the roadway on a suspension bridge like Mackinac in such a seemingly fragile manner form a natural curve called a catenary. You will get this same curve if you fix a chain between two points and let it hang. Gravity itself “draws the curve.” [1]

catenary

Source: Wikipedia

Since this curve is slightly different from a classic parabola, mathematicians struggled for a long time to figure out how to derive it with a simple formula. One of my favorite mathematicians, Gottfried Leibniz (1646–1716) didn’t quite get to the simplest formula for this curve, but he did recognize that this was a very physical expression of mathematical logarithms, which were very difficult to calculate at that point in history. He figured out how to use the curve of the chain itself to physically calculate logarithms, and he marveled at the “wonderful and elegant harmony of the curve.”

Once Euler’s number, usually designated as just e (with a never-resolving value of 2.71828…), was figured out as the “natural logarithm” base, this “more elegant” version of the formula for a basic catenary emerged:

y = (ex + e-x)/2

When you graph this function on an x-y coordinate system it looks like this:

Catenary graphed

One simplistic way to look at this equation is that the hang of the chain, the y value, is the average (the division by 2) of the effect of each of the two end “anchor” posts “pushing up” against gravity, with the x value being the horizontal distance from the center of the chain. In the middle, where x is zero, gravity has the “maximum gravitational pull” allowed by the two anchors and the length of the chain.

The great physicist Erwin Schrödinger (1887–1961), he of “Schrödinger’s cat” fame who defined many key understandings of quantum physics, suggested that the value e itself likely comes from the behavior of the smallest things in the universe, the forces making up atoms. It defines a kind of “natural curvature factor” for our universe that shows up in everything from a seashell on the beach to your bank account interest calculation.

As I noted earlier, “gravity itself” is drawing this curve, pulling down on the chain but constrained by the posts into an equilibrium of forces. Gottfried Leibniz’s great rival, Isaac Newton (1642–1727) used their new invention of calculus to suggest that gravity was an “invisible force” of some kind that brought apples to the ground and made the planets orbit the sun in a predictable ellipse. Albert Einstein (1879–1955) then re-defined gravity as a “curvature in space-time,” which remains for most of us nearly impossible to conceptualize.

But maybe not. I’d like to suggest that the reason we see a suspension bridge like Mackinac as so elegantly graceful is because we are seeing the “natural curvature” of gravity itself. Something deep inside us recognizes how amazing that is, and we are in awe of nature’s beauty, in this case re-constructed by human mathematicians, engineers, and builders willing to walk those hanging wires.

Here is a great simulation of gravity as the “curvature of space-time,” with stretchable fabric representing this “space-time” thing. Apparently, God’s universe is really a giant trampoline. Also, a confession: I have “bent” space-time myself by writing this last week from 1500 miles south of Mackinac just off that same Interstate 75.


Notes:

  1. While a free-hanging cable naturally falls into the shape of a catenary, a flat heavy roadway may distort the curve closer to the shape of a classic parabola. The Mackinac Bridge roadway, however, is slightly arched, somewhat counteracting that force.

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