All the world's a stage, Shakespeare tells us. Which is wrong of course, but that's hardly his fault. Shakespeare died the year John Wallis was born, English mathematician extraordinaire working in the area we would later call calculus forty years before Newton. Or at least poking around the suburbs, until Newton's bright star and what can be politely described as an unhealthy obsession with preeminence identified the true stage the world is built upon, eclipsing all who came before, during, or after.
For as any engineer can tell you, all the world is really constrained optimization. Ya makes the best ya can with the whats yous given. A universal truth operating in every institution known. Suspension bridges to airplane wings, microchip design to antibiotics, universities to investment banks. If you want to succeed, you take the cards you're dealt and you do it lightest, you do it fastest, you do it cheapest.
Fine words, and if words were all that is needed we already had Shakespeare. But we don't need words; we have calculus.
Optimization problems in calculus start off simple enough. We have a function (think: wiggly line) and we would like to identify the location where it is maximal (or minimal). Attach a real-world consequence like "profit" to the function and what mathematicians call maximization engineers call optimization. This curve describes how much profit we get for making so many widgets. We want to find the number of widgets that maximizes our profit.
We can also look for an optimal solution subject to a side condition that must be satisfied. Compare: the lightest airplane wing v. the lightest airplane wing that won't break. The smallest length of fence that encloses the biggest area. Or the biggest sphere that fits inside a given cone. (Calculus textbook authors are simply mad with putting spheres inside cones for some reason. It's almost Freudian.) The calculus tool used to solve this kind of optimization problem is called Lagrange multipliers. I have gushed about Lagrange multipliers elsewhere.
Finally, rather than searching for a value that optimizes a given function, we can be asked to find the optimal function that solves a given problem. A lifeguard needing to get to a drowning swimmer chooses the quickest path over sand and sea. The optimal function describes the path taken. Or we might ask after the flight path that gets us to cruising altitude fastest, or using the least amount of fuel. In Greek mythology, Queen Dido was tasked to find the shape of the greatest land area she could cover using a single ox hide (that shape turned out to be Carthage. Mythical bull, indeed).
Some of the most interesting optimal function problems involve no human intervention whatsoever. My favorite: if you make a loop of wire and dip it in bubble juice (dilute glycerin, say, or soapy water) the bubble that forms inside the loop is a minimal surface. That is, out of all the possible surfaces that could form, the one nature "picks" has the smallest surface area. If things are simple, you can probably guess the answer. If the wire loop is just a circle, you don't need me to tell you what you're going to see when you pull the loop out of the bubble juice: a flat soap film filling the interior.
But change things just a little and be humbled. Consider a slightly more difficult problem. Take two circular loops. Hold the loops reasonably close together and stick them in the bubble juice. Remove. We ask: What surface forms between the loops?
You might guess the surface is a cylinder, like the side of a tin can. After all, the shortest distance between two points is a straight line, and if we think of a surface as just lots and lots of lines connecting corresponding pairs of points on two loops, it seems reasonable to assume the result would be a cylinder.
It turns out that is wrong, as this handy YouTube demonstration demonstrates:
What forms is a cylinder that curves inward at the midpoint. If we draw a curve between two representative points on the left wire and right wire that lies on the surface, that curve is called a catenary. The minimal surface is called a catenary of revolution. The catenary is longer than a straight line connecting two points, but nature figured out the savings in surface area obtained by making the middle part skinnier more than makes up for the penalty of not using the shortest path between the wires.
This problem is nice because it's complicated enough that you can't guess the answer (well, I couldn't) but not so complicated that you can't derive the answer using math. The applicable mathematics is called variational calculus. It usually comes after three or four semesters of regular calculus, but if you're interested in an introduction, some crazy person wrote a friendly 10,000 word primer. Nonetheless, I will show you what deriving the catenary minimal surface looks like using variational calculus. Below is the solution to the slightly more general problem in which the wire loops have different diameters:
In mathematics, a catenary is formally called a "hyperbolic cosine" -- written "cosh" -- which is what emerges here. If that means nothing to you, that's okay. Look at the scribbling as abstract art. Modern hieroglyphics. Even if the literal meaning escapes you, it still has an austere beauty that can be appreciated by anyone.
Now, compare the above to how nature derives the catenary:
You can make any wire shape you like and nature will find the minimal surface. Even when the shape gets too complicated for pencil-and-paper solutions and we humans must turn to the computer. Even when the shape gets so complicated we can't compute the minimal surface even using a computer. Nature still finds it. That's spooky. How do it know? Can we make a shape so complicated nature can't find a minimal surface for it? The answer is no.* Nature always figures it out.
* Eventually you may make a shape that will support more than one stable soap film. Each might represent the minimal surface in some regions but not in all. In math speak, we say that the surface is a local minimum but not necessarily a global minimum.
The master engineer, Nature is. Plum obsessed with optimization. It is a rare physical law indeed that cannot be expressed in the language of variational calculus. The minimal surface area, the minimal length, the minimal weight, the minimal time, the minimal difference, the minimal action. Einstein called it reading the mind of God. Fine words. And if words were all that is needed we already had Shakespeare.
But we don't need words; we have calculus.
For as any engineer can tell you, all the world is really constrained optimization. Ya makes the best ya can with the whats yous given. A universal truth operating in every institution known. Suspension bridges to airplane wings, microchip design to antibiotics, universities to investment banks. If you want to succeed, you take the cards you're dealt and you do it lightest, you do it fastest, you do it cheapest.
Fine words, and if words were all that is needed we already had Shakespeare. But we don't need words; we have calculus.
Optimization problems in calculus start off simple enough. We have a function (think: wiggly line) and we would like to identify the location where it is maximal (or minimal). Attach a real-world consequence like "profit" to the function and what mathematicians call maximization engineers call optimization. This curve describes how much profit we get for making so many widgets. We want to find the number of widgets that maximizes our profit.
We can also look for an optimal solution subject to a side condition that must be satisfied. Compare: the lightest airplane wing v. the lightest airplane wing that won't break. The smallest length of fence that encloses the biggest area. Or the biggest sphere that fits inside a given cone. (Calculus textbook authors are simply mad with putting spheres inside cones for some reason. It's almost Freudian.) The calculus tool used to solve this kind of optimization problem is called Lagrange multipliers. I have gushed about Lagrange multipliers elsewhere.
Finally, rather than searching for a value that optimizes a given function, we can be asked to find the optimal function that solves a given problem. A lifeguard needing to get to a drowning swimmer chooses the quickest path over sand and sea. The optimal function describes the path taken. Or we might ask after the flight path that gets us to cruising altitude fastest, or using the least amount of fuel. In Greek mythology, Queen Dido was tasked to find the shape of the greatest land area she could cover using a single ox hide (that shape turned out to be Carthage. Mythical bull, indeed).
Some of the most interesting optimal function problems involve no human intervention whatsoever. My favorite: if you make a loop of wire and dip it in bubble juice (dilute glycerin, say, or soapy water) the bubble that forms inside the loop is a minimal surface. That is, out of all the possible surfaces that could form, the one nature "picks" has the smallest surface area. If things are simple, you can probably guess the answer. If the wire loop is just a circle, you don't need me to tell you what you're going to see when you pull the loop out of the bubble juice: a flat soap film filling the interior.
But change things just a little and be humbled. Consider a slightly more difficult problem. Take two circular loops. Hold the loops reasonably close together and stick them in the bubble juice. Remove. We ask: What surface forms between the loops?
You might guess the surface is a cylinder, like the side of a tin can. After all, the shortest distance between two points is a straight line, and if we think of a surface as just lots and lots of lines connecting corresponding pairs of points on two loops, it seems reasonable to assume the result would be a cylinder.
It turns out that is wrong, as this handy YouTube demonstration demonstrates:
What forms is a cylinder that curves inward at the midpoint. If we draw a curve between two representative points on the left wire and right wire that lies on the surface, that curve is called a catenary. The minimal surface is called a catenary of revolution. The catenary is longer than a straight line connecting two points, but nature figured out the savings in surface area obtained by making the middle part skinnier more than makes up for the penalty of not using the shortest path between the wires.
This problem is nice because it's complicated enough that you can't guess the answer (well, I couldn't) but not so complicated that you can't derive the answer using math. The applicable mathematics is called variational calculus. It usually comes after three or four semesters of regular calculus, but if you're interested in an introduction, some crazy person wrote a friendly 10,000 word primer. Nonetheless, I will show you what deriving the catenary minimal surface looks like using variational calculus. Below is the solution to the slightly more general problem in which the wire loops have different diameters:
In mathematics, a catenary is formally called a "hyperbolic cosine" -- written "cosh" -- which is what emerges here. If that means nothing to you, that's okay. Look at the scribbling as abstract art. Modern hieroglyphics. Even if the literal meaning escapes you, it still has an austere beauty that can be appreciated by anyone.
Now, compare the above to how nature derives the catenary:
You can make any wire shape you like and nature will find the minimal surface. Even when the shape gets too complicated for pencil-and-paper solutions and we humans must turn to the computer. Even when the shape gets so complicated we can't compute the minimal surface even using a computer. Nature still finds it. That's spooky. How do it know? Can we make a shape so complicated nature can't find a minimal surface for it? The answer is no.* Nature always figures it out.
* Eventually you may make a shape that will support more than one stable soap film. Each might represent the minimal surface in some regions but not in all. In math speak, we say that the surface is a local minimum but not necessarily a global minimum.
The master engineer, Nature is. Plum obsessed with optimization. It is a rare physical law indeed that cannot be expressed in the language of variational calculus. The minimal surface area, the minimal length, the minimal weight, the minimal time, the minimal difference, the minimal action. Einstein called it reading the mind of God. Fine words. And if words were all that is needed we already had Shakespeare.
But we don't need words; we have calculus.
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