I. Introduction
In the following we propose an open/lidless box volume problem, please note that we are using that problem to crystallize several points about optimization in the created order—that is our focus—the actual method of solution is quite secondary to the points that we hope to address (this study is not a problem-solving exercise). If the reader cannot solve the problem, no worries, that is okay, it is not readily solvable by most, it does require a certain training that the reader may not have been exposed to; the whole idea is to follow the points of the discussion concerning optimization in the created order—the open box volume problem is our vehicle for discussing that optimization, no more.
The open box volume problem proposed below is an example of optimization. It is a very simple problem compared to the discernible optimization that happens on a grand scale in our universe, both biologically in organisms and nonbiologically in materials and the physical systems they comprise. If the universe were not optimized at all levels of its functionality, it could not exist. Our universe, the created order, optimizes. Some comparisons and examples: At the heart of artificial intelligence lies optimization. Our universe’s conservation laws are decidedly a function of optimization. Christ-centered ministries, businesses, economies, militaries, and societies alike optimize in one form or another. Computational science and engineering “work” because nature is predictable, and nature is predictable because it optimizes—the natural laws have this optimization coded into them. In another Creation Corner article, in the parabola section, we discussed the catenary shape, for example a chain hanging between two posts, this shape minimizes tensile and compressive forces, themselves reacting to other forces, typically gravitational—it is an optimization. The catenary shape is assumed quite naturally throughout nature. Jehovah’s Golden Ratio, again found throughout nature, is an optimization of form and shape, and on and on it goes. Human beings, indeed, organisms per se, have an optimization bent coded into their being, without it one could not survive; without it, humankind would not only have vanished long ago, but it would never have gotten too far out of Eden to begin with.
The crux of optimization is the maximizing or minimizing of some function or parameter thereof. Sometimes “optimal” is a maximum, sometimes it is a minimum; optimization is very much problem specific. By God’s grace and good pleasure there is a special branch of mathematics that arose centuries ago that addresses such problems.
II. Why Does the Created Order Optimize?
Why in fact does the created order optimize? It is not just for the sake of biological systems, which get tangled up in all manner of fantastic evolutionary survival arguments, it is that way nonbiologically as well, from the micro level at atomic scales all the way out to the macro. Of course, the nonbiological has no need whatsoever to “survive”—please note that carefully—for example, a planet orbits optimally because of physical laws, not because it has an innate survival motive or mechanism. And therefore, optimization must surely be primitive, and hierarchically superior to survival. Said differently, optimization drives (biological) survival. Optimization is how the Creator secured survival for living organisms, not least humankind, who is of His utmost concern. Some living organisms are better optimizers than others, that is clear. And the key point is that optimization is a code, an algorithm, as for example the optimization computation behind our open box maximum volume problem. The created order optimizes because optimization was coded into it by a Grand Optimizer, even the exceedingly elegant Creator, the Word, from whom all codes are derived, even Jesus Christ, praised be His elegant Name.
III. The Open Box Volume Problem
Suppose the reader had a twelve-inch square piece of cardboard and wanted to construct an open box of maximum volume by cutting its corners and turning up its sides along the dashed lines as shown in the figure. What should cutout dimension x be to maximize the volume of the lidless/open box thus formed?
A. Solution Methodology and Mind
What should dimension x be in our box-optimization problem, can the reader get at the answer by chance, by way of random guesses? Sure, maybe you could get it right by blind guessing, or some sort of informed guess. But is that any way to build a universe with staying power? Who in their right mind would go about it in such a low-browed way? (By the way, the universe did not build itself, let us please dismiss that empirical-law defying “get something from nothing” nonsense.) So how would the reader solve this problem? For sure you would use your brain and think your way through it. Even if you made a loosely informed guess, your brain is parsing best-guess possibilities, you are thinking; your brain, animated by your mind, is in some manner computing here. (As touching the denial of “mind” by some deluded, is not a brain without a mind largely a chunk of useless gray matter sad to say? Though the individual may still be alive and quite well spiritually, without a mind their reasoning capacity is certainly not alive and well.) To solve this problem, anything other than blind guesswork would force the reader to think computationally in some manner or another, because optimization is the manifestation of a computation, be it at the scale of the individual or the universe, and Computation, bespeaks Mind.
IV. The Solution
The volume of any box is its length times its width times its height. In this problem, the length is the same as the width, and each is (12-2x) inches after corner-cutting and turning up the sides, and the height after corner-cutting and turning up the sides is x inches (figure 1). Therefore, length times width times height after turning up the sides gives us this volume result:
V=144x-48x^2+4x^3.
We are now out of the starting gate and ready to roll, and at this point, utilizing the above equation, we have a couple of options to find the dimension x that gives us the maximum volume, an intellectually rich man’s approach that is fast, accurate, and timesaving, and an intellectually poor man’s approach, that is slow, yet accurate, but time-consuming (and painful), either way, we can get at the answer. The intellectually rich man’s approach is called “WOW,” and utilizes the God-endowed special branch of mathematics mentioned above, the intellectually poor man’s approach is called “PLUG AND CHUG,” he sees many possibilities for dimension x. O my, the possibilities are endless he gasps; he experiences heart palpitations realizing that only one can be optimum as concerns maximizing the volume; he nervously and quite painfully proceeds to plug in one x value after another into the equation and records the volume result, and then searches for the maximum volume and attendant x out of a whole bunch of tests, and yea gratefully stops the process just short of a serious case of writer’s cramp and eye fatigue/ failure (Figure 2 uses a gracious 0.25 step). Nevertheless, each man going about their business finds that precisely x=2 inches is the volume-maximizing cutout dimension. And with that x cutout dimension the maximum volume is seen to be 128 cubic inches (Figure 3—the long-coming poor man’s answer agrees with the thumb-twiddling rich man’s answer; notice the not quite parabolic shape of the curve, the somewhat left-right symmetry across the optimum dimension x=2; notice that the volume =0 when x =0 and when x=6, the latter being the midpoint of a given side figure 1). It is clear enough, even to the intellectually poor man, that x must be greater than 0 (he realizes that immediately from the equation he is stuffing), but it is also necessary that x be less than the midpoint 6 in this problem (x exactly equal to 6 means not having any cardboard left to fold, and x greater than 6 is nonsensical; x must lie somewhere between 0 and 6; it is not immediately obvious that plugging x=6 into the equation=0)—both cases—x not equal 0 and x less than 6, are critical constraints in the rich man’s approach (Jehovah is the intellectually rich man in this solution drama put forth, amen and amen); we point this constraint matter out to highlight the fact that the Creator must have imposed myriad constraints in order to optimize His creation just right, i.e., it didn’t just mindlessly fall together optimally on its own “out of the blue,” there is a staggering amount of thought behind it all. In that regard, please consider that optimization is more than equation construction, which in itself bespeaks Mind, optimization goes one step further, it requires deduced constraints that seamlessly shake hands with the equations (=Mind^2). That being so, we have moved from the domain of raw equations to the domain of an algorithm—a systematic, rule-based block of computations that serves a specific function, namely optimization in this context. And so it is in the (intellectually) Rich Man’s approach. Did He not a priori consider, and compensate for, the following and infinitely more, for a post-Fall, degenerate universe:
How much water pressure can a cell membrane withstand without bursting; how fast can the earth spin on its axis without flying apart; how much of a load can the collar of a tree limb bear and still hold the limb intact; what is the maximum spacing between teeth for chewing efficacy; what is the minimum allowable water content on the earth to maintain a robust hydrological cycle; what is the optimum reproduction rate for humans, the animals, plants, etc., to comfortably live together on the earth—what should the minimum surface area of the earth be to sustain its planned number of inhabitants; what is the optimum predator-survivor ratio in the sad aftermath of the Fall; what is the optimum neutron-proton ratio for the elements and what is the optimum radioactive decay rate for the unstable elements; for whomever you and I were created to be, what is our optimum height, weight, strength, speed, reasoning capacity. (We were created to serve God and for His glory, let us not overlook that exceeding blessing.) What is the optimum thus and so…? The list of questions is endless, yet each question has been perfectly addressed by our blessed Creator, the Grand Optimizer.
A. The Big Picture of Creation
The big picture of creation is ever so much more than mundane mass-energy. What we see in the Creation is mundane mass-energy dancing according to the good pleasure of the Grand Optimizer, like a puppet on a string dancing to the good pleasure of the marrionettist pulling the strings just right to achieve a pleasing, satisfying, optimal performance.
Finally, and quite aside, we hold that the fundamental computation in nature is feedback, negative and positive feedback—a highly optimized, accurate, exhaustive, efficaciously simple computation. Thus it pleased the Creator. He utilized this elegant mechanism near everywhere again and again as far as we can tell.
Praised be your exceedingly elegant Name intellectually rich Creator God Jesus, amen.