Book summary “When Least Is Best”: summary

Introduction

You constantly face situations where you need to find the absolute best outcome under strict constraints, whether you manage a business, design an engineering system, or solve a theoretical puzzle. Optimization problems surround you in everyday life, where the difference between success and failure lies in hitting the exact minimum or maximum point.

Many of the most profound problems in mathematics, physics, and everyday life are solved not by adding more, but by finding the exact point where things reach their absolute minimum or maximum.

Paul J. Nahin guides you through the rich history of extremum problems, starting with basic algebra and geometry before moving into advanced calculus and modern programming. You will discover how thinkers across the centuries tackled these challenges without modern tools, and how rigorous methods eventually emerged to tame complex systems.

As you read, you will explore questions such as:

• How can you solve maximum and minimum problems without using calculus?
• What physical laws compel light and nature to always choose optimal paths?
• How do you handle multi-variable constraints that threaten to overwhelm your calculations?

1. Optimization Without Calculus

Setting derivatives to zero is not always necessary or even possible when you solve maximization and minimum problems. You can explore many complex optimization puzzles using simple algebra, geometry, and inequalities.

You can solve many complex optimization problems using simple algebra, geometry, and inequalities rather than relying on derivatives.

Algebraic and Geometric Minimization

Ancient mathematicians proved that of all rectangles with a given perimeter, the square has the largest area. You see this by showing that the area expression can be bounded by a constant minus a squared term.

When you minimize the sum of absolute distances from a point $x$ to a set of ordered points $x_1$ through $x_n$, you can pair outer points. Placing any point between them minimizes their combined distance.

For an even number of points, the sum of distances is minimized by placing $x$ anywhere in an interval between the two middle points. For an odd number of points, $x$ must be placed uniquely at the single middle point.

By treating path segments as vectors and applying the triangle inequality, you can solve civil engineering path problems without calculus. You find the optimal bridge placement between towns $A$ and $B$ on opposite sides of a river of constant width $w$ using similar triangles and basic geometry.

Computers can visually and numerically solve complex optimization problems where analytical differentiation is impractical. Plotting objective functions across feasible solution sets allows you to observe how variables scale and where extrema occur without solving difficult transcendental or multivariable equations.

Analyzing a man's optimal travel time across a circular lake or across a right-angle beach involves plotting travel time curves for varying speeds. In these computer runs, parameters include a critical value of $n = 1.57$ for the lake crossing and $v_2 = 5$ in the beach crossing example.

To solve the water filtration station problem along a river edge, you can follow specific geometric steps:

• Assign the coordinate of zero to the point on the river edge directly below town $A$.
• Express the coordinate of the facility as $x$.
• Express the total pipeline length $T$ as the sum of $l_1$ and $l_2$.
• Set the first derivative of $T$ with respect to $x$ to zero.
• Use the law of cosines and trigonometric identities to establish that $\cos(\alpha)$ equals $\cos(\beta)$ at the minimum.

For specific values where $a = 10$, $b = 8$, and $c = 20$, a computer plot confirms that the optimal position $x$ is $11$, using $1000$ assumed $x$ values uniformly spaced for computer plotting.

The AM-GM Inequality

The arithmetic mean-geometric mean inequality provides a powerful tool for solving optimization problems without calculus. The AM-GM inequality states that the arithmetic mean of $n$ positive numbers is greater than or equal to their geometric mean, with equality holding if and only if all the numbers are equal.

The arithmetic mean-geometric mean inequality determines maximum areas and minimum surface areas without derivatives.

G. Ehlers provided a notable proof of the AM-GM inequality in the year 1954. You can use it to find the dimensions of a cylindrical food can with a given volume that minimizes its surface area, resulting in the height equaling the diameter.

A geometric inequality regarding the side lengths of any triangle can be proven using the properties of angle bisectors and the incenter. The challenge inequality states that for any triangle with side lengths $a$, $b$, and $c$, the product $abc$ is greater than or equal to the product of $(a + b - c)$, $(b + c - a)$, and $(a + c - b)$.

To verify this geometric relationship, you follow these steps:

• Identify $a$, $b$, and $c$ as the lengths of the sides of any triangle.
• Verify the inequality using the common intersection point of the three interior vertex angle bisectors known as the incenter.

For specific $45-45-90$ and $30-60-90$ triangles, the inequality reduces directly to the claims that $9 > 8$ and $4 > 3$.

Geometry and computer analysis allow you to calculate the maximum distance required to ensure interception with a straight road or shoreline when completely disoriented. A lost hiker walks one mile in a random direction and then walks along a one-mile-radius circle centered on their starting point to find their car.

If you are a fisherman lost in the woods, you can guarantee a return to a linear boundary by walking a specific composite path:

• Walk one mile in a randomly selected direction from the starting point.
• Walk along the circular path with a one-mile radius centered on the starting point.

These paths result in an absolute maximum distance of $7.2832$ miles for the original solution, while an angle $\theta$ of $16.61$ degrees reduces the maximum path length for the fisherman.

2. Ancient Geometry and Isoperimetric Problems

Ancient mathematicians wrestled with area and perimeter extremes, laying foundations that exposed subtle logical flaws in assuming solutions exist.

The ancients knew intuitively that a straight line is the minimum length joining two points and a circle is the maximum area for a perimeter. However, rigorous proofs for these facts were not established until the eighteenth and nineteenth centuries. Apollonius of Perga lived from 262 to 190 B.C., and Archimedes lived from 287 to 212 B.C. Polybius and Proclus noted that people often incorrectly equated perimeter with area, such as assuming a triangle with a smaller perimeter must have a smaller area.

The First Extremal Problems

The von Koch snowflake demonstrates that a simple closed curve can enclose a finite area within an infinite perimeter. Created through an iterative process where the middle third of each side is replaced by an equilateral triangle, the perimeter grows by a factor of 4 while side length decreases by a factor of 3 at each step. Helge von Koch published the von Koch snowflake curve in 1906, which increases the initial enclosed area by just 60% in the limit while pushing the perimeter to infinity.

The isoperimetric quotient, or I.Q., quantifies the ratio of a figure's area to the area of a circle with the same perimeter, reaching a maximum of 1 for circles. The I.Q. allows comparison of efficiency across regular polygons and geometric shapes, establishing that all closed curves have an I.Q. less than or equal to 1. Queen Dido used strips from a bull's hide along a straight seashore to enclose the maximum possible area, founding the city of Carthage and the classic Dido problem. Thomas Hales proved the honeycomb conjecture in 1999.

Dido's Problem and Steiner's Stumbles

Jakob Steiner provided an elegant geometric solution to Dido's problem and the isoperimetric theorem in 1842, but his proof assumed the existence of a solution without proving it. Peter Dirichlet pointed out that assuming a solution exists is dangerous because certain optimization problems lack a maximum or minimum solution. Abram Besicovitch showed in 1928 that there is no smallest area in which a line segment of unit length can be rotated through 360 degrees, defying intuition through the Kakeya problem.

Fagnano's problem seeks the inscribed triangle of minimum perimeter within any given acute-angled triangle, constructed using the altitudes of the original triangle. Giulio Carlo Toschi di Fagnano demonstrated the existence of a solution, and his son Giovanni Francesco Fagnano completed the minimization using calculus in 1775, while Lipót Fejér found a geometric proof. Lipót Fejér discovered a geometric proof for Fagnano's problem while still a student at the University of Budapest.

You can construct the minimum-perimeter triangle of Fagnano, also known as the pedal or orthic triangle, by following specific steps:

• Draw the three altitudes of the original acute triangle ABC.
• Locate the points U, V, and W where the altitudes intersect the sides.
• Connect U, V, and W to form the inscribed pedal or orthic triangle.

3. Medieval Maximization and Physical Extrema

Historical puzzles from painting views to light refraction demonstrate how physical constraints and optical principles drive extremal mathematics.

You can explore early maximization problems through Regiomontanus formulated a maximization problem in 1471 to find the optimal viewing distance from a wall-hung painting to maximize the viewing angle. By setting up the viewing angle as the difference between two angles formed by the top and bottom of the painting, the problem can be solved without calculus using the AM-GM inequality by minimizing the reciprocal of the tangent function.

A bug on the floor viewing a painting with a bottom edge at 8 feet and top edge at 20 feet should position itself at a distance of the square root of 216, or approximately 14.70 feet from the wall.

The Saturn problem determines the latitude on a spherical planet where the angular width of its ring appears at its maximum. Using the law of cosines and considering the visibility limits where the line of sight becomes tangent to the planet's surface, the problem requires computer analysis to plot the observed angular width across latitudes.

For Saturn with radius 56,900 km, outer radius 138,800 km, and inner radius 88,500 km, a computer simulation shows a maximum viewing angle of 18.44 degrees occurring at a latitude of 33.5 degrees.

The pipe-and-corner problem calculates the required minimum width of a second intersecting tunnel to successfully transport a pipe of a given length and outside diameter around a right-angle corner. Accounting for a nonzero diameter of the pipe rather than treating it merely as a zero-width line segment yields a more realistic and larger required tunnel width.

A 100-foot-long pipe with a 1-foot outside diameter being moved through a first tunnel of width 10 feet requires the second tunnel to be at least 71.68 feet wide.

Optics and the Principle of Least Time

Fermat resolved the correct mathematical form of Snell's law by discovering the principle of least time for light propagation.

Fermat generalized Heron's reflection principle by positing that light follows the path that minimizes total travel time across different media rather than minimum distance, correctly proving that light travels slower in denser media. Fermat used special algebraic manipulations of square-root transit time equations to minimize travel time T, successfully deriving the ratio of speeds v1/v2 as the constant in Snell's law. This occurred in 1658: when Fermat successfully concluded his quest with the principle of least time. Later milestones include 1675: when Olaus Roemer made the first experimental measurement of the finite speed of light, and 1850: when Hippolyte Fizeau and Jean Foucault experimentally measured light to be slower in water than in air.

Calculus explains that primary and secondary rainbows form at the angular extrema of light ray deviations caused by refraction and internal reflections inside spherical raindrops. Light rays concentrate at a broad maximum deviation angle of approximately 42 degrees for the primary rainbow due to one internal reflection, and at about 52 degrees for the secondary rainbow due to two internal reflections. René Descartes used a glass spherical globe as an artificial drop in 1637 to trace incident rays and discover the concentration of light at the rainbow angles.

You can apply specific steps to analyze these optical systems:

• Drop the perpendicular from the curve point B to the x-axis, landing at point C.
• Measure the length CD = s, where D is the coordinate system origin.
• Determine the point E on the x-axis by making DE equal to s.
• Connect B and E with a straight line to establish the tangent.

You can also use numerical techniques to solve equations by following these instructions:

• Select an initial starting value x0 near the expected root.
• Calculate the next approximation using the iterative formula x_n+1 = x_n - f(x_n) / f'(x_n).
• Repeat the calculation steps until the desired decimal accuracy is achieved.

4. The Power of Calculus and Second Derivatives

The invention of calculus and the second derivative test provided rigorous tools to classify stationary points and solve practical engineering dilemmas.

Calculus Foundations and Second Derivatives

Isaac Newton introduced the second derivative test in 1665. When the first derivative is zero, a negative second derivative indicates a local maximum. A positive second derivative indicates a local minimum.

British philosopher George Berkeley published The Analyst in 1734. He attacked the logical foundations of early calculus and famously referred to fluxions as ghosts of departed quantities.

Practical Extrema and Gunnery Problems

Calculus reveals that a cylindrical package maximizes volume under postal constraints. This leads to a paradox where a larger volume cylinder is accepted while a smaller cubical package is rejected.

• Use UPS constraints where the maximum length plus maximum girth is 130 inches.
• Set the derivative of the volume equation to zero.
• Determine that the optimal length of a cylindrical package is one-third of the maximum allowable size.

A cubical package with edge length 5S/24 exceeds the maximum allowable size of 130 inches. It still has a smaller volume than an acceptable maximum-volume cylindrical package. Maximum allowable length for a UPS package is 108 inches.

The optimal launch angle to maximize the distance of a projectile on a flat surface in a gravitational field is 45 degrees when release height is zero. A strong golfer driving a golf ball with an initial speed of 160 feet per second at a 45-degree angle achieves a maximum driving distance of 800 yards.

Edmond Halley applied calculus to military gunnery to find the minimum initial speed required to hit an elevated target. This minimized impact kinetic energy. Halley published his solution regarding gravity and projectile motion in 1688 in the Philosophical Transactions of the Royal Society, backdated to 1686.

Guillaume-Francois-Antoine de L'Hospital hired Johann Bernoulli to teach him calculus. He subsequently published the world's first calculus book, Analyse des Infiniment Petits, in 1696. De L'Hospital's pulley problem demonstrates that the stable equilibrium position of a pulley system depends only on geometric cable lengths rather than the suspended weight.

5. Calculus of Variations and Modern Programming

Advanced variations, minimal surfaces, and computational linear programming resolve complex multi-variable constraints and combinatorial explosion.

The Brachistochrone and Minimal Surfaces

Johann Bernoulli challenged mathematicians in June 1696 to find the curve of swiftest descent. You can solve this brachistochrone problem by applying Fermat's principle of least time and Snell's law of refraction to a layered optical medium. The resulting path is a cycloid. This curve is 1.5% faster than Galileo's quarter-circle descent.

Johann Bernoulli challenged Newton with the brachistochrone problem to prove Newton's inability to solve it, prompting Newton to solve it anonymously in a single night.

Euler solved the surface of revolution minimum area problem, discovering the catenoid. Joseph Plateau studied soap film geometry experimentally using wire frames, leading to rules where three films meet at 120-degree angles or six films meet at approximately 109 degrees. Soap films minimize their surface area and energy. Jean Taylor proved Plateau's rules mathematically in 1976.

Jesse Douglas and Tibor Radó provided existence proofs in 1931 and 1933 showing that a minimal surface always exists for any single closed contour in space.

Linear Programming and Dynamic Programming

Brute-force enumeration becomes impossible for modern optimization problems because the number of possible solutions grows factorially or exponentially with the problem size.

In the traveling salesman problem, the number of closed-loop candidate paths grows as $n$ factorial. For 6 cities, 6! yields 720 paths. George Dantzig noted that for 70 cities, 70! routes exceed $10^{100}$. This makes brute-force checking computationally impossible even with massive parallel computing across the 15 billion years since the Big Bang.

George Dantzig developed the simplex algorithm in 1947 to solve linear programming problems. The constraints carve out a convex region in $n$-dimensional space called the feasible solution set, and the optimal solution is always located at one of its extreme vertices.

George Stigler formulated the diet problem in 1945 to find a minimum-cost nutritionally sound diet using 77 foods. The solution cost $39.93 yearly, which was later reduced to $39.69 by the National Bureau of Standards using Dantzig's simplex algorithm requiring nearly 17,000 multiplications and divisions.

Richard Bellman developed dynamic programming and published his book Dynamic Programming in 1957. You can solve multistage decision processes by working backwards in time using the principle of optimality.

To apply dynamic programming to a complex process:

• Formulate the problem as a time-ordered sequence of decisions.
• Derive the characteristic functional equation for the problem.
• Apply the principle of optimality to work backwards from the final stage to the initial stage.
• Solve the functional equation analytically or numerically to find the optimal policy.

An earth-moving machine manufacturer used dynamic programming on a directed graph with nodes representing months to determine the least-cost production schedule, generating a maximum four-week profit of $1,340,800. Richard Bellman received the IEEE Medal of Honor in 1979 for these contributions.

10 Key Ideas

1. You can solve many complex optimization problems using simple algebra, geometry, and inequalities rather than relying on derivatives.
2. The arithmetic mean-geometric mean inequality determines maximum areas and minimum surface areas without derivatives.
3. Ancient mathematicians wrestled with area and perimeter extremes, laying foundations that exposed subtle logical flaws in assuming solutions exist.
4. Historical puzzles from painting views to light refraction demonstrate how physical constraints and optical principles drive extremal mathematics.
5. Fermat resolved the correct mathematical form of Snell's law by discovering the principle of least time for light propagation.
6. The invention of calculus and the second derivative test provided rigorous tools to classify stationary points and solve practical engineering dilemmas.
7. Isaac Newton introduced the second derivative test in 1665 to classify stationary points as local maxima or minima using negative or positive second derivatives.
8. Johann Bernoulli challenged mathematicians to find the brachistochrone curve of swiftest descent, which solved the problem by showing the path is a cycloid.
9. Brute-force enumeration becomes impossible for modern optimization problems because the number of possible solutions grows factorially or exponentially with the problem size.
10. George Dantzig developed the simplex algorithm in 1947 to solve linear programming problems where constraints carve out a convex region in multi-dimensional space.