Table 16.1 Principal Frequencies of Membranes of Equal Area
16
Circle, Sphere, Symmetrization, and Some Classical Physical Problems
GEORGE PÓLYA
PROFESSOR OF MATHEMATICS
STANFORD UNIVERSITY
16.1 Introduction
This chapter consists of three parts. The first part offers mainly plausible (intuitive, inductive, heuristic) considerations. Engineers and physicists use such considerations quite frequently; if they used them more explicitly, more consciously, they could use them, perhaps, more efficiently. And so, it is hoped even the tone of presentation of this first part may possibly do some good. The second part brings the main mathematical idea. The third part offers additional remarks.
THE HEURISTIC ASPECT
16.2 Observations
The essential ideas with which this chapter is concerned are applicable to various problems of classical mathematical physics. Yet, for the sake of concreteness, we prefer to single out one typical problem, namely, that of the vibrating membrane. We shall discuss this particular problem most of the time, but we shall try to discuss it so that the essential ideas appear in a form readily applicable to other problems.
The tone of presentation in this first part is similar to that of Ref. 7, where also some of the points here treated are mentioned in a different context; cf. vol. 1, pages 168 to 171, and vol. 2, pages 9 to 12.
Let Λ denote the principal frequency (which belongs to the characteristic tone of lowest pitch) of a uniform membrane stretched over the plane domain D. We shall consider various domains, but all the membranes we mention are supposed to have the same thickness, density, and elasticity, and also the same uniform tension when at rest. Therefore, Λ depends only on the shape and size of the domain D, so that it is a functional of D; we postpone the formal definition of Λ until Sec. 16.8.
The computation of the principal frequency Λ for a given domain D is a typical problem of mathematical physics. It is typical, for instance, that “exact” solutions are exceptional. There are only a few simple domains for which Λ can be “exactly” or “explicitly” computed. These are the circle, an arbitrary section of the circle, an arbitrary rectangle, and just three triangles, namely, the triangles with angles 60°, 60°, 60°; 45°, 45°, 90°; and 30°, 60°, 90°. For other domains, such as an ellipse or an arbitrary triangle, the computation of Λ is much more difficult: We have to deal with little-known transcendental functions or resort to approximations the error of which may be difficult to estimate.
To start or to check such approximations, it would be desirable to know more about the dependence of the frequency Λ on the shape of the domain D. In order to examine this dependence, we consider, with a few additions, a table computed by Lord Rayleigh (Ref. 10, vol. 1, page 345 [211]). Table 16.1 exhibits the numerical value of the principal frequency for a dozen different shapes: the circle, four circular sectors (semicircle, quadrant, sextant, and octant—i.e., the sectors with opening 180°, 90°, 60°, and 45°, respectively), four rectangles [with sides in proportion 1:1 (square), 3:2, 2:1, 3:1], and the three triangles mentioned above. All these figures have the same area = 1; they are so arranged that the pitch increases as we read down.
Table 16.1 Principal Frequencies of Membranes of Equal Area
Table 16.2 Perimeters of Figures of Equal Area
Table 16.1 offers a highly interesting indication. The reader should try to catch by himself the suggestion offered—this is a challenge to his faculty of observation. To strengthen that indication, we place Table 16.2 next to Table 16.1. Table 16.2 lists the same 12 figures of equal area as Table 16.1, but instead of the principal frequency Λ, it exhibits the length of the perimeter of each figure, and the figures are so ordered that the perimeters increase as we read down. Therefore, the 12 figures are differently arranged in our two tables (but not too differently). Before passing to the next section, the reader should try to unravel by himself some of the mystery that the comparison of our two tables is about to reveal.
16.3 Conjectures
The main suggestion offered by Tables 16.1 and 16.2 arises from the fact that the circle tops both: Of the twelve figures considered, the circle has both the lowest principal frequency and the shortest perimeter. Now, as the reader probably knows (for full knowledge, he should also know a proof), the circle has the shortest perimeter not only among the twelve figures of equal area listed in Table 16.2 but among all figures of equal area. This geometrical minimum property of the circle and the resemblance between Tables 16.1 and 16.2 jointly suggest a physical minimum property of the circle: Of all membranes of equal area, the circle has not only the shortest perimeter but also the lowest principal frequency. This beautiful property of the circle was first stated by Lord Rayleigh as a conjecture (Ref. 10, vol. 1, page 339 [210]).
Yet the comparison of Tables 16.1 and 16.2 has still other suggestions to offer. Some of these indications become clearer if we collect from these tables figures of the same kind and rearrange them; see Table 16.3. (To the rectangles and triangles listed in Tables 16.1 and 16.2, Table 16.3 adds their limiting cases: the “infinitely narrow” rectangle, with area 1 and sides a and b, where b/a tends to 0, and “infinitely narrow” triangles; a triangle is so called if, 1 being its area, a its longest side, and h its altitude perpendicular to a, the ratio h/a tends to 0. The corresponding approximate, or “asymptotic,” values of principal frequency and perimeter are given in Table 16.3.)
Table 16.3 Principal Frequencies and Perimeters of Figures of Equal Area
Observe that, within each of the three sets of figures displayed by Table 16.3, principal frequencies and perimeters vary in the same direction. Now, the study of the lengths of perimeters presents much easier and much more elementary problems than the study of principal frequencies. For perimeters, we can easily verify the observed regularities. For principal frequencies, the mathematical problems are much less accessible. Yet we can hope that the regularities observed remain valid beyond the narrow limits of the observational material collected in Table 16.3. And so we suspect, besides the great conjecture of Lord Rayleigh, further physical analogues of elementary geometrical facts:
Of all triangles with a given area, the equilateral triangle has not only the shortest perimeter, but also the lowest principal frequency.
Of all quadrilaterals with a given area, the square has not only the shortest perimeter, but also the lowest principal frequency.
Consider the regular polygons with a given area. As the number of sides increases, not only the perimeter but also the principal frequency steadily decreases.
16.4 A Line of Inquiry
The three foregoing statements are, of course, merely conjectures at this stage of our inquiry, although they are conjectures reasonably motivated by the observational material collected in our tables, by analogy, and by our whole previous consideration. This consideration also suggests a procedure of research.
Our aim is to learn something about the dependence of the principal frequency Λ on the shape of the domain D. Yet there is doubtless some sort of parallelism between the frequency Λ and the perimeter of D. Now the study of the length of the perimeter is more elementary and more accessible than the study of Λ. Let us, therefore, study first the dependence of the length of the perimeter on the shape of the figure in the hope that this study may bring us some useful suggestion about Λ.
This leads us to certain geometrical maximum and minimum problems. Such problems have been studied already by some ancient Greek geometers (Zenodorus, Pappus) and by innumerable mathematicians after them.
Of the various approaches known to the author, the most fruitful for our line of inquiry seems to be a highly original idea due to the Swiss geometer Jacob Steiner. He invented a geometrical transformation that we call today symmetrization or, more specifically, Steiner symmetrization. This transformation can be performed in a plane or in space. See Ref. 13, vol. 2, pages 75 to 91; see also vol. 2, pages 264 to 269, which form a part of the highly stimulating extensive memoir on geometric maxima and minima, vol. 2, pages 177 to 308.
16.5 Plane
In a plane, we symmetrize a figure with respect to a specified straight line, given in advance and called the line of symmetrization. For instance, symmetrization with respect to the line A*H* of Fig. 16.1 transforms the pentagon ABFHE into the octagon A*B*D*F*H*G*E*C*. We conceive the original figure (in our particular case, the pentagon) as consisting of “matches” (of various lengths—slender rods, line segments) parallel to each other and perpendicular to the given line of symmetrization (in our case, A*H*). Each match (segment) is pushed along its own line into a position where it is bisected by the line of symmetrization; thus, the segment BC is shifted into the position B*C*. In their new positions, the segments fill the new, transformed, symmetrized figure: It has, by construction, a line of symmetry, the line A*H* in the case of Fig. 16.1. We consider here only figures having not more than two boundary points in common with any line perpendicular to the line of symmetrization. This simple case is enough to bring out the decisive ideas. For the general case, see Ref. 9, page 5.
Fig. 16.1 Steiner symmetrization.
Figure 16.2 shows three successive symmetrizations. The original Fig. 16.2a is an irregular (general) quadrilateral with a vertical diagonal. Symmetrized with respect to the horizontal line α, Fig. 16.2a is transformed into Fig. 16.2b, a quadrilateral with one axis of symmetry, α. Symmetrized with respect to the vertical line β, Fig. 16.2b is changed into Fig. 16.2c, a rhombus. Symmetrized with respect to γ, a line perpendicular to two of its sides, the rhombus (Fig. 16.2c) is transformed into Fig. 16.2d, a rectangle.
Jacob Steiner found interesting properties of symmetrization that are essential to our purpose. Here is the first:
I. Symmetrization leaves the area unchanged.
This property is obvious: In pushing the matches, we do not change their dimensions. It will be enough to give a more formal proof for a polygon; in fact, we could treat a curved line afterward, as a limiting case. We divide the polygon that we are about to symmetrize by parallels passing through its vertices and perpendicular to the line of symmetrization. Thus, the polygon is dissected into two triangles (ΔABC and ΔHGF in Fig. 16.1) and two trapezoids (BDEC, DFGE). A triangle is symmetrized into another triangle (ΔABC into ΔA*B*C*) with the same base and the same altitude and, therefore, the same area. A trapezoid is symmetrized into another trapezoid (BDEC into B*D*E*C*) with the same lower base, the same upper base, the same altitude, and, therefore, the same area. By summing up the contributions of all parts, we prove property I.
Fig. 16.2 Repeated Steiner symmetrization.
II. Symmetrization decreases the length of the perimeter.
In proving this less obvious property, we can use the same parallels we have used in proving property I. We need a theorem from elementary geometry: Of all triangles with the same base and the same altitude, the isosceles triangle has the shortest perimeter. For instance, in Fig. 16.1, the broken line B*A*C*, formed by two sides of the isosceles ΔA*B*C*, is shorter than the broken line BAC, formed by the two corresponding sides of ΔABC, which is transformed into ΔA*B*C* by symmetrization.† The same theorem may help us when we consider the combined length of the two slanting sides of a trapezoid before and after symmetrization: It is shorter after. Thus in Fig. 16.1, we have
since ΔD*B*I* is isosceles, but ΔDBI need not be isosceles. By summing up the contributions of all triangles and trapezoids into which our two polygons, the original and the symmetrized, are divided, we see that the latter, the symmetrical one, has the shorter perimeter, and so we have proved property II.
We take the term “decreases” in the statement of the property just proved in its wider sense, in which it means “does not increase,” and we keep this interpretation also in the sequel. In the present case, it is easy to point out those (trivial) exceptional cases in which the perimeter is not diminished; but it is much harder to keep track of the exceptional cases in some of the following proofs.
16.6 Space
In space, we symmetrize a solid with respect to a specified plane, given in advance and called the plane of symmetrization. Again, we conceive the original solid as consisting of matches (line segments) parallel to each other and perpendicular to the plane of symmetrization. Each match (segment) is shifted along its own line into a position where it is bisected by the plane of symmetrization. In their new positions, the segments form the new, transformed, symmetrized solid: It has, by construction, a plane of symmetry.
For instance, in Fig. 16.3, the plane of symmetrization is vertical, perpendicular to the horizontal plane of the drawing, which it intersects along the line A*C*. The original solid is a hill, represented by contour lines on the left; the symmetrized solid is a symmetric hill, halved by the vertical plane through A*C*.
In any plane that is perpendicular to the plane of symmetrization, we have the geometric relationships described in Sec. 16.5. To the properties I and II proved in that section, there correspond the following properties, also discovered by Jacob Steiner:
III. Symmetrization leaves the volume unchanged.
IV. Symmetrization decreases the surface area.
Property III is obvious (the matches do not change their dimensions when shifted). By elementary geometry, the proof of III is easy, but that of IV is a little more troublesome. By integral calculus, the proof of III is immediate and that of IV very short, but we shall save space and omit these proofs; see Ref. 9, pages 182 to 184.
Fig. 16.3 Steiner symmetrization in space.
16.7 Applications
The following two applications of symmetrization are presented without complete rigor (it would be much more difficult to render the second rigorous than the first), yet even so they seem to offer valuable suggestions.
a. Figure 16.2 shows how any quadrilateral a can be transformed into a rectangle d by three successive symmetrizations. By symmetrizing the rectangle d with respect to a perpendicular to one of its diagonals, we obtain another rhombus e (draw it!). By symmetrizing this rhombus e with respect to a perpendicular to two of its sides, we obtain another rectangle f—we pass from e to f as we have passed from c to d in Fig. 16.2. By repeating our last steps, we pass from rectangle to rhombus, from rhombus to rectangle, and so on, generating an infinite sequence of quadrilaterals
By the two properties proved in Sec. 16.5, each quadrilateral in this infinite sequence has the same area as, but a shorter perimeter than, the preceding quadrilateral, so that all have the same area but the perimeter decreases steadily as we advance in the sequence.
The reader is asked to concede that our infinite sequence has a limit. The limiting figure, as a limit of rhombi, must be a rhombus, and, as a limit of rectangles, it must be a rectangle; thus, being both a rhombus and a rectangle, it is a square. This limiting square still has the same area as all quadrilaterals of the sequence but a shorter perimeter than any quadrilateral of the sequence. We compare the limiting square with a, the initial figure of the sequence, which is an arbitrary quadrilateral, and find: A square has a shorter perimeter than any other quadrilateral with the same area. We have proved here a theorem of which we have supposed knowledge in formulating one of our conjectures on principal frequency at the end of Sec. 16.3.
b. In the plane, we are given the figure F, the boundary of which is an arbitrary closed curve, and we are also given two straight lines, 1 and 2, which include the angle θ.
By symmetrizing F with respect to 1, we obtain F1.
By symmetrizing F1 with respect to 2, we obtain F2.
By symmetrizing F2 with respect to 1, we obtain F3.
By symmetrizing F3 with respect to 2, we obtain F4.
And so on. We thus generate the infinite sequence
from the initial figure F by successive symmetrizations with respect to the two given axes 1 and 2 alternately; F1, F3, F5, …, F2n−1, … are symmetric with respect to 1, and F2, F4, F6, …, F2n, … with respect to 2.
The reader is asked, as before, to concede that our infinite sequence has a limit L. This limiting figure L, as a limit of F2n−1, must admit the line 1 as axis of symmetry; and, as a limit of F2n, it must admit the line 2 as axis of symmetry; and so L is symmetric with respect to both lines, 1 and 2.
Draw the line 3 as the mirror image of 1 with respect to 2. This line 3 —which, by the symmetry of L with respect to 2, is “located in L in the same way as 1 is located in L”—must be an axis of symmetry for L, just as 1 is. Draw the line 4 as the mirror image of 2 with respect to 3; by using both the result and the method of the foregoing remark, we see that also the line 4 must be an axis of symmetry for L; and so on. The lines so obtained, 1, 2, 3, 4, …, all pass through a common point, and the angle between any two successive lines is θ.
This is so, independently of the choice of θ. Yet we may choose θ from the start so that the ratio θ/π is irrational. Then, as is easily seen, the lines 1, 2, 3, 4, … are all different from each other, they are “everywhere dense,” and all of them are axes of symmetry for L. What figure can L be? It is intuitively obvious (and can be proved) that L must be a circle.
Now, by the properties I and II proved in Sec. 16.5, all the figures F, F1, F2, …, generated by repeated symmetrization, have the same area, whereas the length of their perimeters steadily decreases as we advance in the sequence. Hence the limiting circular figure L has still the same area as any one of them, for instance F, but L has a perimeter shorter than any one of them, for instance F. Yet the boundary of F was an arbitrary closed curve, and so we see: A circle has a shorter perimeter than any other plane figure with the same area. This is the theorem that has suggested, by analogy, Rayleigh’s conjecture formulated in Sec. 16.3.
THE KEY IDEA OF THE PROOF
16.8 Definition
Our previous work was merely heuristic; observations, conjectures, and analogies may be interesting and stimulating, they may be in some respects even more important than proofs, but they prove nothing. And between the geometric theorems that we have proved (or nearly proved) and the conjectures concerning Λ that we wish to prove, we perceive for the moment only the loose link of analogy.
One thing is certain: If we wish to prove anything about Λ, we cannot remain on the merely intuitive level, but we must introduce, and use, a mathematical definition of Λ. Now, Λ can be defined in various ways.
It is usual to define Λ2 as the first eigenvalue of a boundary-value problem. There is a function w vanishing along the boundary of the domain D, continuous on the domain plus its boundary, not vanishing identically in the interior of D, and in the interior of D satisfying the differential equation
By these conditions, Λ2 would not yet be determined, but Λ2 is defined as the least value of this kind.
For our purpose, however, it is more advantageous to define Λ by a minimum principle. Let f = f(x,y) be any function, defined, sufficiently “smooth,” nonnegative, and not identically vanishing in the interior of D, but vanishing along the whole boundary of D; then
Of course, fx and fy denote partial derivatives:
The double integrals on the right-hand side of the inequality (16.1) are extended over the whole domain D. The case of equality can be attained in the relationship (16.1) (when f = w), so that Λ2 is the minimum of the quotient (usually called the Rayleigh ratio) on the right-hand side of the inequality (16.1); this defines Λ. For the equivalence of this definition with the former, see Ref. 9, especially pages 89 to 91.
The definition of Λ by a minimum principle is of great importance in practice. It yields upper approximations to Λ and forms the basis of the widely used Rayleigh-Ritz method.4,14
16.9 From Surface Area to Dirichlet Integral
The definition of Λ by a minimum principle is also important for our more theoretical purpose. It may direct our attention to the function f(x,y), which, as we have said, vanishes along the boundary of the domain D and is positive inside D. The surface
z = f(x,y)
and the xy plane include a solid that we call the hill; see Fig. 16.3, on the left, where the outermost contour line, corresponding to the elevation z = 0, forms the boundary of the domain D that we wish to call the base of the hill.
We symmetrize the hill (see Sec. 16.6) with respect to the vertical plane that intersects the horizontal xy plane in the line A*C*. We thus obtain a new solid, the symmetric hill (see Fig. 16.3, on the right) contained between the xy plane and a new surface the equation of which we write in the form
z = f*(x,y)
The base of the symmetric hill, i.e., that part of its boundary that lies in the xy plane, is a symmetric plane domain that we denote by D*; it is surrounded by the outermost contour line on the right in Fig. 16.3. Of course, the function f*(x,y) vanishes along this outermost contour line, the boundary of D*,
By property III, formulated in Sec. 16.6, the volume remains unchanged by symmetrization:
Yet, by property IV (again see Sec. 16.6) the surface area is diminished by symmetrization. The base, however, which is the part of the surface area contained in the xy plane, remains unchanged by symmetrization, by virtue of property I proved in Sec. 16.5:
Thus the remaining (curved) part of the surface area is responsible for the diminishing of the whole:
We are approaching the turning point of our discussion. We shall attain it by realizing the physical significance of the numerator of the Rayleigh ratio; see the right-hand side of the inequality (16.1). This numerator is a double integral extended over the domain D (the equilibrium position of the membrane), usually called the Dirichlet integral;2,11 the integrand is the square of the gradient of the function f. The Dirichlet integral is proportional to the potential energy of the membrane, since it is proportional (as we shall see in a moment) to the change of area that the membrane undergoes when its equilibrium position (fully in the xy plane) is disturbed. The change of area must be considered, as are all displacements in the classical theory of elasticity,12 as being “very small” or “infinitesimal”; cf. Ref. 10, page 307 [194].
Starting from an arbitrary function f, we make it “small” by multiplying it by a “small” positive constant 
. Let us apply the inequality (16.4) to f, instead of f. We thus obtain
Expand both sides in powers of the small quantity :
By virtue of Eq. (16.3), the initial terms cancel. After division by 2, we pass to the limit by letting tend to 0 and obtain
We have added a new property to the properties I, II, III, and IV that were discovered by Jacob Steiner:
V. Symmetrization decreases the Dirichlet integral.
The foregoing transition from the surface area to the Dirichlet integral contributes a good deal to elucidating certain analogies between geometrical and physical quantities such as we have observed at the beginning between the perimeter and the principal frequency of a membrane.†
16.10 A Minor Remark
Property V is the key to the physical applications of symmetrization. We need, however, one more (easy) remark for our next conclusion.
Let Φ(t) be a steadily increasing positive-valued function of the positive variable t; we are concerned here with such simple examples as Φ(t) = t2 or Φ(t) = t, where is a positive constant. We assert: If the symmetrization of f(x,y) yields f*(x,y), then the symmetrization of Φ[f(x,y)] yields Φ[f*(x,y)].
Obviously, the functions f(x,y) and Φ[f(x,y)] have the same level lines: If
and vice versa. Of course, the constants in these two equations are in general different: Those congruent level lines are at different elevations. Still, the two functions, f(x,y) and Φ[f(x,y)], are represented by the same contour map; Fig. 16.3, made to represent the symmetrization of f(x,y), can just as well represent the symmetrization of Φ[f(x,y)], and so the inspection of Fig. 16.3 renders the truth of the advanced assertion obvious.
Symmetrization, which, transforming f(x,y) into f*(x,y), transforms f2 into f*2, preserves volume (Sec. 16.6, property III). Therefore, besides Eq. (16.2), we also have
We shall need this fact in the next section.
16.11 Symmetrization and Principal Frequency
We have now collected the facts we need for proving the following property:
VI. Symmetrization decreases the principal frequency.
We are given an arbitrary plane domain D. The corresponding principal frequency is defined (see Sec. 16.8) as the minimum of the Rayleigh ratio; there exists a function f for which the case of equality is attained in the inequality (16.1):
We symmetrize in space with respect to a plane of symmetrization perpendicular to the xy plane; see Fig. 16.3. By symmetrizing the domain D, we obtain the domain D* to which there corresponds the principal frequency Λ*. This symmetrization transforms the function f arising in Eq. (16.7), which vanishes along the boundary of D, into f*, which vanishes along the boundary of D*; see Sec. 16.9. By combining the relationships (16.5) and (16.6), we obtain
Now Λ*, the principal frequency corresponding to D*, is defined by the minimum property stated in Sec. 16.8, and the function f*, since it vanishes along the boundary of D*, is appropriate for computing an upper bound for Λ*: Just as we obtained the inequality (16.1), we similarly can establish the inequality
Combination of the relationships (16.7) to (16.9) yields
Λ2 ≥ Λ*2
which proves property VI.
16.12 Scope of the Proof
The foregoing proof hinges on the work of Sec. 16.9, the symmetrization of the Dirichlet integral. Therefore, it applies essentially also to some other physical and geometrical quantities defined, as Λ, by a minimum principle involving the Dirichlet integral (in the plane or in space). We mention here only two such quantities:
The torsional rigidity P of a uniform elastic cylinder the cross section of which is the domain D
The electrostatic capacity C of a metallic plate (with vanishing thickness) coextensive with the domain D
For more details and more analogous quantities, see Ref. 9, especially pages 1 to 3.
It is convenient to add here a few more notations:
The area A of D
The length L of the perimeter of D
The polar moment of inertia I of D with respect to an axis perpendicular to D and passing through its center of gravity
Properties I and II and (what is the main point) the result proved in Sec. 16.9 are contained in the following comprehensive proposition:
THEOREM 16.1. Symmetrization leaves A unchanged, decreases L, I, Λ, and C, and increases P.
For A and L, see Sec. 16.5; for Λ, see Sec. 16.11. The proof for P and C is closely related to the proof for Λ; the proof for I is much simpler. For details concerning Theorem 16.1 and the rest of this section, see Ref. 9, pages 151 to 161 and passim.
In possession of Theorem 16.1, we perceive that the argument b in Sec. 16.7 extends much beyond its original scope. In fact, it comes pretty close to proving the following comprehensive proposition:
THEOREM 16.2. Of all plane domains with a given area A, the circle yields the lowest value for L, I, Λ, and C and the highest value for P.
In so far as this proposition refers to P, the domain in question is supposed to be simply connected. The argument b of Sec. 16.7 can be modified, developed, and rendered rigorous in various ways; see especially Ref. 9, pages 189 to 193. There is an extremely short and elementary argument for I.6
Also the argument a in Sec. 16.7 extends beyond its original scope and proves, in fact, the following result:
THEOREM 16.3. Of all quadrilaterals with a given area A, the square yields the lowest value for L, I, Λ, and C and the highest value for P.
For some of these quantities, the argument a of Sec. 16.7 can be greatly simplified; see Exercises 5 and 6 below. There is an analogous theorem for triangles of which we have stated a particular case (concerning Λ) as a conjecture toward the end of Sec. 16.3; we leave the proof as another exercise to the reader (Exercise 7).
Yet we have no means at our disposal for proving an analogous proposition for pentagons. The reason is that, by symmetrizing a polygon, we increase, in most cases, the number of vertices. For instance, in Fig. 16.1, the original polygon has five vertices, but the new polygon, obtained by symmetrization, has eight vertices. In dealing with triangles or quadrilaterals we can, by a suitable choice of the line of symmetrization, avoid increasing the number of vertices; yet, in dealing with pentagons or hexagons or higher polygons, we cannot. And so decades may pass before the last conjecture stated in Sec. 16.3 (about regular polygons) will be proved.
Yet we can extend the foregoing considerations to three-dimensional problems without essential difficulties. We thus obtain, among others, the following theorems (see Refs. 8 and 9, passim):
Symmetrization decreases the electrostatic capacity of a condenser.
Of all solids with a given volume, the sphere has the smallest electrostatic capacity.
Of all tetrahedra with a given volume, the regular tetrahedron has the smallest electrostatic capacity.
ADDITIONAL REMARKS
16.13 Alternative Symmetrization
We are given a plane domain D. By an appropriate straight line l, which is not a line of symmetry for D, we cut D into two “unequal halves.” We adjoin to each “half” its mirror image with respect to l and thus obtain two symmetric domains D′ and D″ (In the top row of Fig. 16.4, the two “halves” of the trapezoid D, along with their images in the dashed line, form the rhombus D′ and the hexagon D″.) Now, we face an alternative: We have to choose between D′ and D″; if an appropriate choice leads us to one of these two figures, we say that it has been derived from D by alternative symmetrization. †
Let A, A′, and A″ denote the area, and L, L′, and L″ the length of the perimeter, of D, D′, and D″, respectively. Obviously,
A′ + A″ = 2AL′ + L″ = 2L
If the “unequal halves” into which D has been cut by l are of equal area, then
A′ = A″ = A
If the notation is so chosen that L′ ≤ L″, then also L′ ≤ L; we select D′ and in so doing we derive from D, by alternative symmetrization, a symmetric figure with the same area and a shorter, or possibly equal, perimeter. The perimeter will definitely be shortened if L′ ≠ L″.
We can put this procedure to good use if we wish to determine the figure with a given area A the perimeter of which is a minimum. Any straight line that bisects the area of the desired figure must also bisect its perimeter; if this were not so, alternative symmetrization would yield a figure with the same area A and a shorter perimeter, which would contradict the definition of the desired figure. This hints very strongly that the desired figure (if it exists) must be the circle.
Fig. 16.4 Alternative symmetrization.
The argument just sketched can be developed more fully (especially, we can take care of more intricate possibilities; cf. the middle row of Fig. 16.4) and, what is more interesting for us, it can be adapted to some physical problems.
16.14 Uniqueness
“Of all plane domains with a given area, the circle has the lowest principal frequency.” (Cf. Theorem 16.2 in Sec. 16.12.) Yet is the circle the only figure with that given area and the minimum principal frequency? Or is there some noncircular figure that has the same area and the same principal frequency as the circle? Steiner symmetrization, the transition from the surface area to the Dirichlet integral, and the connected ideas sketched in the foregoing argument give no obvious answer to this question and for this reason the formulation of Theorem 16.2 leaves a gap: Intentionally, it leaves open the question of the uniqueness of the extremal figure. Fortunately, alternative symmetrization can fill this gap.
If a domain D is not a circle, we have to show that there is another domain 
with the same area as D but a lower principal frequency than D; being given D, we have to construct such a domain .
If a connected domain (consisting of one piece) is neither a circle nor an annulus contained between two concentric circles, it cannot have a line of symmetry in every direction. Let us assume that our domain D (for example, the trapezoid in the top row of Fig. 16.4) has no vertical line of symmetry. Yet there is a vertical line l that bisects the area of D; this can be shown by a continuity consideration. Alternative symmetrization performed on D with respect to this line l yields two symmetric domains D′ and D″ (the rhombus and the nonconvex hexagon in our example), each of which is equal in area to D.
We consider now the function f for which equality is attained in the inequality (16.1) of Sec. 16.8. We define two new functions f′ and f″ (the primes do not denote derivatives!). The function f′ is defined in the domain D′. In the “half” of D′ that coincides with a certain “half” of D, the function f′ coincides with the function f, and in any two points of D′ that are mirror images of each other with respect to the line l, the function f′, takes the same value. Visibly, the function f′ vanishes along the boundary of D′ (as f vanishes along the boundary of D) and f′ is continuous throughout D′, in particular along the line l, although its normal derivative on l may not exist. The function f″ is analogously defined in D″; as f′ extends f from one “half” of D into the full domain D′ by symmetry, so f″ extends f from the other “half” of D into D″.
Since equality is attained by our f in the inequality (16.1), we have Eq. (16.7) of Sec. 16.11. By the definition of f′ and f″, we can write Eq. (16.7) in the form
by first multiplying the numerator and the denominator on the right-hand side of Eq. (16.7) by 2.
Consider the two Rayleigh ratios:
By an elementary property of fractions, the ratios (16.11) cannot both be larger than the right-hand side of Eq. (16.10). We assume the notation so chosen that
Now, as we have observed above, f′ is continuous in D′ and vanishes along the boundary of D′. Therefore, the right-hand side of the inequality (16.12) yields an upper bound for Λ′2 [of course, Λ′ is the principal frequency of D′; we have to apply the inequality (16.1) to D′ instead of D]. Yet more is true; we actually have
Exclusion of the equality in the relationship (16.13) (> instead of ≥) is the cardinal point of our argument.
Equality in the relationship (16.13) could be attained only if f′ represented the true shape of the vibrating membrane stretched over D′ (if it satisfied the corresponding differential equation). As such, f′ would have continuous partial derivatives (even of higher order) and, since it coincides with f in one half of D′, it would, by virtue of the differential equation, fully coincide with f. Therefore, D′ would also coincide with D, and so D would have l as a line of symmetry—yet this is precisely the circumstance that we have excluded from the start. Thus, we have fully proved the inequality (16.13), with >, not only with ≥.
Obviously, the inequalities (16.12) and (16.13) now yield
Λ2 > Λ′2
and so D′ is a domain such as we have desired: It has the same area as D but a lower principal frequency than D.
We can take care of more complicated cases such as the one represented by the middle row of Fig. 16.4, extend the argument (with appropriate changes) to P and C, and so we can prove that the circle, and only the circle, yields the extremal value in all five cases considered in Theorem 16.2. The extension to three dimensions presents no additional difficulty.†
16.15 Where the Alternative Symmetrization Leaves No Alternative
We have still to discuss the case illustrated by the bottom row of Fig. 16.4. The rectangle and the vertical should represent to us any domain D with a center of symmetry and a line l passing through this center that is not a line of symmetry for D. Alternative symmetrization applied to D with respect to l yields two congruent figures. The argument of the foregoing Sec. 16.14 and elementary considerations lead us to the following counterpart of Theorem 16.1 (in which “symmetrization” means “Steiner symmetrization” and the terms “decrease” and “increase” are used in the wide sense) :
Fig. 16.5 Higher rotational symmetrization.
Alternative symmetrization of a domain that has a center of symmetry, with respect to a line that passes through this center but is not an axis of symmetry of the domain, leaves A and L unchanged, decreases I or leaves it unchanged, definitely decreases Λ and C, and definitely increases P.
We shall not stop to explain in detail the extension of the foregoing remark to centers of symmetry of higher order, which is hinted by Fig. 16.5. See, however, Exercise 9, below.
16.16 One More Inequality Suggested by Observation
Consider, in Table 16.3, the quotients of which the denominator is a number in the last column and the numerator the corresponding number in the preceding column. All these quotients are pretty close to 1 and none is greater than the value π/2 yielded by the infinitely narrow rectangle. For this last statement to be true not only for the cases collected in Table 16.3 but generally, we would have
This inequality will be proved by the author in a paper that makes essential use of a foregoing research of E. Makai.
EXERCISES
1. By Steiner symmetrization, show that an ellipse is transformed into an ellipse and an ellipsoid into an ellipsoid.
2. Steiner symmetrization, in plane or space. The figure F, which has a center of symmetry, is transformed by symmetrization into F*. Show that F* also has a center of symmetry.
3. Steiner symmetrization in the plane. The figure F, which has two axes of symmetry perpendicular to each other, is transformed by symmetrization into F*. Show that F* also has two axes of symmetry perpendicular to each other.
4. Steiner symmetrization in the plane. The figure F, which has three different axes of symmetry, is transformed by symmetrization into F*. Show by an example that F* need not have more than one axis of symmetry.
5. Prove that of all quadrilaterals of equal area, the square minimizes the perimeter.
In showing this you need not the whole (infinite) process a of Sec. 16.7 (only its first steps, indicated by Fig. 16.2), provided that you can prove directly that the square has a shorter perimeter than any other rectangle with the same area (which can be done by very simple algebra).
6. Prove that of all quadrilaterals with a given area, the square has the lowest principal frequency.
In showing this, you need not the whole (infinite) process a of Sec. 16.7, provided that you use the expression
for the principal frequency of a rectangle with sides a and b. (It is not so easy to eliminate the infinite process from the proof for the full Theorem 16.3 of Sec. 16.12.)
7. Show that of all triangles with a given area, the equilateral triangle has the lowest principal frequency.
(Steiner symmetrization of ΔABC with respect to the perpendicular bisector of AB yields the isosceles ΔA′B′C′ with A′ = A, B′ = B, A′C′ = B′C′. Now use the perpendicular bisector of A′C′ as line of symmetrization, then that of A″ B″, then that of A′″C′″, and so on, alternately.)
8. Let a denote the horizontal side and h the vertical side of a rectangle. The sum of the two parallel sides of an isosceles trapezoid is 2a; the altitude (perpendicular to these parallel sides) is h. A parallelogram has the same base a and the same altitude h as the rectangle, and its four angles are equal to the four angles of the isosceles trapezoid (the four angles are, of course, differently disposed around the two figures). Show that, of these three figures, the last one (the parallelogram) has the highest principal frequency.
9. The octagon in Fig. 16.5 has a center of symmetry of order 4 (coincides with itself when rotated about this center through the angle 2π/4). We obtain the hexagon in Fig. 16.5 from the octagon by replacing two quarters of the octagon by their respective mirror images. Show that, in the transition from the octagon to the hexagon, A, L, and I remain unchanged, Λ and C are decreased, and P is increased.
REFERENCES
1. Courant, R., Beweis des Satzes, dass von allen homogenen Membranen gegebenen Umfanges und gegebener Spannung die kreisförmige den tiefsten Grundton Gibt, Math. Z., vol. 1, pp. 321–328, 1918.
2. Hestenes, Magnus R., Elements of the Calculus of Variations, chap. 4 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
3. Jenkins, J. J., “Univalent Functions and Conformal Mapping,” Ergeb. der Mathematik, new series, vol. 18, Springer-Verlag, Berlin, 1958.
4. Morrey, Charles B., Jr., Nonlinear Methods, chap. 16 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
5. Pólya, G., Torsional Rigidity, Principal Frequency, Electrostatic Capacity and Symmetrization, Quart. Appl. Math., vol. 6, pp. 267–277, 1948.
6. ——, Remarks on the Foregoing Paper, J. Math. Phys., vol. 31, pp. 55–57, 1952.
7. ——, “Mathematics and Plausible Reasoning,” Princeton University Press, Princeton, N.J., 1954.
8. —— and G. Szegö, Inequalities for the Capacity of a Condenser, Amer. J. Math., vol. 67, pp. 1–32, 1945.
9. —— and ——, “Isoperimetric Inequalities in Mathematical Physics,” Princeton University Press, Princeton, N.J., 1951.
10. Rayleigh, Lord J. W. S., “The Theory of Sound,” 2d ed., London, 1894. Reprinted by Dover Publications, New York, 1955. In references, the section number in square brackets follows the numbers of the pages quoted.
11. Schiffer, Menahem M., Boundary-value Problems in Elliptic Partial Differential Equations, chap. 6 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
12. Sokolnikoff, Ivan S., The Elastostatic Boundary-value Problems, chap. 7 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
13. Steiner, Jacob, “Gesammelte Werke,” G. Reimer, Berlin, 1881–1882.
14. Tompkins, Charles B., Methods of Steep Descent, chap. 18 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
† A physicist may be tempted to prove this theorem of geometry by an optical argument: Regard the line AA* as a mirror. Then B*A*C* is the path of light, starting from a source at B*, being reflected at A*, and proceeding hence to the eye of an observer at C*. Yet BAC cannot be a path described by light, since the lines BA and CA do not include the same angle with the mirror AA*. And, as we know, light chooses the shortest path—hence B*A*C* is shorter than BAC. If we go back to the figure on which Heron of Alexandria, discoverer of the optical principle of the shortest path, based this principle, we obtain a beautiful proof for the geometric theorem. See Ref. 7, vol. 1, pages 142–144.
† In the seminar of Professor Timoshenko, I was about to present another proof for property V (cf. Ref. 8 or 9, pages 153 to 157) as he intervened with the remark that considerations of energy should have some bearing on the matter—and then, quite abruptly, the idea of the foregoing proof presented itself to my mind. A shorter description of this event in my paper,5 where the foregoing argument was first presented, should not give rise to misinterpretations about the authorship of my proof; cf. Ref. 3, page 133. (A few short passages of my paper5 have been reproduced here with the kind permission of the editor of the Quarterly of Applied Mathematics.)
† The geometric operation for which we have introduced the term “alternative symmetrization” has been employed by Steiner; see Ref. 13, vol. 2, pages 193 and 194, 299, etc. Its first application to a physical problem is due to R. Courant, Ref. 1. Further physical applications have been sketched by L. E. Payne and H. F. Weinberger; see two abstracts in Bulletin of the American Mathematical Society, vol. 59, pages 244 and 363, 1953. The application of alternative symmetrization to the proof of the uniqueness in Rayleigh’s problem has been briefly indicated by Payne and Weinberger in a conversation with the author and is published here with their kind permission; of course, the author assumes full responsibility for the exposition in Sec. 16.14, for which he had to supply the details by himself. The remarks in Sec. 16.15 have been known to the author since about 1950. (Cf. the abstract of an article by J. Hersch, “Une symétrisation differénte de celle de Steiner,” Enseignement Math., vol. 5, pp. 219–220, 1959.)
† The process hinted by Fig. 16.4 may yield n connected symmetric domains D′, D″, … , D(n) (n = 2 in the top row, n = 3 in the middle row). None of these n domains has, however, a larger area than D. We replace Eq. (16.10) by another in which there are n terms both in the numerator and in the denominator of the right-hand side, and we consider correspondingly n Rayleigh ratios, not only two as in formulas (16.11). Among these n ratios, there will be one (belonging to a domain that, with suitable notation, may be called D′) for which the inequality (16.12) holds. If the area of D′ happens to be less than that of D, we “inflate” D′ to a similar figure that has the same area as D; this operation diminishes the principal frequency. Eventually, by the argument that leads to the inequality (16.13), we obtain a domain with the same area as, but a lower principal frequency than, D, quod erat faciendum. We can take care of the case of the annulus in various ways—for instance, by Steiner symmetrization followed by alternative symmetrization.