Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions

where S is the shape factor, $ΔT$ (K) is the temperature difference, and k (W/m K) is the object's thermal conductivity. In two-dimensional problems, S is dimensionless and Q is the heat flow per unit distance normal to the plane. The shape factor is scale invariant and determined exclusively by the particular object's geometry and boundary conditions. Shape factor theory is discussed in detail in Refs. [1–3].

The heat flow can be calculated by solving the Laplace equation in the domain of interest, and then integrating the heat flux exiting the high-temperature portion of the boundary. The shape factor conveniently summarizes the result of such an analysis. In most cases, the shape factor inside a two-dimensional object is easier to compute (analytically or numerically) than the shape factor outside the object, even for an apparently simple object like a square (see, e.g., Ref. [4], Sec. 7.2). Thus, a relationship between the interior and exterior shape factors allows a potentially difficult exterior calculation to be replaced by a more tractable interior calculation or an already known result.

This paper examines the shape factor in two dimensions using techniques from complex analysis and conformal mapping. To proceed mathematically, we precisely define the “object” boundary as a Jordan curve: a closed nonself-intersecting curve that divides the plane into an interior region and an unbounded exterior region. For example, the interior and exterior problems relative to a rectangular arc are illustrated in Figs. 1(a) and 1(b). The boundary is partitioned into a finite number of sections, each prescribed a boundary condition that is either adiabatic (a zero Neumann condition) or isothermal (a constant Dirichlet condition) at one of two temperatures. Sections with different temperatures must be separated by an adiabatic section to ensure a finite rate of heat transfer. However, any number of sections may be at each temperature.

To mathematicians, the shape factor is the reciprocal of the extremal distance—a conformal invariant of interest in function theory that is equal to the reciprocal of the Dirichlet energy² [5]. Useful properties of the extremal distance are outlined by Ahlfors [5]. When the boundary curve is split into precisely four segments, the extremal distance is equal to the conformal modulus, which was studied by Hersch [6,7] in geometries possessing certain symmetries. By exploiting the Schwarz reflection principle, Hersch derived analytical results for a variety of geometries.

More recently, Lienhard [8] made the striking claim that interior and exterior shape factors are always equal. To reach this conclusion, he first used the Riemann mapping theorem to argue that the interior and exterior of a Jordan curve may always be related by a conformal map. Indeed, such a map is guaranteed to exist [9].

Lienhard proceeded by further assuming that each boundary section possessing a given boundary condition is invariant under the interior-to-exterior mapping (up to an overall reflection); we label this assumption A. When true, assumption A does indeed imply that the interior and exterior shape factor problems are conformally equivalent (see Sec. 2), and thus they possess the same shape factor, S_i = S_e [5]. Based on assumption A, Lienhard [8] conjectured that S_i = S_e must always hold. He subsequently analyzed the shape factors for a disk and a square, under various boundary conditions, and found in each case that S_i = S_e, thus gaining support for the conjecture. However, we now demonstrate that S_i = S_e need not hold in more general cases than those considered by Lienhard.

Consider the following counterexample: a rectangle with isothermal sides of width a and adiabatic sides with height b. For the case of a square, when a = b, it is clear that $Si=Se=1$ as was demonstrated previously [8]. However, for rectangles with $a≠b$⁠, the two shape factors diverge rapidly, as seen in Fig. 1(c).

The purpose of this paper is twofold. First, we shall elucidate why Lienhard's conjecture is not universally true. Second, we provide an additional criterion that does ensure the equality of interior and exterior shape factors. The new criterion relies on a little-known method due to Hersch [7], and we thus take this opportunity to explain that method in an expository manner. In addition, we derive a simple formula for the shape factor of many geometries that satisfy our symmetry criterion.

The failure of assumption A ultimately spoils the conformal equivalence between the interior and exterior problems, as we now describe.

Consider two Jordan curves $∂C$ and $∂D$⁠. Then, let C be the domain defined as either the interior or exterior of $∂C$⁠. Let $∂D$ be the domain defined as either the interior or exterior of $∂D$⁠. Suppose that each boundary curve is partitioned into a finite number of sections with each section assigned either a zero Neumann condition or a constant Dirichlet condition at one of two temperatures. Again, sections at different temperatures must be separated by an adiabatic segment to ensure a finite rate of heat transfer.

We say that the two shape factor problems, in the domains C and D, are conformally equivalent if the following criteria hold: (1) the two problem domains are related by a conformal map; and (2) under that mapping, the boundary condition at every image point matches that at the corresponding pre-image point. If two shape factor problems are conformally equivalent, then they possess the same shape factor [5]. The case of interest in this paper is that for which C is the interior and D the exterior of the same Jordan curve.

The Riemann mapping theorem guarantees the existence of a conformal map f(z) between the interior and exterior of a Jordan curve, thus guaranteeing the satisfaction of criterion 1. However, the mapping theorem only offers three degrees-of-freedom: one can always construct a map that sends three points on the original domain boundary to three chosen points on the target domain's boundary [9]. Note that the choice of the three image points is not completely arbitrary: their order along the boundary must be preserved under the mapping. Once the image locations of three pre-image points are chosen, the map becomes fully specified so that the location of any other point cannot be chosen at will.

Shape factors, however, are determined by a minimum of four points, one at the end of each isothermal section. As a result, Riemann's theorem does not guarantee that the boundary conditions are preserved when mapping from the interior to the exterior domain. It follows that the interior and exterior boundary value problems are not necessarily conformally equivalent, meaning that the interior and exterior shape factors are not necessarily equal. An additional constraint or symmetry must be present to ensure that the map leaves the boundary conditions invariant, as will be discussed in Sec. 3. By chance, the specific cases examined by Lienhard each tacitly embodied such symmetries, which led to the incorrect deduction that S_i = S_e must always hold.

In this section, we apply a powerful method for analyzing Laplace problems that was described by Hersch in another context [7]. The method builds upon the Schwarz reflection principle. By application of this method, we deduce a simple criterion that ensures S_i = S_e.

After assuming that the domain boundary $∂B$ (Jordan curve) possesses a certain reflectional symmetry, we proceed to construct a conformal map that translates the interior shape factor problem, defined on $int(∂B)$⁠, into a shape factor problem interior to the unit disk. We similarly translate the exterior shape factor problem into a problem interior to the unit disk. Under further symmetry conditions on the boundary condition placements, we demonstrate that both shape factor problems (interior and exterior to $∂B$⁠) map to the same problem on the interior of the unit circle, and thus possess identical shape factors.

The Schwarz reflection principle may be stated as follows [9].

We now illustrate how the Schwarz reflection principle applies to conformal maps of sectors of the complex plane. Consider a circular sector occupying $π/4$ rad of the interior of the unit circle, which we denote $Wπ/4−$⁠, as depicted in Fig. 2(a). The Riemann mapping theorem guarantees that this sector can be transformed into any other simply-connected region by a conformal map. In particular, the mapping theorem ensures that there exists a mapping function f(z) to the region $Ω−$ depicted on the right side of Fig. 2(a), whose boundary we denote as Γ. Figure 2(a) illustrates just one particular choice of $Ω−$⁠. If $Ω−$ is taken instead to be the shaded element in Fig. 3(b), the shape obtained by Schwarz reflections in Fig. 2(d) becomes a regular pentagon. A large class of shapes can be obtained through Schwarz reflections of various choices of $Ω−$⁠, a few of which are illustrated in Fig. 4.

Our analysis applies generally to situations where the boundary curve Γ has the following properties. Consider a wedge sector of the complex plane spanning some angle, with its boundaries defined by two straight lines emanating from the origin. Now choose one point on each sector boundary and call the straight segments from the origin to each of these two points L₁ and L₂. Now connect the open endpoints of L₁ and L₂ to one another with a nonself-intersecting curve γ which only intersects the sector boundaries at its endpoints. We call the outer curve, γ, a primitive edge. Then, we define the primitive curve, $Γ≡L1∪L2∪γ$⁠. We call the shape bounded by Γ, $int(Γ)$⁠, the primitive shape. Primitive shapes are shaded in all figures herein.

In addition, we can choose the image location of three pre-image points under the mapping f(z), all of which are labeled as black circles in Fig. 2(b). Then, f(z) maps the domain $Wπ/4−$ onto its image $Ω−$ and maps the three pre-image points to the three image points.

Now consider the region $W=Wπ/4+∪Wπ/4−$⁠, in the pre-image domain, which is symmetric about the real axis and illustrated on the left of Fig. 2(b). It is clear that W meets the criterion of the Schwarz reflection principle, and so it is guaranteed that the domain of analyticity of f(z) may be extended from $Wπ/4−$ to W. Moreover, it is guaranteed that the analytic extension of f(z) defined over W satisfies Eq. (2). It is thus clear that the image of W, under the analytic extension of f(z), is $Ω=Ω−∪Ω+$⁠, as drawn on the right of Fig. 2(b).

To extend the mapping domain further, we first rotate the pre-image W from Fig. 2(b) about the origin by $π/4$ rad clockwise: specifically, the new map is written in terms of the old as $exp(−iπ/4)f(exp(iπ/4)z)$⁠. We next apply the Schwarz reflection principle about the real axis—again taking $W−$ to be a sector of $π/4$ rad in the unit circle (below the real axis) and $Ω−$ to be one primitive sector below the real axis—from which we obtain the extended map illustrated in Fig. 2(c). At this point, we have constructed a map that takes a wedge spanning $3π/4$ rad in the unit circle to an object containing three primitive shapes (see Fig. 2(c)).

Repeating the procedure of a rotation followed by a sector reflection, one may achieve a mapping of the full unit circle onto the shape in the right panel of Fig. 2(d). Generally, if a body is composed of N primitive shapes, one must perform N − 1 sector reflections of the primitive shape to construct a map from the unit circle. For example, a total of seven reflections are used to generate the map in Fig. 2(d). Note the geometric requirement that N be even.

In Fig. 2(d), the equal length boundary segments labeled 1–8 on the unit circle boundary are mapped by f(z) to the corresponding labeled segments in the right panel. Despite the fact that the Riemann mapping theorem generally only guarantees three degrees-of-freedom in a conformal map, symmetry has allowed us to fix the location of nine points: eight boundary points and the origin in Fig. 2(d).

More generally, if a Jordan curve $∂B$ may be constructed from N sector reflections of a primitive edge, then there exists a conformal map from the interior of the unit circle to $B≡int(∂B)$ which maps each of N equal length arcs on the unit circle to a primitive edge in $∂B$⁠.

It follows that if boundary conditions are specified (and unchanging) along each of the N primitive edges in $∂B$⁠, then the shape factor problem in B is conformally equivalent to the problem interior to the unit circle having the same boundary conditions specified on N arcs of equal length. The conformal transformation between the two domains is given by f(z) as constructed in Fig. 2(d).

We now move to consideration of the boundary value problem on the exterior region to B, $ℂ\B$⁠, satisfying the same boundary conditions on $∂B$⁠. We proceed to show the conformal equivalence between the interior and exterior boundary value problems under the following assumptions: (1) B may be constructed by sector reflections of a primitive shape and (2) the boundary conditions on $∂B$ do not change along any primitive edge.

Consider again a Jordan curve $∂B$ which may be constructed by N − 1 sector reflections of a primitive edge. For illustration, let this be the same boundary curve $∂B$ as considered in Sec. 3.1 (Fig. 2(d)). We now examine the heat transfer problem exterior to that curve, in the region $ext(∂B)$⁠, which is shaded on the right of Fig. 5(c). Since the shape factor is invariant under conformal mapping, we may establish the equality of the interior and exterior shape factors, relative to the curve $∂B$⁠, by showing that both the exterior and interior problems can be mapped onto the same problem on the interior of the unit disk (illustrated in Fig. 2(d)). We now proceed to demonstrate this fact.

We now construct the conformal map from the unit circle interior to $ext(∂B)$⁠, as outlined in Fig. 5. First, consider the curve labeled $1/Γ$ on the right of Fig. 5(a). This curve is constructed by transforming each point in the primitive curve Γ, from Fig. 2(a), according to the mapping $z→1/z$⁠; the resultant primitive curve is denoted $1/Γ$⁠. By applying the arguments of Sec. 3.1 to this new primitive curve, one can construct a mapping g(z) from the unit circle interior to the interior of the shape in the right side of Fig. 5(b), which is itself constructed through sector reflections of $1/Γ$⁠. The boundary correspondence under the mapping g(z) is labeled. Note that the boundary correspondence in Fig. 5(b) is now clockwise (compared to counterclockwise in Fig. 2(d)) because here the Schwarz reflection principle was applied to the sector $1/Γ$ in Fig. 5(a) which was above the real axis. In a final step, the reciprocal map $1/g(z)$ sends the interior of the unit circle to $ext(∂B)$⁠, as illustrated in Fig. 5(c). The boundary correspondence is labeled, the reciprocal map having reinstated the clockwise order.

As before, if boundary conditions are unchanging over each primitive edge, the exterior shape factor problem is conformally equivalent to the problem interior to the unit circle having the same boundary conditions on equiangular arcs. Figures 2(d) and 5(c) thus reveal that, under the same boundary conditions on $∂B$⁠, the interior and exterior shape factor problems may both be mapped to the same problem on the unit circle interior. Thus, the interior and exterior shape factor problems are conformally equivalent and possess the same shape factor and S_e = S_i.

Note that one may apply similar arguments to construct a mapping directly between the interior and exterior problems. However, we find it instructive to introduce the auxiliary problem on the unit circle interior and to show the conformal equivalence there. It should be clear to the reader that the map $1/g(f−1(z))$ directly maps the interior problem (right side of Fig. 2(d)) to exterior problem (right side of Fig. 5(c)) while preserving the locations of the primitive boundary segments.

The preceding Secs. 3.1 and 3.2, outline a set of criteria whose fulfillment implies the equality of interior and exterior shape factors. At this stage, the more mathematical reader will beg the authors to state these criteria concisely in the form of a theorem. Consider first a definition.

Consider a partition of the complex plane, $ℂ$⁠, into some positive whole number, N, of equiangular wedge sectors centered on the origin. For example, the case of N = 8 is depicted in the left of Fig. 2(d), provided each gray line extends to complex infinity (⁠$|z|→∞$⁠). We call this arrangement an N-sector partition of the complex plane.

Theorem 2 (Interior/Exterior Shape Factor Equality). Consider an N-sector partition of the complex plane, for an even value$N≥4$. Now consider a Jordan curve$∂B$constructed from N − 1 sector reflections of a primitive edge. Suppose that each primitive edge of$∂B$is subjected to either a zero Neumann (adiabatic) or a constant Dirichlet (isothermal) boundary condition, with the Dirichlet constant restricted to one of two temperatures. Additionally, Dirichlet sections at different temperatures must be separated by at least one adiabatic section. Then, the shape factors for heat transfer in the domains$int(∂B)$and$ext(∂B)$are equal.

Note that we require N to be greater or equal to four because the boundary conditions may not change along any primitive edge. Since the simplest heat transfer problem having a finite shape factor comprises two isothermal segments separated by two adiabatic segments, a minimum of N = 4 is required to specify all four segments in accordance with the boundary condition criterion.

We now explain, using the theoretical tools developed in Secs. 3.1 and 3.2, why the boundary value problems analyzed by Lienhard [8] happened to have S_i = S_e. We shall show that all cases considered therein were within the scope of Theorem 2. We also discuss more general shapes and Yin–Yang bodies.

We now describe how the rectangle violates the assumptions laid out in Theorem 2. Any rectangle may be constructed from successive reflections of a smaller interior similar rectangle (the primitive shape) as depicted in Fig. 6(a). Theorem 2 thus guarantees the equality of interior and exterior shape factors if boundary conditions are specified along the associated primitive edge, as illustrated in Fig. 6(b). For the rectangle shown in Fig. 1(a), however, each boundary conditions is specified along portions of two primitive edges (e.g., the entire top), and so Theorem 2 is not applicable.

The square interior domain comprises eight of the primitive triangles shaded in Fig. 3(a). It follows from the developments of Secs. 3.1 and 3.2 that the mapping from the square's interior to its exterior will preserve the eight primitive edges (segments connecting each pair of adjacent black circles in Fig. 3(a)). Lienhard [8] numerically tested two configurations of boundary conditions on the square. Both examples considered therein specified boundary conditions unchanging along each primitive edge, in accordance with the assumptions of Theorem 2, and it was indeed found that S_i = S_e.

Nonetheless, the equality of interior and exterior shape factors does not generalize to arbitrary problems. Symmetries of both the domain boundary shape and the boundary condition specification are required to guarantee the equality of interior and exterior shape factors: (1) the shape interior $int(∂B)$ must be constructable from sector reflections of a primitive shape; and (2) boundary conditions must be unchanging along each primitive edge.

A regular K-gon can be constructed from $N=2K$ copies of a primitive triangle; we illustrate the primitive triangles for the cases K = 4 and K = 5 in Fig. 3. If only one boundary condition is specified as unchanging along each primitive edge, then the interior and exterior shape factors are guaranteed to be equal, in accordance with Theorem 2.

Specifically for regular K-gons, primitive edges are half-faces. Thus, a single boundary condition must be specified along the entirety of each half-face to ensure that S_i = S_e. For example, the square of Sec. 4.2 has K = 4, but boundary conditions are allowed to vary between its $N=2K=8$ primitive edges (half-faces), for Theorem 2 to be applicable and thus to ensure the equality of interior and exterior shape factors.

To validate this somewhat surprising result, we use finite element method (FEM) to compute the interior and exterior shape factor of the hexagon with half-heated faces in the last row of Table 1. Here, the top and bottom faces of the hexagon have opposite halves at different temperatures, while the remainder of the hexagon boundary is taken to be adiabatic. The computed interior and exterior shape factors are shown together with the analytical result for the interior problem given by Hersch [6], $Si=1/3$⁠. The interior value is within 0.03% of the exact result, and the exterior value is within 1.3%. While the exact solution $Si=1/3$ given by Hersch was computed for the interior problem, Theorem 2 indicates the same exact result for the exterior problem, as is supported by our numerical calculations.

For the disk, the map $1/z$ takes the interior problem to the exterior problem with the same boundary conditions (up to a reflection, which does not affect the shape factor). Thus, for any alternating sequence of constant Dirichlet and Neumann conditions around the boundary of a circle, the interior and exterior problems are conformally equivalent and S_i = S_e must hold.

In light of Sec. 4.3, one may also rationalize the equality of the interior and exterior shape factors of the circle in the following manner. Consider the circle as the limit of a regular K-gon, whose vertices lie on the unit circle, as $K→∞$⁠. For the K-gon, the interior and exterior shape factors are guaranteed to be equal as long as boundary conditions are constant along each of the 2K primitive edges. In the limit of $K→∞$⁠, the length of each primitive edge tends to zero. In this limit, any finite segment of the circle can be decomposed into a union of primitive segments. Thus, for any shape factor problem on the circle, with finite boundary segments possessing different boundary conditions, it is guaranteed that S_e = S_i.

Note that our criteria, which guaranteed S_i = S_e, do not require shapes to be regular polygons—or any other simple shape for that matter. The first criterion (geometric criterion) is that the domain may be constructed from sector reflections of a primitive curve confined to a sector of angle $2π/N$ with $N≥4$ being some even integer. Some examples of more complex objects are given in Fig. 4, with primitive shapes shaded. Note that for Theorem 2 to apply, ensuring the equality of interior and exterior shape factors, applied boundary conditions must be unchanging along each primitive edge.

We here demonstrate how Schwarz reflections may be used to prove another interesting symmetry of shape factors, the Yin–Yang property [10], which is outlined as follows.

Suppose a Jordan curve $∂B$ has an axis of reflectional symmetry, as shown in Fig. 7. The boundary on one side of the axis contains an isothermal segment and an adiabatic segment; on the reflected side, the isothermal and adiabatic conditions have been interchanged. When a shape factor problem embodies all the aforementioned symmetries, we say it embodies the Yin–Yang symmetry. Lienhard [10] stated that Yin–Yang problems have a shape factor of unity, although no formal proof was given.

Here, we use Schwarz reflections to provide a proof that all such shape factor problems are conformally equivalent to a problem on the square which possesses S = 1. Consider a shape factor problem embodying the Yin–Yang symmetry as in Fig. 7(a). Now consider single half of the domain (shaded), as defined by a cut made along the axis of symmetry (gray dotted line). By the Riemann mapping theorem, there exists a mapping between the shaded region in Fig. 7(a) and the shaded half of the square (Fig. 7(b)), wherein we are free to dictate the image location of three boundary points. In particular, we choose to specify the images of the three following boundary points (shown with black circles): the two intersections of $∂B$ with the reflection axis, and the point where the isothermal boundary segment adjoins the adiabatic boundary segment. We choose as the images of these three points the three vertices of the square also indicated by black circles. The boundary condition correspondence is colored in both panels. The Schwarz reflection principle then implies the image of the remainder of the domain (white region), for which the boundary correspondence is also indicated in Fig. 7. Since the shape factor of Fig. 7(b) is trivially S = 1, the conformal invariance of the shape factor shows that S = 1 in Fig. 7(a) as well.

By a similar argument to Sec. 3.2, the proof also holds for the exterior Yin–Yang problem, so that the interior and exterior Yin–Yang shape factor problems must have $Si=Se=1$⁠.

Finally, we note that the Yin–Yang problem shows that, although our criteria in Theorem 2 are sufficient to guarantee S_i = S_e, they are not necessary criteria: many Yin–Yangs do not meet the criteria of Theorem 2, but all have equality of their interior and exterior shape factors.

We now show several explicit results to support our conclusion.

Table 1 contains several cases for which exact interior shape factors are known. Interior and exterior simulations are shown, along with the computed shape factors. The interior FEM results are accurate to 0.05% or better (see Appendix  A). The exterior FEM results are accurate to about 1.5%, with most of the error attributable to a finite outer domain. In all cases, the exact, interior, and exterior results agree to within the apparent accuracy of the FEM computations. Additional numerically validated K-gons are shown in Appendix  D.

Note that in the third row in Table 1 the two upper segments have the same temperature, but are not connected. Theorem 2 applies to such cases without limitation.

The exact values for the first, second, and fourth cases in Table 1, S = 1, arise because these problems have Yin–Yang symmetry (Sec. 4.6). The exact results for third and fifth cases were found by Hersch for the interior geometries [6]. We have proven herein that Hersch's exact values for the interior domain are also those for the exterior problem. Thus, we label these exact results $Sexact$ since there is distinction between the interior and exterior shape factors. Our numerical results support our proof.

The compass rose is a design traditionally used to display the cardinal directions on a map, consisting of four orthogonal arms of one length and four of another at a 45 deg rotation. In the language of this paper, the compass rose has an N = 8 symmetry with a primitive element containing one-half arm of each length (Fig. 3(a)).

To further demonstrate the applicability of our theorem, we have simulated heat conduction inside and outside the rose for two arrangements of boundary conditions compatible with Theorem 2 (see Table 2). The first case specifies isothermal boundary conditions on four identical sections of the boundary, with edges in the upper right-hand and lower left-hand quadrants at different temperatures. This case embodies the Yin–Yang symmetry so that $Sexact=1$⁠.

The second case has isothermal boundary conditions specified in an asymmetric fashion. The two primitive elements in the upper right-hand quadrant have edges at one temperature, and one primitive element in the lower right-hand quadrant a different temperature. (The theoretical value for this case is given in Sec. 6.)

The numerical value of S_i for the first case is within 0.12% of the theoretical value. The numerical value of S_e is 2.6% below the theoretical value, primarily owing to the difficultly in capturing the very strong singularities of the heat flux at the tips of the long arms. A further difficulty with this, and other, exterior computations is the need to truncate the infinite domain with an outer boundary of sufficient size (see Appendix  A.2). If nothing else, these difficulties illustrate the great value in knowing that S_e can be found through the much easier computation of S_i. The interior agreement is similar for second case, with the outside value lower than the theoretical value by 3.5%.

The explicit Schwarz–Christoffel transformations between the unit circle and the K-gon interior and exterior show that the interior and exterior problems are conformally equivalent when each K-gon face adopts a single boundary condition [11]. These transforms are discussed briefly in Appendix  B. While the Schwarz–Christoffel maps show that S_e = S_i when boundary conditions do not vary over each full face, our results in Sec. 3.2 indicate a less restrictive requirement: boundary conditions must not vary over each half-face.

Note that the definition of the elliptic function in Eq. (5) varies between authors; for example, the elliptic function in mathematica, $KMA$⁠, is related to our definition by $K(x)=KMA(x2)$⁠. In the context of Laplace problems, we find the definition presented in Eq. (5) to be most common. This formula is proved in Appendix  C.

To demonstrate the utility of our formula, we now compute the shape factors for several nontrivial problems; the values are verified numerically using FEM analysis in Appendix  D. For example, the first pentagon in Table 3 has N = 5, N_H = 1, N_A = 2, and N_C = 1, in which case our formula gives $S(1,2,1,5)=0.896256$⁠, which agrees with the FEM calculation to 0.005%. As another example, the last hexagon in Table 3 has $S(2,1,2,6)=1.27926$⁠, within 0.02% of the numerical result. The compass rose problem illustrated in the bottom row of Table 2 has N = 8, N_H = 1, N_A = 3, and N_C = 2, and our formula (3) returns $S(1,3,2,8)=0.819644⋯$⁠, within 0.12% of the numerical value in Table 2. The compass rose has many exterior corner singularities which are hard to resolve with FEM, as was noted in Sec. 5.2, emphasizing the utility of Eq. (2).

where we have used $K′(2−1)/K(2−1)=2$ [12]. Since $1/2=0.707107⋯$⁠, the numerical result in Table 4 agrees with the analytical result to within 0.03%.

The new analytical solution, Eq. (3), is not restricted to polygons or star shapes. As long as a body has an N-fold symmetry, and the quadrilateral boundary conditions are specified along full entire segments of the primitive element, Eq. (3) is valid. For example, consider the puffy snowflake in Fig. 4(b). The shape has 16 primitive edges. Suppose one edge is set to T₁. Then, proceeding clockwise, let the next five primitive edges be adiabatic. Continuing clockwise, let the next four primitive edges be isothermal at T₂, with the remainder adiabatic. The shape factor is then $S(1,5,4,16)=0.6869⋯$⁠. One may proceed similarly to find the shape factor in a wide variety of problems including all of those in Fig. 4.

Our analytical solution holds whenever shapes have an appropriate symmetry, but it does not rely on the specific details of the primitive elements. Thus, in even very complicated geometries that are extremely difficult to tackle numerically (for example, Fig. 4(c)), Eq. (3) holds. The formula may thus be used as a benchmark for numerical methods meant to deal with such extreme geometries [13]. Further, as a result of Theorem 2, Eq. (3) yields the shape factor exterior to such complex shapes, for which numerical solutions would be very challenging to obtain.

The results presented in Sec. 3 have precise mathematical analogues in fluid flow through porous media and in electrical capacitance, which we now briefly outline.

The present results apply to flow through a porous media after the following modifications: k is replaced by the hydraulic conductivity σ; $ΔT$ is replaced by the pressure differential between isobaric segments $ΔP$⁠; the heat flux Q is replaced by the volume flux Q_f; and the shape factor is replaced a flow shape factor S_f. Note also that in the fluid flow context, thermal conductors and insulators are replaced by isobaric segments and impermeable boundaries, respectively. In the fluid flow context, Theorem 2 guarantees the equivalence of the flow shape factors interior and exterior to a body.

Our results apply to the electrical capacitance, C, after the following substitutions: $ΔT$ is replaced by electrical potential difference between conducting segments $ΔV$⁠; Q/k is replaced by the conductor surface charge q; and the shape factor is replaced by the electrical capacitance C. Thermal conductors and insulators must be replaced by electrical conductors and insulators, respectively. Theorem 2 guarantees the equivalence of the electrostatic capacitances interior and exterior to a body.

Interior and exterior shape factors are not universally equal, contrary to a previous report. In this paper, we first demonstrated why the interior and exterior shape factors are not always equal. We then provided a set of criteria sufficient to guarantee equality. In particular, the interior and exterior shape factors will be equal if specific symmetries are present: (1) the shape boundary must be constructed from sector reflections of a primitive sector edge; and (2) the boundary conditions—either zero Neumann (adiabatic) or constant Dirichlet (isothermal)—must not change along each primitive edge. This result is stated precisely in Theorem 2.

Objects that meet the conditions of the theorem include not only regular polygons and disks but also much more complex shapes like those shown in Fig. 4. These results have direct analogies in flow through porous media and in electrostatic capacitance, as outlined in Sec. 7.

We have further derived an exact expression for the shape factor of appropriately symmetric N-sided objects that have contiguous sides at one temperature separated from contiguous sides at another temperature by one or more adiabatic sides. This expression, Eq. (3), applies not only to regular polygons and star shapes but also to any other shape that has N-fold symmetry. We have validated this powerful result with FEM analysis of multiple configurations.

We have tested our theorem against independent results from finite element simulations and from Schwarz–Christoffel maps. Additionally, a brief tabulation of shape factors in geometries that meet the conditions of Theorem 2 is provided in Appendix  D.

We note that this paper may be useful also as an accessible tutorial for applying symmetry methods to Laplace problems through the exploitation of the Schwarz reflection principle.

K.M. would like to thank Alex Cohen and Darren Crowdy for valuable discussions. Both authors would like to thank Nick Trefethen for useful comments and for pointing out the work of Hersch.

• Massachusetts Institute of Technology (MathWorks Fellowship; Funder ID: 10.13039/100006919).

• STEM Chateaubriand Fellowship, Ambassade de France aux États-Unis

The authors attest that all data for this study are included in the paper.

a, b =

dimensions of rectangle

B =

a planar body (interior of a Jordan curve, $∂B$⁠)

b_k =

prevertices on unit disk

$ℂ$ =

the set of complex numbers

$c1,c2,c3$ =

constants, Appendix  B

$CR$ =

cross-ratio, Eq. (C3)

$C(NH,NA,NC,N)$ =

see Eq. (4)

ext(∂B) =

region exterior to $∂B$

$f(z),g(z)$ =

conformal maps

$Hmax$ =

maximum mesh size for FEM

$int(∂B)$ =

region interior to $∂B$

k =

thermal conductivity (W m⁻¹ K⁻¹)

K =

number of sides in polygon

K(x) =

complete elliptic integral of the first kind, Eq. (5)

$K′(x)$ =

complementary elliptic integral, $K(1−x2)$

L₁, L₂ =

straight lines along each sector boundary extending from the origin to the ends of the primitive edge, Fig. 2(a) 

N =

number of primitive elements composing a shape

$NA,NC,NH$ =

number of contiguous adiabatic, cold, and hot sides

Q =

heat flow (W m⁻¹)

S =

shape factor (-)

$S(NH,NA,NC,N)$ =

see Eq. (3)

T =

temperature (K)

W =

domain in the complex z plane

x =

coordinate along the bottom wall, Appendix  A.1

z =

coordinate in the complex plane

Greek Symbols

α =

angle on disk, see Fig. 8 

γ =

outer section of boundary denoted a primitive edge, Fig. 2(a) 

Γ =

closed curve defined by $Γ=L1∪L2∪γ$⁠, and denoted the primitive curve

δx =

$Hmax/2$

$ΔT$ =

temperature difference (K)

η₁ =

K/[2(K + 2)]

μ_i =

exterior angles as fraction of π

Ω =

range in the mapped plane

Superscripts and Subscripts

e =

exterior value

h =

hexagon

i =

interior value

p =

pentagon

$z¯$ =

complex conjugate of z

For a rectangle with isothermal sides of width a and adiabatic sides of height b, the interior shape factor is exactly $Si=a/b$⁠. The exterior shape factor was computed using matlab's FEM tools. Mesh convergence was evaluated by reducing the maximum mesh size, $Hmax$⁠, until machine memory limits were exceeded (Table 5). Localized mesh refinement was applied at each outside vertex and edge, limiting the maximum mesh to $Hmax/40$⁠. The computation is sensitive to the outer boundary condition of the domain, which was set to a circle of radius 20. The far-field temperature was shown to limit to a dipole distribution in Ref. [4], and that boundary condition was used on the outside edge.

for k = 1. We then add $ΔS$ to S.

The horizontal dimension of the rectangle was held fixed at a = 2 and b was varied. Table 5 shows the results. The computed value for $a/b=1$ is within 0.5% of the exact value, 1, when $Hmax=0.02$⁠. Obviously, $Se(a/b)=1/Se(b/a)$⁠. This relationship is satisfied to about 1% over the range considered. For the two largest a/b, the convergence is less, but likely better than 2%. The values for $Hmax=0.02$ are plotted in Fig. 1. In these calculations, the far-field dipole temperature distribution was based on a strength $2Se$ at a separation b.

The exterior shape factors were computed using the same procedure as for the rectangles (with accommodations for the $Sh2-2$ hexagon, Fig. 4, in which an isothermal edge adjoins the bottom edge). The exterior results are within 1.4% of the interior FEM results. The specific choice of outer boundary condition (isothermal, adiabatic, or dipole) affects the results. For the maximum radii and minimum mesh sizes achievable, isothermal outer boundaries run a fraction of a percent higher than dipole boundaries, and adiabatic boundaries run lower by 1–1.5%.

The interior FEM calculations were straightforward, and comparison to known exact values and reciprocal relationships, together with grid convergence studies, places their accuracy at about 0.05%. The reciprocal relationship [6] states that, when isothermal and adiabatic boundaries are interchanged, the resulting shape factor is the reciprocal of the original shape factor. Thus, in Fig. 4, the following relationships are met, as is evident upon comparison of the respective flux plots: $Sp1-1=1/Sp2-1, Sh1-1=1/Sh2-2,and Sh1-1R=1/Sh3-1$⁠. The reciprocal values agree to 0.05–0.1%.

For K > 4, the interior flux has a weak singularity at corners that join isothermal and adiabatic segments, but grid refinement was sufficient to account for these. For the hexagon with half of a side isothermal (Table 1, last item), a stronger interior singularity is present at the transition point from isothermal to adiabatic (see Ref. [4], Sec. 6.2), and a numerical correction was applied using methods similar to Appendix  A.1.

where we take $c1=1$ and $c2=0$⁠.

Since the same points, the prevertices, are conformally mapped to the vertices of either the interior or exterior K-gon, and since isotherms map to isotherms and adiabats to adiabats, the boundary conditions are identical for the interior and exterior problems. It follows immediately see that the shape factor, a conformal invariant, is the same for the interior and the exterior problems on the K-gon. (Note that we have not shown that the interior and exterior K-gons have the same dimensions or the same vertices—neither question is relevant to our result.)

(The shape factor is the inverse of Hersch's conformal modulus.) Equation (C1) is readily evaluated to find the shape factor for any angle α.

Now consider a quadrilateral with N-fold symmetry, where boundary conditions do not change along each primitive edge. Four points demarcate the junctions of isothermal and adiabatic sections on the quadrilateral. By the construction of Sec. 3, each primitive edge of the shape under consideration will be mapped onto a segment on the unit disk spanning $2π/N$ rad. This mapping takes four boundary points on the quadrilateral onto four distinct points of the form $exp(i2πm/N)$⁠. The four values of m are determined as follows. Suppose that, in clockwise order, the boundary conditions are N_H primitive sides isothermal at T₁, followed by N_A adiabatic primitive sides, followed by N_C primitive sides isothermal at T₂, followed by $N−(NH+NA+NC)$ adiabatic sides. Then, the four boundary points are located at $z1=1, z2=exp(−i2πNH/N), z3=exp [−i2π(NH+NA)/N]$⁠, and $z4=exp [−i2π(NH+NA+NC)/N]$⁠.

For some value of α, the unit disk can be mapped onto itself while taking each boundary point of the quadrilateral to the four boundary points in Fig. 8. Since the Riemann mapping theorem only allows one to set the image of three points, the mapping is only possible for a specific value of α that is determined as follows.

In this appendix, we provide a brief tabulation of shape factors for a few additional objects for which interior and exterior shape factors are equal. This catalog is by no means exhaustive.

Yin–Yang bodies. A Yin–Yang body has an axis of symmetry about which isothermal and adiabatic boundaries are interchanged (Sec. 4.6). Every Yin–Yang body is conformally equivalent to a square and has $Si=Se=1$⁠.

Disks. A disk with symmetric isothermal angular sectors is shown in Fig. 8. Equation (C1) provides the shape factor as a function of the angle α. Shape factors for several angles are given in Table 6; and the angles that yield several shape factors are given in Table 7. For all disks, $Si≡Se$⁠.

The case α = 30 deg has the analytical solution $S=1/3$ [6]; and, because the case α = 150 deg simply interchanges the boundary conditions, reciprocity [6] shows it to have the analytical solution $S=3$⁠. Similarly, the shape factor for α is the reciprocal of the shape factor for $180deg−α$⁠. The case α = 90 deg is a Yin–Yang body with S = 1. The results are accurate to the number of digits shown in the tables.

Regular$K$-gons. Interior flux plots for all unique regular pentagon and regular hexagon problems are shown in Table 3. The isothermal sides are indicated with heavy lines, and the other sides are adiabatic. In all cases, the bottom side has a different temperature than the top sides. The specific boundary conditions are apparent from the flux plots. The exact shape factors from Eq. (3) are shown alongside the numerical results obtained through FEM analysis.

The FEM results are accurate to 0.03% or better. The exact value for $Sh2-1$ stems from the Yin–Yang symmetry, while the exact result for $Sh2n-1$ is due to Hersch [6].

The exterior shape factors were computed numerically and, in each case, we found an agreement of 1.4% or better with the interior FEM value; the difference is within the accuracy of the exterior computation. (FEM convergence was discussed in Appendix  A.) These results support the equality of interior and exterior shape factors for regular K-gons, and they serve to further confirm Eq. (3).

Shape factors on half-squares. If each side of a square is divided into two equal parts, we have eightfold symmetry. Consider the shape factor when two half-sides are at different temperatures and are separated by one or more adiabatic sides. Four unique arrangements exist, with other arrangements being simple rotations of the others.

To construct the four unique cases, we hold the bottom-left half-side at T₁ and vary the side at T₂. Table 4 shows the possible combinations with the interior shape factors calculated by FEM as before.

Cases TL and UR both have $Si=1/2$ because these two shapes are conformally equivalent. By mapping the square problems onto the unit circle, using the method laid out in Sec. 3, it is readily seen that both TL and UR correspond to the same problem on the unit circle and are thus conformally equivalent. Conformally equivalent problems have the same shape factor.

The half-square geometry is also conformally equivalent to an octagon. Therefore, these shape factors also apply to octagons with two isothermal faces separated by the appropriate number of adiabatic faces.