Casimir Energy of Sierpinski Triangles

Home , Sierpinski carpet, Vicsek fractal

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Cross section Dirichlet Neumann EM with a numerical value bc 0.0237188, ∼ 0.0237 0.0613 0.0375 Equilateral Tr. − − a2 a2 a2 3√3 3 1 b = , b = , b = , (6) 0.0756 0.0944 0.0187 2 2 1 0 Hemiequilateral Tr. − − 8π −8π 6π a2 a2 a2 m 0.0263 − 0.0454 − 0.0190 of Eq.(1), where ψ( )(z) is the polygamma function of Isosceles Tr. 2 2 2 a a a order m and ζ(z) is the Riemann zeta function. The 0.00483 − 0.0429 − 0.0381 above evaluation was achieved using the mode summa- Square 2 2 2 a a a tion method, which presumes that the Casimir energy of a closed Dirichlet cavity is completely determined by Rectangle Refer Ref. [3]. the modes in the interior of the cavity alone [3]. This should be contrasted with field theoretic methods of Lif- TABLE I. Casimir energy per unit length for cylinders of five shitz et al. [6] and Schwinger et al. [7] that incorporate cross sections from Ref. [3], referred to as E∆ and E in this both the interior and exterior modes in the evaluation. paper. The cutoff independent finite part is presented. The The Casimir energies for the five geometries of Ref. [3], numbers correspond to the constant bc in Eq. (1) for the re- for empty triangles and rectangles or squares, due to in- spective cross sections, presented here to three significant dig- terior modes only, referred to as and in this paper, its without rounding. The second, third, and fourth, columns E∆ E correspond to the boundary conditions imposed on the fields. are summarized in Table I. Before we proceed with our discussion, we present the results for the finite part of the Casimir energy for each of the five geometries presented in Fig. 1, which are ex- where δc is the fractal dimension of the Casimir energy pressed in terms of and of Table I. The expression per unit length of the fractal cavity. E∆ E s = 4 /11 is universal for all Sierpinski triangles, The central theme of this paper is to use the scaling E − E∆ and s = 9 /71 is universal for all Sierpinski rect- arguments and the property of self-similarity introduced angles,E and− areE not restricted to the five geometries of in Ref. [4] to derive the Casimir energies of Sierpinski Ref. [3]. However, we often confine the analysis to the triangles and Sierpinski rectangles. We also introduce a five geometries of Ref. [3], because exact expressions were class of fractals for which the energy does not scale as in- derived for the Casimir energy per unit length for these verse length square, which leads us to introduce a fractal geometries there. dimension for the Casimir energy. One usually associates fractal dimensions to geometrical quantities like perimeter and area, but being able to introduce fractal dimensions to Casimir energy for a class of fractals directly re- lates to the conventional wisdom that Casimir energy of II. SIERPINSKI TRIANGLE cavities, satisfying perfectly conducting boundary conditions (or Dirichlet boundary conditions for scalar fields), The Sierpinski triangle is self-similar. That is, it con- is purely geometrical. It should not be very surprising, sists of copies of the scaled-down versions of itself. Fig- because energy has been shown to exhibit fractal nature ure 2 shows the Sierpinski triangle of side length a, which before. For example, the Hofstadter butterfly is a fractal may be viewed as comprised of three Sierpinski triangles that represents the energy of Bloch electrons in a mag- of side length a/2. netic field [5]. We shall consider an equilateral triangle even though most of our discussion holds true for an arbitrary triangle. The Casimir energy of an equilateral triangular cylinder on which a scalar field satisfies Dirichlet boundary conditions was calculated exactly, in closed form, in Ref. [3]. This involves the Casimir energy of five cylindrical cross sections, namely an equilateral triangle, a hemiequilateral triangle, an isosceles right triangle, a square, and a rectangle, see Fig. 1. For all five geometries the authors of Ref. [3] have shown that the Casimir energy per unit length obeys the Weyl law in Eq. (1). Using Ref. [3], the FIG. 2. Sierpinski triangle. The white regions in the inte- Casimir energy per unit length of an equilateral triangle rior are triangular cavities, each of which contributes to the is described by the parameters Casimir energy per unit length of the Sierpinski triangle. The matter bounding each of the triangles (in blue) are perfectly conducting for the case of electromagnetic fields. 1 √3 (1) 1 (1) 2 8 c b = ψ 3 ψ 3 ζ(3) , (5) −72 " 9 − − π # n o 3

A. Area and δ2 C. Corner angle and δ0

Using the self-similarity of a Sierpinski triangle we can The interior corner angles of a Sierpinski triangle sat- write the following recursion relation for the area As(a) isfy of the cavities inside a Sierpinski triangle: a a a a Cs(a)= C∆ +3Cs , (16) As(a)= A +3As , (7) 2 2 ∆ 2 2 which leads to the series where A∆(a) is the area of an equilateral triangle of side a a 2 a length a. Using Eq. (7) recursively in itself we obtain the Cs(a)= C∆ +3 C∆ +3 C∆ + .... (17) series 2 22 23 a a 2 a The corner term for a triangle is given by [3] As(a)= A +3 A +3 A + .... (8) ∆ 2 ∆ 22 ∆ 23 π αi Then, using the scaling relation of the area of a triangle, C∆(a)= , (18) i αi − π a 1 X A∆ = A∆(a), (9) 2 22 where αi are the angles of a triangle, is independent of the scale a. Thus, we can derive we obtain the area of the Sierpinski triangle in Fig. 2 to be exactly equal to the area of a triangle. That is, 1 Cs(a)= C (a), (19) −2 ∆ As(a)= A∆(a). (10) 2 2 after using the divergent sum 1 + 3 + 3 + ... = 1/2. Since the area of a triangle A∆(a) scales like a , dimen- − sional analysis of Eq. (10) implies that the fractal dimen- We learn from Eq.(19) that the fractal dimension for the sion of the area, deﬁned in Eq.(3), of the Sierpinski tri- corner angle of the Sierpinski triangle in Fig. 2 is angle is δ0 =0, (20) δ2 =2. (11) because the corner angle of a triangle C∆(a) is scale independent. B. Perimeter and δ1

D. Casimir energy and δc Similarly, the interior perimeter of the cavities inside a Sierpinski triangle satisﬁes the recursion relation Using the decomposition of Casimir energies into a a single-body energy and the respective interaction en- Ps(a)= P∆ +3Ps , (12) 2 2 ergy between the bodies [9], the Casimir energy per unit length of a Sierpinski triangle s(a) can be decomposed where P∆(a) is the interior perimeter of a triangular cav- E ity of side length a and Ps(a) is the sum of the inte- as rior perimeter of all the individual cavities constituting a a s(a)= +3 s , (21) the Sierpinski triangle. The series constructed from the E Eint 2 E 2 recursion relation, after using the scaling argument for s P∆(a), is divergent and leads to where 3 (a/2) is the single-body Casimir energy of three SierpinskiE triangles of side a/2 in Fig. 2, and (a) is Eint Ps(a)= P∆(a) (13) the interaction energy between the three Sierpinski tri- − angles. Arguably, in general, the interaction energy int after using the divergent sum depends on both the interior and exterior modes. But,E 1 3 32 the Casimir energies of the cavities we are considering are + + + ... = 1, (14) all due to interior modes. Thus, for consistency, we shall 2 22 23 − presume the interaction energy involves only the interior which can be deduced using the property of self-similarity modes. We shall further justify the consistency of this of the series [8]. Ignoring the counterintuitive nature of presumption in the following discussion. a negative perimeter, we learn from Eq.(13) that the Using Eq. (21) recursively we obtain the series fractal dimension for the perimeter, deﬁned in Eq. (3), of a a 2 a the Sierpinski triangle in Fig. 2 is s(a)= +3 +3 +.... (22) E Eint 2 Eint 22 Eint 23 δ1 =1, (15) Thus, the evaluation of the Casimir energy reduces to because the perimeter of a triangle P (a) scales like a. computing the interaction energy . ∆ Eint 4

The Casimir energy per unit length for the Sierpinski extension of a cylinder with arbitrary triangular cross section will also be given using Eq. (25). The expression for ∆, without the cutoff dependent part, for the three cylindersE with triangular cross sections, for which closed- form solutions has been achieved, has been summarized in Table I. FIG. 3. Four triangles constituting the Sierpinski triangle. III. INVERSE SIERPINSKI TRIANGLE Dirichlet boundary conditions requires a scalar field to be zero on the boundary. This restriction essentially sep- We define the inverse Sierpinski triangle as the object arates the physical phenomena on the two sides of the obtained by swapping the empty space with the perfectly boundary. Thus, the modes and the associated physical conducting material in the Sierpinski triangle. In Fig. 2 phenomena inside Dirichlet cavities are essentially inde- this is obtained by swapping the white color representing pendent of its surroundings. Extending this argument to empty space with blue color representing perfectly con- Sierpinski triangles we learn that the interaction energy ducting material. See Fig. 4. The outer region in Fig. 4 between two or more Sierpinski triangles is independent is unbounded. of the internal structure of each of them. We can thus infer that the interaction energy of the three Dirichlet Sierpinski triangles in Fig. 2 is identical to the interaction energy of the three Dirichlet triangles in Fig. 3. We can determine the total energy of the four triangles in Fig. 3 in two independent methods. In the first method we argue that the energy is the sum of the four trian- a gular cavities, 4 ∆ 2 . In the second method we argue that the total energyE is the sum of the energies of the three outer triangles, 3 a , plus the interaction en- E∆ 2 ergy int(a/2) between the three triangles. That is, E FIG. 4. Inverse Sierpinski triangle. It is obtained from the a a a Sierpinski triangle in Fig. 2 by swapping the empty space with 4 ∆ =3 ∆ + int . (23) perfectly conducting material there. The blue region extends E 2 E 2 E 2 to infinity. This immediately suggests that the interaction energy of three outer triangles is completely given by the energy of The area of the inverse Sierpinski triangle satisfies the the inner triangle, relation a a n a = =22 (a), (24) As(a)=3 As n , (28) Eint 2 E∆ 2 E∆ 2 for any non-negative integer n. Presuming that this area where in the second equality we used the fact that the scales like aδ2 we obtain the relation Casimir energy of a triangle ∆(a) scales like the inverse n square of length. E 3 As(a)= As(a), (29) Using Eq. (24) in Eq. (22) we derive the Casimir energy (2n)δ2 of the Dirichlet Sierpinski triangle in terms of the Casimir energy of the equilateral triangle as for any positive integer n. The central idea of non-trivial fractal dimensions stems from Eq. (29) and its solutions. 4 The trivial solution is As(a) = 0, which can be envisioned s(a)= (a) (25) E −11E∆ to be a possible scenario by extending Fig. 4 (drawn to 5 iterations) to infinite iterations. This trivial solution using the divergent sum agrees with the notion that the perfectly conducting ma- 1 terial in Fig. 4 fills all space in this limit. But, Eq. (29) 1+12+122 + ... = . (26) also admits a non-trivial solution, namely −11 n 3 Since the energy per unit length of a triangle ∆(a) scales 1= , (30) E 2δ2 like inverse length square, Eq. (25) implies that the energy per unit length of the Sierpinski triangle also scales for any positive integer n, which immediately implies that similarly, that is, ln 3 δc = 2. (27) δ2 = 1.58496. (31) − ln 2 ∼ 5

the Casimir energy of a square enclosed between the eight surrounding squares. Thus, we have

a a 2 = =3 (a), (35) Eint 3 E 3 E where (a) is the Casimir energy per unit length of a squareE of side length a. Hence, the Casimir energy of the Sierpinski square is evaluated as 9 c(a)= (a). (36) E −71E The calculation for the Sierpinski rectangle goes through the same procedure, because the length and width have a fixed aspect ratio. The Casimir energy per unit length FIG. 5. Vicsek fractal. for the Sierpinski extension of a cylinder with arbitrary rectangular cross section will also be given using Eq. (36). The expression for , without the cutoff dependent The non-trivial solution here is probably a consequence of part, for the cylindersE with rectangular cross sections, for the non-trivial solution of a divergent series as a regular- which closed-form solutions have been found, are summa- ized sum, which has the trivial solution to be infinity [8]. rized in Table I. The perimeter also satisfies the relation in Eq. (28) The vacuum energy of the inverse Sierpinski square or with the areas A now replaced with perimeters P . The rectangle satisfies the relation corner angles also satisfy Eq. (28). Further, the Casimir n a energy per unit length also satisfies Eq. (28). Thus, we c(a)=8 c n , (37) learn that E E 3 ln 3 which implies δc = ln 8/ ln 3. Since the area, the perime- δc = δ2 = δ1 = δ0 = , (32) ter, and the corner angles, satisfy the same relation, of ln 2 Eq.(37), we also learn that δc = δ2 = δ1 = δ0. Thus, unless we confine to the trivial solution that area, perime- using the same arguments used to derive Eq. (33), the ter, angles, and the energy per unit length are all zero. Casimir energy per unit length of the inverse Sierpinski The fact that the fractal dimensions of all the rele- rectangle or square will not have divergent terms. vant physical quantities, the area, the perimeter, the cor- We can also extend our discussion to non-Sierpinski ner angle, and the Casimir energy per unit length, scale fractals. The Vicsek fractal, illustrated in Fig. 5, is ob- the same way, in conjunction with the generalized Berry tained by starting from a square, dividing it into nine conjecture of Eq.(4) implies that there are no divergent equal squares of one-third side, and removing four of terms in the Casimir energy per unit length of the inverse them. The inverse Vicsek fractal is obtained by swap- Sierpinski triangle. That is, ping the empty space with perfectly conducting material,

δc δc δc such that the Casimir energy is given in terms of the en- δc δc ′ a ′ a ′ a (a)= bc a + lim τ b2 + b1 + b0 , ergies of the individual cavities. The vacuum energy of E τ→0 τ τ τ the inverse Vicsek fractal satisfies the relation ′ ′ ′ δc c = (b + b2 + b1 + b0)a , (33) n a v(a)=5 v n , (38) where b′ , b′ , and b′ are redefined constants with respect E E 3 2 1 0 to the constants b2, b1, and b0, in Eq. (4), to accommo- and the area, the perimeter, and the corner angles, of the date numerical constants inside A(x), P (x), and C(x), Vicsek fractal also satisfy the same relation. Like in the respectively. case of the inverse Sierpinski triangle, Eq. (38) for energy and the corresponding relations admits the non-trivial solution IV. SIERPINSKI CARPET ln 5 δc = δ = δ = δ = 1.46497. (39) 2 1 0 ln 3 ∼ The Sierpinski carpet, or a Sierpinski rectangle or square, is the rectangular version of a Sierpinski trian- Thus, using the generalized Berry conjecture of Eq.(4) gle, see the rectangle and square version in Fig. 1. A we can conclude that the Casimir energy per unit length Sierpinski carpet satisfies the energy decomposition of the inverse Vicsek fractal will not have divergent terms. a a c(a)= int +8 c , (34) E E 3 E 3 V. DISCUSSION which leads to c(a)= int(a/3)/71. Again, to be con- sistent with theE fact that−E we are including only the in- We list the fractal dimensions of the geometries dis- terior modes, the interaction energy (a/3) is equal to cussed here in Table II. For the Sierpinski triangle and Eint 6

Geometry δ2 δ1 δ0 δc the fact that its perimeter scales like a fractal while its Sierpinski triangle 2 1 0 -2 area scales normally. In the Sierpinski triangle, and in the inverse Sierpinski triangle, the cavities are compart- ln 3 ln 3 ln 3 ln 3 Inverse Sierpinski triangle mentalized. These geometries are not simply connected. ln 2 ln 2 ln 2 ln 2 This feature in conjunction with the idea of self-similarity Sierpinski carpet 2 1 0 -2 was the key to our evaluation of the Casimir energies of ln 8 ln 8 ln 8 ln 8 the fractal geometries considered here. Without the fea- Inverse Sierpinski carpet ture of compartmentalization we are unable to calculate ln 3 ln 3 ln 3 ln 3 the Casimir energy of a Koch snowflake as yet. Thus, we ln 5 ln 5 ln 5 ln 5 Inverse Vicsek fractal are unable to analyze the Berry conjecture for a Koch ln 3 ln 3 ln 3 ln 3 snowflake. ln 4 In the literature, the Berry conjecture was formulated Koch snowflake 2 0 ? ln 3 and has been studied for the distribution of the modes of a cavity. Casimir energy is directly related to the sum TABLE II. Fractal dimensions of the area of cavities, δ2, of the of all modes of a cavity, and gets divergent contributions perimeter of cavities, δ1, of the corner angles of cavities, δ0, dictated by the distribution of the modes for large fre- and of the Casimir energy per unit length, δc, for a few geome- quencies. There seems to be an extensive literature on tries. We note that ln 3/ ln 2 ∼ 1.58496, ln 8/ ln 3 ∼ 1.89279, the Weyl law and the associated Berry conjecture. We ln 5/ ln 3 ∼ 1.46497, and ln 4/ ln 3 ∼ 1.26186. The question mark indicates the value that remains to be calculated, the found the review in Ref. [11] very resourceful. Neverthe- Casimir energy per unit length of a cylinder with the cross less, to our knowledge, Berry’s conjecture has not been section of the shape of a Koch snowflake. addressed in the context of Casimir energies before. The only exception seems to be the discussion in Ref. [12] where fractal geometries are discussed in the context of the Sierpinski carpet we have shown that the Berry con- heat kernels, which is a powerful technique used to ex- jecture holds true. But, these are relatively trivial cases tract divergent terms in Casimir energy. Our method because the fractal dimensions for these cases is equal exploits the formalism of Green’s functions, that was de- to the respective topological dimensions. For the in- scribed in the context of self-similar plates in Ref. [4], verse Sierpinski triangle, the inverse Sierpinski carpet, and its correspondence with the heat kernel method is and the inverse Vicsek fractal, even though we encounter well known in the community. In Ref. [3], the method non-trivial fractal dimensions, these dimensions (for area, of mode summation was used, so the connection with perimeter, corner angle, and Casimir energy,) turn out to heat-kernel methods should not be so remote. In our un- be all equal. This, in turn, remarkably, gives no room for derstanding, there is no direct overlap between the dis- the divergent terms in the Weyl expansion. Thus, in the cussions we have presented here with those in Ref. [12], absence of the divergent terms, even though the Berry’s but the connections should be worth pursuing. conjecture holds in principle, we can not conclude this to In the studies on distribution of modes, without con- be a non-trivial verification of the conjecture. cerning Casimir energy, it has been argued in Ref. [13] Berry’s conjecture was probably motivated for fractals that the dimensions of the regions and surfaces should like the Koch snowflake [10], a simply connected domain, be interpreted as the Minkowski-Bouligand dimension in which the perimeter encloses a single continuously con- instead of the Hausdorff-Besicovitch dimension as orig- nected region. The area and the corner angles of a Koch inally proposed by Berry. This was further promoted curve scales normally, but the perimeter scales like a frac- in Ref. [14]. Counterexamples involving domains that tal with a fractal dimension of ln 4/ ln 3 1.26. This dif- are not simply connected were presented in Ref. [15], ference in the scaling behavior between area∼ and perime- but it has been suggested that the conjecture is ex- ter, we believe, will be suitable for studying the Berry pected to hold for simply connected fractals like the Koch conjecture. The methods presented here do not seem to snowflake. These conclusions seem to be in agreement yield the Casimir energy of a Koch snowflake. with our results here in the context of Casimir energies. No doubt, Berry’s conjecture needs to be explored fur- The perimeter of a Koch snowflake scales like Pk(a)= n n ther. 4 Pk(a/3 ), which implies that for a Koch snowflake δ1 = ln 4/ ln3. In contrast, the area of a Koch snowflake Ak(a) ′ is given by Ak(a)= A∆(a)+3Ak(a), expressed in terms of ′ a reduced area Ak(a) that satisfies the recursion relation ACKNOWLEDGMENTS ′ ′ Ak(a) = A∆(a/3)+4Ak(a/3), which leads to Ak(a) = 8A∆(a)/5 and implies that for a Koch snowflake δ2 = We thank Kimball A. Milton for discussions, and for 2. Further, the corner angles ck(a) of a Koch snowflake directing our attention to the Casimir energy calcula- satisfies ck(a)=6π +4ck(a/3), which leads to ck(a) = tions using interior modes alone that was essential for 2π. Thus, for a Koch snowflake δ0 = 0. this work. We thank Mathias Boström for discussions. − It seems that it would be more appropriate to analyze We acknowledge support from the Research Council of the Berry conjecture for a Koch snowflake, because of Norway (Project No. 250346). I.C.P. acknowledges sup- 7 port from Centro Universitario de la Defensa (Grant No. CUD2015-12) and DGA-FSE (Grant No. 2015-E24/2).

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