Fractal bodies invisible in 2 and 3 directions

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [A. Aleksenko and A. Plakhov. Bodies of zero resistance and bodies invisible in one direction. Nonlinearity 22, 1247-1258 (2009)], [A Plakhov and V Roshchina. Invisibility in billiards. Nonlinearity 24, 847-854 (2011)], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.


Introduction
Invisibility has fascinated people's imagination since ancient times: the idea is exploited in folklore, fiction and movies. This "magical" concept is, however, rapidly migrating into the scientific domain. On the cutting edge of the modern developments is the design of metamaterials with special refractive properties, which could, amongst other important applications, ultimately lead to the creation of a real invisible cloak. For an overview of the recent works in this field we refer the reader to our recent article [10]. The aforementioned developments deal with the wave nature of light, and metamaterials are engineered at the nanoscale level. The effects specific to geometrical optics, however, also remain important in modern technology, mostly in cases where the objects are large enough for geometrical optics to dominate the wave effects. Examples include fiber optics, design of lenses (e.g. for photography or DVD readers), and many others.
In this article we are concerned with invisibility in billiards. We consider bodies with a perfectly mirrored surface in a beam of light, or, equivalently, in a flow of non-interacting billiard particles. Invisibility in a direction v (where v is a unit vector) means that any light ray which initially moves along a straight line in this direction, after several reflections from the body's surface will eventually move along the same straight line. Invisibility in a set of directions means that the above is true for any direction from this set. This problem is closely related to the problem of minimal resistance going back to Newton [7]. The latter consists of finding a body, from a given class of bodies, that experiences the smallest possible force of flow pressure, or resistance force. Since the 1990s, many interesting results on this problem have been obtained by various authors (see, e.g., [2,3,4,5,6,8,9]). Bodies of zero resistance in one and two directions are described in [1] and [10] respectively. In both cases it is possible to construct an invisible body by arranging several such bodies together in a specific way. In [10] it was shown that there exist bodies invisible in two mutually orthogonal directions in the three-dimensional setting and that bodies invisible in all directions do not exist.
In this work we continue the study of invisibility and construct bodies invisible in any two directions in two-dimensional space and in three orthogonal directions in three dimensions. Each body in the construction is a union of infinitely many pieces of varying size going to zero, where each piece is a domain with a piecewise smooth boundary. By slightly abusing the terminology, such a union will be called a fractal body, or a solid fractal body. In a preliminary construction (Section 3.1) and in a limiting case of the basic construction (Section 4) some pieces comprising the body are smooth curves (in the 2D case) or pieces of surfaces (in the 3D case). The corresponding body will be called a thin fractal body.
The article is organized as follows: we first reintroduce some definitions and briefly revisit earlier results in Section 2. In Sections 3 and 4 we explain the construction of bodies invisible in two and three directions respectively. Section 5 contains some final remarks and a brief discussion of open problems.

Bodies of zero resistance and invisible bodies
Consider a parallel flow of point particles in R n moving with unit velocity v ∈ S n−1 towards a body B at rest. The flow is so rarefied that the particles do not mutually interact. Particles reflect elastically when colliding with the body's surface and move freely between consecutive collisions.
We deal with bounded (not necessarily connected) bodies composed of a (possibly infinite) number of piecewise smooth fragments. When a particle moving along a straight line and with a constant velocity hits the boundary of B, it is reflected from the latter without loss of speed, and keeps moving along the new linear trajectory until the next collision. All reflections are specular: the angle of incidence just before the collision is equal to the angle of reflection just after the collision (see Fig. 1). In general, it is possible that the particle never leaves the body and keeps bouncing off its sides infinitely; however, Figure 1: The broken line through the points ξ and ξ + is a billiard trajectory in the complement of B.
we only consider such bodies and velocities of incidence v for which almost every particle makes a finite number of reflections. Also note that in some cases the particle may hit a singular point of the boundary. In this case the further movement of the particle is not defined. We consider such bodies and velocities v that the set of initial points ξ for which the motion is undefined (i.e. such that a particle starting the movement at a point ξ in the direction v, eventually hits a singular point) has zero measure.
In view of this description, for almost any ξ ∈ R n , the particle that initially moves freely according to x(t) = ξ + vt, after a finite number of reflections from B moves freely again according to x(t) = ξ + + v + t, where ξ + = ξ + B,v (ξ) and v + = v + B,v (ξ) are measurable functions defined almost everywhere. Fig. 2 (a)).
(ii) We say that the body B is invisible in the direction v, if it has zero resistance in this direction and, additionally, ξ + B,v (ξ) − ξ is always parallel to v (see Fig. 2 (b)).
(iii) Let A ⊂ S n−1 . The body B is said to be invisible/have zero resistance in the set of directions A, if it is invisible/has zero resistance in any direction v ∈ A.
Observe that invisibility is a symmetric notion, i.e. if a body is invisible in a direction v, it is also invisible in −v. This follows directly from the fact that billiard dynamics is Examples of bodies invisible in one direction constructed with the use of thin mirrors can be found on the Wikipedia page on invisibility [11], and one of them is reproduced in Fig. 3  videos of prototypes designed by Karl Bednarik (one of them that uses only flat mirrors is plotted in Fig. 3 (b)). A solid body (i.e. without use of any thin mirrors) of zero resistance in one direction was constructed in [1] (see Fig. 4). It was also shown in the same work that there exist connected (and even homeomorphic to the ball) bodies of zero resistance in one direction. An invisible body is obtained by using two such bodies consecutively (see [1] for details).
In [10] a body of zero resistance in two directions in a three-dimensional setting was described. The body is sketched in Fig. 5: it employs 8 fragments of congruent parabolic cylinders with two mutually orthogonal focal lines (each line corresponds to 4 fragments). An invisible body is constructed by putting 4 such bodies together, as shown in Fig. 5   For more details we refer the reader to [10].

A thin fractal body invisible in two orthogonal directions
We start with a two-dimensional body invisible in two orthogonal directions. For the clarity of exposition, we assume that the directions of invisibility are parallel to the xand y-axes. We construct our body inside the square Consider a thin parabolic mirror (i.e. having zero thickness) p 1 given by the equation Observe that (−1, 1) ∈ p 1 , (1, 1) ∈ p 1 , and the focus of the parabola y = 1 2 x 2 + 1 2 is located at (0, 1). The axis is then given by the equation x = 0. Also note that the mirror is located above the graph of the function y = |x|.
A particle moving "downwards", i.e., along the direction (0, −1), would be reflected towards the focal point due to the reflective property of a parabola (see Fig. 6 (a)). We add one more parabolic mirror p 2 to our construction. This mirror is similar to the previous one, only two times smaller, while the foci of the corresponding parabolas coincide at (0, 1). We let Now all particles that go in the downward direction and pass the line segment , are first reflected towards the focal point, move towards p 2 , and after the collision with p 2 are redirected downwards (because of the aforementioned property of a parabola and coinciding foci of the two parabolas). If we remove a piece that is directly behind p 2 from p 1 , the resulting body is not obstructing the further movement of the particles, and they leave the body with the same velocity as they had before entering the body (see Fig. 6 (b)). If we repeat this construction process infinitely, adding figures similar to p 1 and cutting out the middle sections of the relevant parabolas, we obtain a sequence {p ′ k } ∞ 1 of parabolic mirror segments: This mirror sequence is plotted in Fig. 7 (a). The union of these segments is denoted by Take the sequence of parabolic mirror pieces q k symmetric to p ′ k with respect to the x-axis, and let Q = ∪ ∞ k=1 q k ; the union P ∪ Q is a fractal body invisible in the direction (0, ±1) (see Fig. 7 (b)). Indeed, since the lower part Q of our body is symmetric to the upper part P , it is redirecting the particles back to their original trajectories.
Observe that the area (greyed in Fig. 7 (b)) is completely "shaded" from the particles moving parallel to the y-axis. We can hence use this area to make our body invisible in a second direction. We simply add one more construction identical to the original one, but rotated by π 2 . It is not difficult to observe that it fits into G, and also makes the body invisible in the direction (±1, 0). Also observe that the four grey blocks in the corners of the square (see Fig. 8), are never accessed by the particles moving in the directions of invisibility. We can hence include these areas into our body.

A body invisible in two arbitrary directions
In this section we go further and construct a two-dimensional fractal body (without thin parts) invisible in any two directions.
This time we construct our body inside a rhombus ABCD with sides parallel to the directions of invisibility. Assume that we are given two non-parallel directions, not necessarily perpendicular. We rotate the coordinate system in a way that one of the directions is vertical. Our rhombus has got two sides parallel to the y-axis, and the other two parallel to the second direction of invisibility. The centre of the rhombus coincides with the origin.
This time our construction requires some preparatory work. Denote by −c the abscissa of A and select two infinite sequences of positive values a 0 , a 1 , a 2 , . . . and c 0 , c 1 , c 2 , . . . recursively according to the following rules.
Let c 0 = c and a 0 = +∞, and choose arbitrary c 1 satisfying 0 < c 1 < c 0 . On the ith step of the procedure, i = 1, 2, . . . select arbitrary a i satisfying the inequalities Figure 9: Preliminary construction: A body inside a rhombus invisible in one direction. and put Using (1) and (2), one easily gets that c i > 0 and derives by induction that c i+1 < c i . Now we are ready to draw the parabolas of our construction. In the description below, the points on the side AB are marked by the values of their abscissas: −c 0 , −c 1 , −c 2 , . . . , a 1 , a 2 , . . . (see Fig. 9).
For any i = 0, 1, 2, . . . consider the parabola through −c i with a vertical axis and with the focus at a i+1 . Denote by p i the arc of this parabola bounded by the points with abscissas −c i and −c i+1 . Then for any i = 1, 2, . . . consider the parabola through −c i with a vertical axis and with the focus at a i . The arc of this parabola situated between the points with abscissas −c i and −c i+1 is denoted by q i . Also denote by q 0 the part of the diagonal AC between A and the point with abscissa −c 1 . Each arc p i is situated above q i ; more precisely, both p i and q i are the graphs of functions p i (−t) and q i (−t), t ∈ [c i+1 , c i ] with p i (−t) > q i (−t) for any t ∈ (c i+1 , c i ]. Denote by P i , i = 0, 1, 2, . . . the set bounded by the arcs p i and q i from above and below and by a segment of the vertical line x i = −c i+1 on the right. Similarly, denote by −c ′ i and a ′ i , respectively, the points on the side CD whose abscissas are −c i and a i (the same values as above). Denote by p ′ i (q ′ i ) the arc of parabola through −c ′ i with vertical axis and with focus at a ′ i+1 (a Proof. Note that the parabolas containing p i−1 and q i have the same focus at a i ; therefore they are homothetic with the centre at this point. Since the points −c i−1 and −c i (which are the left endpoints of the arcs p i−1 and q i ) are homothetic, one readily concludes that the ratio equals r i , and therefore, the homothety is H i . Further, the right endpoint of p i−1 has abscissa −c i ; using (2), one verifies that its image under H i has the abscissa −c i+1 , and therefore, coincides with the right endpoint of q i . Thus, the arcs p i−1 and q i belong to homothetic parabolas, and their right and left endpoints are homothetic; therefore they are also homothetic under H i .
Consider the fractal body (greyed in Figure 9) Now we are in a position to prove the following Proposition.
Proposition 2. The body A L is invisible in the vertical direction.
Proof. Consider a particle falling vertically downward with the velocity (0, −1) along a line with abscissa x. We assume that x ∈ [−c, 0]; otherwise the particle does not hit the body and there is nothing to prove. If x = −c i (i = 0, 1, 2, . . .), the particle hits the body at a singular point and the motion is not defined from this point. Otherwise, x belongs to an interval (c i−1 , c i ), i = 1, 2, . . . (i = 1 in Fig. 9). The particle is first reflected by p i−1 and then moves to the focus a i ; then it makes the second reflection from q i and moves along a vertical line. Since the line through the two reflection points contains the focus, these two points (and therefore the vertical lines through these points) are homothetic under H i .
The particle then makes the third and fourth reflections from q ′ i and p ′ i−1 , and finally, moves freely downwards. Repeating the above argument, one concludes that the vertical By adding the set A R symmetric to A L with respect to the centre of the rhombus, one gets the body A = A L ∪ A R , which is also invisible in the vertical direction (0, −1) ( Fig. 10 (a)).

Remark 1. The arcs
. . are graphs of functions; denote these functions by Then the sets P i , P ′ i can be represented as In particular, if the directions of invisibility are orthogonal, we have p' i (x) = −p i (x), q' i (x) = −q i (x), and the body A is analytically described as This representation will be used in the next section.
Observe that the trapezoids AA ′ D ′ D and BB ′ C ′ C bounded by the lines x = ±c 0 , x = ±c 1 and by the diagonals of the rhombus are completely "shaded" from the particles falling in the vertical direction. We can therefore use this area to make our body invisible in the second direction.
Namely, the body B symmetric to A with respect to the line BD (or AC) is contained in the union of the trapezoids and is invisible in the direction −→ AB. The union A ∪ B is then invisible in both directions ( Fig. 10 (b)).
We have proved the following result. Observe that it includes Theorem 1 as a particular case.
Theorem 2. For any two directions v 1 and v 2 ∈ S 1 , there exists a solid fractal body invisible in these directions.

Remark 2.
In the limiting case c i = 2 −i c, a i = 0 (i ≥ 1) one gets a thin fractal. In particular, if the two directions are orthogonal, we have the construction described in the previous Subsection 3.1.

A body invisible in 3 directions
Using the two-dimensional construction described in the previous section, we can now describe a three-dimensional body invisible in 3 orthogonal directions.
In the 3D case the construction is more complicated and intuition is less reliable; therefore we provide here a more detailed argument than in the previous section. Some of the accompanying figures, for better visibility, correspond to the limiting case of thin fractal body.
Let us first introduce some notation. We consider Euclidean space R 3 with orthogonal coordinates x, y, z and the cube Q = [−c, c] 3 centered at the origin O = (0, 0, 0). The pyramids with the vertex at O and with the bases z = ±c, |x| ≤ c, |y| ≤ c (the upper and lower faces of the cube Q) are denoted by Π ± z . In other words, The pyramids Π x and Π y are defined in the same way. Notice that the interiors of Π x , Π y , and Π z are mutually disjoint.
Let 0 < c 1 < c. For each ε x ∈ {−1, 1} and ε z ∈ {−1, 1} we define the gallery ; it is a horizontal parallelepiped adjacent to an edge of Q parallel to the y-axis. The union of 4 such parallelepipeds is The sets G x and G z are defined similarly.
We are going to define three bodies B x , B y , B z , invisible in the directions x, y, z, respectively, and then take their union.
First we take the two-dimensional body A = A yz in the yz-plane, as described in the previous subsection 3.2. It corresponds to 2 orthogonal directions (parallel to the y-and zaxes) and is inscribed in the square [−c, c] 2 . (Notice that the body A shown in Fig. 10 (a) corresponds to 2 non-orthogonal direction and is inscribed in a rhombus.) The body A yz invisible in the z-direction is determined by the parameters c 0 = c, c 1 , c 2 , . . . ; a 1 , a 2 , . . . satisfying (1) and (2).
Take the direct productÃ the resulting three-dimensional bodyÃ yz is also invisible in the z-direction. Notice that it is contained in the union of y-galleries, The bodyÃ yz is shown in Fig. 11 (a); for the sake of better visualization we chose to draw the limiting case of "thin fractal" with a i = 0 and c i = 2 −i c (i = 1, 2, . . .). Let B yz =Ã yz ∩ Π z (Fig. 11 (b)). Then we analogously define the body B xz and take The bodies B x and B y are defined in a similar way, and finally, The bodies B z , B y and B x are shown on Fig. 13 (a), Fig. 13 (b) and Fig. 14 (a) respectively. The body B is shown on Fig. 14 (b). All the pictures correspond to the "thin" fractal case.
Remark 3. In terms of the functions p i and q i introduced in Remark 1, the setÃ yz can be written as Taking the intersection ofÃ yz with Π z = {(x, y, z) : |z| ≥ |x|, |z| ≥ |y|} and using that q i (|y|) ≥ |y|, one gets and therefore, Now we are prepared for the proof of the following theorem.
Theorem 3. There exists a solid fractal body invisible in 3 mutually orthogonal directions.
Proof. We are going to show that B is invisible in the directions parallel to the coordinate axes. It suffices to prove that B is invisible in the z-direction; this will imply invisibility in the x-and y-directions, due to symmetry of the construction under the exchange of variables x, y and z. We will actually show that the body B z is invisible for the incident flow in the z-direction and the bodies B x and B y are shadowed from this flow by B z .
Consider an incident particle with the velocity (0, 0, −1). Our goal is to prove that it makes either 0 or 4 reflections from B z (and no reflections from B x and B y ) and moves freely afterwards; moreover, the initial and final parts of its trajectory belong to the same straight line.
If the orthogonal projection of the coordinate of the incident particle on the xy-plane lies inside the square [−c 1 , c 1 ] 2 or outside the square [−c, c] 2 (see Fig. 12  Due to symmetry of the construction, it suffices to consider the cases where the projection belongs to (i) the triangle c 1 < x < y < c and (ii) the rectangle c 1 < x < c, 0 < y < c 1 (they are shown grey in Fig. 12).
Consider the case (i). Take an auxiliary trajectory corresponding to the particle with the same initial data making reflections fromÃ yz . At the first point of reflection one has z = p 0 (y), at the second point, z = q 1 (y), and the third and fourth reflection points are symmetric to the first two points with respect to the xy-plane. After the fourth reflection the particle moves freely. We know thatÃ yz is invisible in the z-direction; that is, the initial and final parts of the auxiliary trajectory belong to a straight line. It remains to show that the points of impact actually belong to B yz and the auxiliary trajectory does not intersect the rest of the body B \B yz ; this will imply that it is a true billiard trajectory in the complement of B.
Further, the 01-and 12-segments belong to the interior of Π z , and therefore, do not intersect B x and B y . Due to the symmetry with respect to the xy-plane, this is also true for the last two segments of the auxiliary trajectory. It remains therefore to prove that the 23-segment does not intersect the bodies B x = B yx ∪ B zx and B y = B zy ∪ B xy . The sets B zx , B zy and B xy belong to the galleries G z and G x , and therefore cannot have points in common with the 23-segment. It remains to check the set B yx .
Consider now the case (ii). Take i ≥ 1 such that c i+1 < y < c i . (The limiting case y = c i has zero measure in the space of billiard trajectories; in this case the particle hits a singular point of B and the motion is not defined since then.) If x > p i (y), the particle does not hit B. Indeed, by (4), the vertical straight line (x, y, * ) does not intersect B yz , and by (6), it does not intersect B yx . The other sets B xy , B xz , B zx , B zy comprising B belong to the galleries G x and G z , and therefore do not intersect this straight line.
Assume that x < p i (y) and consider an auxiliary trajectory with the same initial data making reflections fromÃ yz . As in the case (i), there are 4 reflections; the first three segments of the trajectory (between the point of entering Q and the 1 st reflection point; between the 1 st and the 2 nd reflection points; between the 2 nd and the 3 rd reflection points) will be referred to as 01-segment, 12-segment, and 23-segment, respectively. The trajectory is symmetric with respect to the xy-plane, and the final velocity, (0, 0, −1), coincides with the initial one.
Repeating the argument of (i), one concludes that the 1 st and 2 nd reflection points and the 12-segment joining them belong to Π z , and by symmetry, the 3 rd and 4 th reflection points, and the segment joining them, also belong to Π z . Therefore all the reflection points belong to B yz . It remains to check that the auxiliary trajectory does not intersect B \ B yz , and therefore, is a true trajectory in the complement of B.
Recall that the sets B xy , B xz , B zx , B zy belong to G x ∪G z , and therefore do not intersect the trajectory. It remains to check the set B yx . Since the 01-and the 12-segments belong to Π z , they do not have points in common with B yx . The same is true for the last two segments symmetric to them. It remains to check the 23-segment.
By (6), (7), and (8), the straight line (x (2) , y (2) , * ) does not intersect B yx . Hence the theorem is proved. Remark 4. We do not know if it is possible to generalize our construction to 3 nonorthogonal directions. The direct generalization does not work even in the case where two non-orthogonal directions lie in the xy-plane and the third one coincides with the z-direction. Indeed, a particle falling vertically down and hitting a mirror in the gallery G y , will then move in a direction orthogonal to the y-axis (and not parallel to the x-axis, which would be desirable), and therefore may fail to hit the opposite mirror in that gallery.

Summary
We have shown that there exist bodies invisible in 2 directions in the two-dimensional case and in 3 directions in the three-dimensional case. It was not known earlier whether such bodies exist. We believe that our construction can be more or less directly generalized to n directions of invisibility for n-dimensional bodies, n > 3. There are, however, many open questions. Can we construct a body invisible in n directions in n-dimensional space without using any fractal constructions? Are there bodies invisible in more than n directions in n-dimensional space? What is the maximal number of directions of invisibility? How to introduce an adequate "measure of invisibility" for a body observed in all directions and find the "most invisible" body?
There is an intriguing observation related to the existing constructions. There exist connected (and even homeomorphic to the ball) bodies invisible in 1 direction [1]. The body invisible in 2 directions found in [10] is disconnected. The body invisible in 3 directions has an infinite number of connected components. We wonder if the increased complexity of the shape is the cost one should pay for the increased number of directions, or whether it is just an artefact of the particular constructions.
These problems are easy to understand, and the existing results can be explained by using only basic school math. However, there are no tools or techniques for constructing invisible bodies, and this makes the subject even more exciting.