• Knight's tour
    Sneak Peeks

    The Knight’s Tour

    The knight’s tour problem is the mathematical problem of finding a knight’s tour, and probably making knight the most interesting piece on the chess board. The knight visits every square exactly once, if the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed; otherwise, it is open.

    The knight’s tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight’s tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight’s tour problem can be solved in linear time.

    Hamiltonian Path Problem

    In Graph Theory, a graph is usually defined to be a collection of nodes or vertices and the set of edges which define which nodes are connected with each other. So we use a well known notation of representing a graph G = (V,E) where V = { v1, v2, v3, … , vn } and E = {(i, j)|i ∈ V and j ∈ V and i and j is connected}.

    Hamiltonian Path is defined to be a single path that visits every node in the given graph, or a permutation of nodes in such a way that for every adjacent node in the permutation there is an edge defined in the graph. Notice that it does not make much sense in repeating the same paths. In order to avoid this repetition, we permute with |V|C2 combinations of starting and ending vertices.

    Simple way of solving the Hamiltonian Path problem would be to permutate all possible paths and see if edges exist on all the adjacent nodes in the permutation. If the graph is a complete graph, then naturally all generated permutations would quality as a Hamiltonian path.

    For example. let us find a Hamiltonian path in graph G = (V,E) where V = {1,2,3,4} and E = {(1,2),(2,3),(3,4)}. Just by inspection, we can easily see that the Hamiltonian path exists in permutation 1234. The given algorithm will first generate the following permutations based on the combinations:
    1342 1432 1243 1423 1234 1324 2143 2413 2134 2314 3124 3214

    The number that has to be generated is (|V|C2 ) (|V| – 2)!


    Schwenk proved that for any m × n board with m ≤ n, a closed knight’s tour is always possible unless one or more of these three conditions are met:

    1. m and n are both odd
    2. m = 1, 2, or 4
    3. m = 3 and n = 4, 6, or 8.

    Cull and Conrad proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight’s tour.

    nNumber of directed tours (open and closed)
    on an n × n board
    (sequence A165134 in the OEIS)

    Neural network solutions

    The neural network is designed such that each legal knight’s move on the chessboard is represented by a neuron. Therefore, the network basically takes the shape of the knight’s graph over an n×n chess board. (A knight’s graph is simply the set of all knight moves on the board)

    Each neuron can be either “active” or “inactive” (output of 1 or 0). If a neuron is active, it is considered part of the solution to the knight’s tour. Once the network is started, each active neuron is configured so that it reaches a “stable” state if and only if it has exactly two neighboring neurons that are also active (otherwise, the state of the neuron changes). When the entire network is stable, a solution is obtained. The complete transition rules are as follows:

    where t represents time (incrementing in discrete intervals), U(Ni,j) is the state of the neuron connecting square i to square j, V(Ni,j) is the output of the neuron from i to j, and G(Ni,j) is the set of “neighbors” of the neuron (all neurons that share a vertex with Ni,j).

    Code For Knight’s Tour

    // Java program for Knight Tour problem
    class KnightTour {
    static int N = 8;
    /* A utility function to check if i,j are
    valid indexes for N*N chessboard */
    static boolean isSafe(int x, int y, int sol[][])
        return (x >= 0 && x < N && y >= 0 && y < N
                && sol[x][y] == -1);
    /* A utility function to print solution
    matrix sol[N][N] */
    static void printSolution(int sol[][])
        for (int x = 0; x < N; x++) {
            for (int y = 0; y < N; y++)
                System.out.print(sol[x][y] + " ");
    /* This function solves the Knight Tour problem
    using Backtracking. This function mainly
    uses solveKTUtil() to solve the problem. It
    returns false if no complete tour is possible,
    otherwise return true and prints the tour.
    Please note that there may be more than one
    solutions, this function prints one of the
    feasible solutions. */
    static boolean solveKT()
        int sol[][] = new int[8][8];
        /* Initialization of solution matrix */
        for (int x = 0; x < N; x++)
            for (int y = 0; y < N; y++)
                sol[x][y] = -1;
        /* xMove[] and yMove[] define next move of Knight.
        xMove[] is for next value of x coordinate
        yMove[] is for next value of y coordinate */
        int xMove[] = { 2, 1, -1, -2, -2, -1, 1, 2 };
        int yMove[] = { 1, 2, 2, 1, -1, -2, -2, -1 };
        // Since the Knight is initially at the first block
        sol[0][0] = 0;
        /* Start from 0,0 and explore all tours using
        solveKTUtil() */
        if (!solveKTUtil(0, 0, 1, sol, xMove, yMove)) {
            System.out.println("Solution does not exist");
            return false;
        return true;
    /* A recursive utility function to solve Knight
    Tour problem */
    static boolean solveKTUtil(int x, int y, int movei,
                            int sol[][], int xMove[],
                            int yMove[])
        int k, next_x, next_y;
        if (movei == N * N)
            return true;
        /* Try all next moves from the current coordinate
            x, y */
        for (k = 0; k < 8; k++) {
            next_x = x + xMove[k];
            next_y = y + yMove[k];
            if (isSafe(next_x, next_y, sol)) {
                sol[next_x][next_y] = movei;
                if (solveKTUtil(next_x, next_y, movei + 1,
                                sol, xMove, yMove))
                    return true;
                        = -1; // backtracking
        return false;
    /* Driver Code */
    public static void main(String args[])
        // Function Call

  • Sneak Peeks


    The Painter’s Paradox is based on the fact that Gabriel’s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel’s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel’s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli’s discovery.

    Gabriel and His Horn

    Gabriel was an archangel, as the Bible tells us, who “used a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)”.

    Torricelli’s Long Horn

    In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel’s horn, which he called the acute hyperbolic solid, has a finite volume. In De solido hyperbolico acuto he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem:

    THEOREM: An acute hyperbolic solid, infinitely long [infinite longum], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body.

    The Queer Volume

    Construct the surface of revolution given by rotating the function f(x) = 1/x on [1,∞) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), de fined on the interval [a; b], around the x-axis is

    where A(x) is the cross-sectional area at x ∈ [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = π (f(x))2, so that

    In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that

    Hence, even though the Horn extends outward along the x-axis to ∞, the improper integral does converge, and hence there is finite volume “inside” the Horn. One can say that one can fill the Horn with π-units of a liquid. (This is oddly satisfying)

    The Infinite Surface Area

    The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating

    This formula basically says that one can nd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2πf(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA):

    This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that

    Thus we can say that, if

    on the interval [1;1), then

    by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we fi nd by comparison that

    But we have already evaluated this last integral in class. We get

    Hence this last integral diverges, and hence by comparison so does the former integral. But
    this implies that the surface area of Gabriel’s Horn is infi nite!

    So we have a surface with infinite surface area enclosing a finite volume.

  • Sneak Peeks


    Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject. Besides the obvious fact that the Fourier transform is itself an oscillatory integral par excellence, one need only recall the occurrence of Bessel functions in the work of Fourier, the study of asymptotic related to these functions by Airy, Stokes, and Lipschitz, and Riemann’s use of the method of “stationary phase” in finding the asymptotic of certain Fourier transforms, all of which took place well over 100 years ago.

    A basic problem which comes up whenever performing a computation in harmonic analysis is how to quickly and efficiently compute (or more precisely, to estimate) an explicit integral. Of course, in some cases undergraduate calculus allows one to compute such integrals exactly, after some effort (e.g. looking up tables of special functions), but since in many applications we only need the order of magnitude of such integrals, there are often faster, more conceptual, more robust, and less computationally intensive ways to estimate these integrals.

    The dyadic decomposition of a function

    Littlewood–Paley theory uses a decomposition of a function f into a sum of functions fρ with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows. If f(x) is a function on R, and ρ is a measurable set (in the frequency space) with characteristic function Xρ(ξ), then fρ is defined via its Fourier transform fρ := Xρ . Informally, fρ is the piece of f whose frequencies lie in ρ. If Δ is a collection of measurable sets which (up to measure 0) are disjoint and have union on the real line, then a well behaved function f can be written as a sum of functions fρ for ρ ∈ Δ. When Δ consists of the sets of the form ρ = [-2k+1,-2k] U [2k, 2k+1] for k an integer, this gives a so-called “dyadic decomposition” of f : Σρ fρ. There are many variations of this construction; for example, the characteristic function of a set used in the definition of fρ can be replaced by a smoother function. A key estimate of Littlewood Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions fρ in terms of the size of f. There are many versions of this theorem corresponding to the different ways of decomposing f. A typical estimate is to bound the Lp norm of (Σρ |fρ|2)1/2 by a multiple of the Lp norm of f. In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions.

    In the case where the integral to evaluate is non-negative, e.g.

    then the method of decomposition, particularly dyadic decomposition, works quite well: split the domain of integration into natural regions, such as dyadic annuli on which a key term in the integrand is essentially constant, estimate each sub-integral (which generally reduces to the geometric problem of measuring the volume of some standard geometric set, such as the intersection of two balls), and then sum (generally one ends up with summing a standard series such a geometric series or harmonic series). For non-negative integrands, this approach tends to give answers which only differ above and below from the truth by a constant (possibly depending on things such as the dimension d). Slightly more generally, this type of estimation works well in providing upper bounds for integrals which do not oscillate very much. With some more effort, one can often extract asymptotics rather than mere upper bounds, by performing some sort of expansion (e.g. Taylor expansion) of the integrand into a main term (which can be integrated exactly, e.g. by methods from undergraduate calculus), plus an error term which can be upper bounded by an expression smaller than the final value of the main term. However, there are many cases in which one has to deal with integration of highly oscillatory integrands, in which the naive approach of taking absolute values (thus destroying most of the oscillation and cancellation) will give very poor bounds. A typical such oscillatory integral takes the form

    where a is a bump function adapted to some reasonable set B (such as a ball), Φ is a real -valued phase function (usually obeying some smoothness conditions), and λ ∈ R is a parameter to measure the extent of oscillation. One could consider more general integrals in which the amplitude function a is replaced by something a bit more singular, e.g. a power singularity |x|−α, but the aforementioned dyadic decomposition trick can usually decompose such a “singular oscillatory integral” into a dyadic sum of oscillatory integrals of the above type. Also, one can use linear changes of variables to rescale B to be a normalised set, such as the unit ball or unit cube. In one dimension, the definite integral

    is also of interest, where J is now an interval. While one can dyadically decompose around the endpoints of these intervals to reduce this integral to the previous smoother integral, in one dimension one can often compute the integrals more directly.

    One dimensional theory

    Beginning with the theory of the one-dimensional definite integrals,

    where J is an interval, λ ∈ R, and Φ : J → R is a function (which we shall assume to be smooth, in order to avoid technicalities). We observe some simple invariances:

    • I(−λ) = I(λ), thus negative λ and positive λ behave similarly;
    • Subtracting a constant from λ does not affect the magnitude of I(λ);
    • If L : R → R is any invertible affine-linear transformation, then IL(J),Φ◦L−1(λ) =| det(L)|IJ,Φ(λ).
    • We haveIJ,Φ(λ) = IJ,Φ(1).

    From the triangle inequality we have the trivial bound |I(λ)| ≤ |J|.

    This bound is of course sharp if Φ is constant. But if Φ is non-constant, we expect I(λ) to decay as λ → ±∞.

    Higher dimensional theory

    The higher dimensional theory is less precise than the one-dimensional theory, mainly because the structure of stationary points can be significantly more complicated. Nevertheless, we can still say quite a bit about the higher dimensional oscillatory integrals

    in many cases. The van der Corput lemma becomes significantly weaker, and will not be discussed here; however, we still have the principle of non stationary phase.

    Principle of non-stationary phase

    Let a ∈ C0 (Rd), and let Φ : Rd → R be smooth such that ∇Φ is non-zero on the support of a. Then Ia,Φ(λ) =ON,a,Φ,d−N) for all N ≥ 0.

    Success! You're on the list.
    Adapted from LECTURE NOTES by TERENCE TAO.                                            No copyright infringement intended.
  • Sneak Peeks


    Modern theories describe physical forces in terms of fields, e.g. the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. For example, if you could measure the color of lead balls and discover that when you change the color, you still fit the same number of balls in a pound, the property of “color” would show gauge invariance. Since any kind of invariance under a field transformation is considered a symmetry, gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory.

    In this Feynman diagram, an electron (e) and a positron (e+annihilate, producing a photon (γ, represented by the blue sine wave) that becomes a quarkantiquark pair (quark q, antiquark ), after which the antiquark radiates a gluon (g, represented by the green helix).

    The idea of a gauge theory evolved from the work of Hermann Weyl. One can find in an interesting discussion of the history of gauge symmetry and the discovery of Yang–Mills theory, also known as “non-abelian gauge theory.” At the classical level one replaces the gauge group U(1) of electromagnetism by a compact gauge group G. The most important Quantum Field Theories (QFTs) for describing elementary particle physics are gauge theories. The classical example of a gauge theory is Maxwell’s theory of electromagnetism. For electromagnetism the gauge symmetry group is the abelian group U(1). If A denotes the U(1) gauge connection, locally a one-form on space-time, then the curvature or electromagnetic field tensor is the two-form F = dA, and Maxwell’s equations in the absence of charges and currents read 0 = dF = d * F. Here * denotes the Hodge duality operator; indeed, Hodge introduced his celebrated theory of harmonic forms as a generalization of the solutions to Maxwell’s equations. Maxwell’s equations describe large-scale electric and magnetic fields and also—as Maxwell discovered—the propagation of light waves, at a characteristic velocity, the speed of light.

    The definition of the curvature arising from the connection must be modified to F = dA + A ^ A, and Maxwell’s equations are replaced by the Yang–Mills equations, 0 = dAF = dA* F, where dA is the gauge-covariant extension of the exterior derivative. These classical equations can be derived as variational equations from the Yang– Mills Lagrangian

    L = 1/4g2 ∫ Tr F ^ F,

    where Tr denotes an invariant quadratic form on the Lie algebra of G. The Yang– Mills equations are nonlinear—in contrast to the Maxwell equations. Like the Einstein equations for the gravitational field, only a few exact solutions of the classical equation are known. But the Yang–Mills equations have certain properties in common with the Maxwell equations: In particular they provide the classical description of massless waves that travel at the speed of light.

    In mathematical terminology, electron phases form an Abelian group under addition, called the circle group or U(1). “Abelian” means that addition commutes, so that θ + φ = φ + θ. Group means that addition associates and has an identity element, namely “0”. Also, for every phase there exists an inverse such that the sum of a phase and its inverse is 0. Other examples of abelian groups are the integers under addition, 0, and negation, and the nonzero fractions under product, 1, and reciprocal.

    Quick note

    As a way of visualizing the choice of a gauge, consider whether it is possible to tell if a cylinder has been twisted. If the cylinder has no bumps, marks, or scratches on it, we cannot tell. We could, however, draw an arbitrary curve along the cylinder, defined by some function θ(x), where x measures distance along the axis of the cylinder. Once this arbitrary choice (the choice of gauge) has been made, it becomes possible to detect it if someone later twists the cylinder.

    The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD). The use of QCD to describe the strong force was motivated by a whole series of experimental and theoretical discoveries made in the 1960s and 1970s, involving the symmetries and high-energy behavior of the strong interactions. But classical nonabelian gauge theory is very different from the observed world of strong interactions; for QCD to describe the strong force successfully, it must have at the quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory:

    • It must have a “mass gap;” namely there must be some constant Δ > 0 such that every excitation of the vacuum has energy at least Δ.
    • It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quark fields, that transform non-trivially under SU(3), the physical particle states—such as the proton, neutron, and pion—are SU(3)-invariant.
    • It must have “chiral symmetry breaking,” which means that the vacuum is potentially invariant (in the limit, that the quark-bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.

    The first point is necessary to explain why the nuclear force is strong but short-ranged; the second is needed to explain why we never see individual quarks; and the third is needed to account for the “current algebra” theory of soft pions that was developed in the 1960s. Both experiment—since QCD has numerous successes in confrontation with experiment—and computer simulations, carried out since the late 1970s, have given strong encouragement that QCD does have the properties cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled lattice gauge theory). But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.

    The Problem

    To establish existence of four-dimensional quantum gauge theory with gauge group G, one should define a quantum field theory (in the above sense) with local quantum field operators in correspondence with the gauge-invariant local polynomials in the curvature F and its covariant derivatives, such as Tr FijFkl(x). Correlation functions of the quantum field operators should agree at short distances with the predictions of asymptotic freedom and perturbative renormalization theory, as described in textbooks. Those predictions include among other things the existence of a stress tensor and an operator product expansion, having prescribed local singularities predicted by asymptotic freedom. Since the vacuum vector Ω is Poincar´e invariant, it is an eigenstate with zero energy, namely HΩ = 0. The positive energy axiom asserts that in any quantum field theory, the spectrum of H is supported in the region [0,∞). A quantum field theory has a mass gap Δ if H has no spectrum in the interval (0, Δ) for some Δ > 0. The supremum of such Δ is the mass m, and we require m < ∞.

    Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap Δ > 0.

    Mathematical Perspective

    Wightman and others have questioned for approximately fifty years whether mathematically well-defined examples of relativistic, nonlinear quantum field theories exist. We now have a partial answer: Extensive results on the existence and physical properties of nonlinear QFTs have been proved through the emergence of the body of work known as “constructive quantum field theory” (CQFT). The answers are partial, for in most of these field theories one replaces the Minkowski space-time M4 by a lower-dimensional space-time M2 or M3, or by a compact approximation such as a torus. (Equivalently in the Euclidean formulation one replaces Euclidean space-time R4 by R2 or R3.) Some results are known for Yang Mills theory on a 4-torus T4 approximating R4, and, while the construction is not complete, there is ample indication that known methods could be extended to construct Yang–Mills theory on T4. In fact, at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions. So even having a detailed mathematical construction of Yang–Mills theory on a compact space would represent a major breakthrough. Yet, even if this were accomplished, no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit T4 ➜ R4.

    Success! You're on the list.
  • Sneak Peeks


    A gyroid structure is a distinct morphology that is triply periodic and consists of minimal iso-surfaces containing no straight lines.

    The gyroid was discovered in 1970 by Alan Schoen, a NASA crystallographer interested in strong but light materials. Among its most curious properties was that, unlike other known surfaces at the time, the gyroid contains no straight lines or planar symmetry curves. In 1975, Bill Meeks discovered a 5-parameter family of embedded genus 3 triply periodic minimal surfaces.

    Photonic crystals were synthesized by deposition of a-Si/Al2O3 coatings onto a sacrificial polymer scaffold defined by two-photon lithography. We observed a 100% reflectance at 7.5 µm for single gyroids with a unit cell size of 4.5 µm, in agreement with the photonic bandgap position predicted from full-wave electromagnetic simulations, whereas the observed reflection peak shifted to 8 µm for a 5.5 µm unit cell size. This approach represents a simulation-fabrication-characterization platform to realize three dimensional gyroid photonic crystals with well-defined dimensions in real space and tailored properties in momentum space.

    Three-dimensional photonic crystals offer opportunities to probe interesting photonic states such as bandgaps, Weyl points, well-controlled dislocations and defects. Combinations of morphologies and dielectric constants of materials can be used to achieve desired photonic states. Gyroid crystals have interesting three-dimensional morphologies defined as triply periodic body centered cubic crystals with minimal surfaces containing no straight lines. A single gyroid structure consists of iso-surfaces described by sin(x)cos(y) + sin(y)cos(z) + sin(z)cos(x) > u(x, y, z), where the surface is constrained by u(x, y, z). Gyroid structures exist in biological systems in nature. For example, self-organizing process of biological membranes forms gyroid photonic crystals that exhibit the iridescent colors of butterfly’s wings. Optical properties of gyroids could vary with tuning of u(x, y, z), unit cell size, spatial symmetry as well as refractive index contrast. Single gyroid photonic crystals, when designed with high refractive index and fill fraction, are predicted to possess among the widest complete three-dimensional bandgaps, making them interesting for potential device applications such as broadband filters and optical cavities.

    Gyroid representation

    The minimal surface of the gyroid can be approximated by the level set equation, with the iso-value, h, set to h = 0. The parameter, a, is the unit cell length. By selecting alternative values for h between the limits (±1.413), the surface can be offset along its normal direction; beyond those limits the surface becomes disconnected, and ceases to exist for |h| > 1.5. Introducing an inequality enables the selection of the regions to either side of the shifted surface, producing the solid-network form. The solid-surface form is generated by selecting the region between two surfaces shifted along the surface normal to either side from their gyroid mid-surface (or alternatively using the inequality. The inequality  produces two interwoven non-connected solid-network forms.

    Solid finite element model

    A voxel based finite element (FE) mesh with eight-node hexahedral elements was generated to approximate the solid-surface gyroid within a cubic bulk form. Firstly, a triangular mesh was constructed within Matlab using the iso-surface for h = 0. This mesh was copied and translated along the local normal to create the thickened solid-surface geometry and enclosed at the boundary edges. A voxelization function was then used to produce the voxel mesh. The voxel element size was set to target a minimum of 4 elements through the thickness direction.

    Success! You're on the list.