slices.md

--- title: Slices lead: The diagonal slice of the solid cube: the 6n census, centered-hexagonal vertices, and the splitting-prime rule. figure: research-slices slug: slices ---

Cut the solid cube of odd side 2k - 1 through its centre, perpendicular to the main diagonal, and the section is a regular hexagon tiled by 6*side^2 unit equilateral triangles. This page is the census of that mesh - triangles, edges, vertices, an Euler characteristic that never moves - and of what the parity designs do to it: which fills partition it, which fall into many pieces, and which pierce it with holes. The mesh itself is classical lattice geometry and no novelty is claimed for it; the designs are where the specific content lives. The back half leaves the slice for the surface census of the same designs in 3D. Lay these hexagons over each other at every odd side and they interfere; that stack, its exact ink laws and the arithmetic of its lines are the hexagon moire.

Every claim carries a tag. Proved means a proof is given or restated here; Verified means recomputed from scratch by a crate test or a lab study; Conjecture means neither. The generators are six::census, six::topology, formulas::six and three::census in ../crates/mrlymath, each test named for the claim it pins, except the profile identity, which lab/py/slice-ladder-controls checks, and the slice ink, which mrlydemo::walsh_spectrum rebuilds and the mrlydemo fixture holds against the counted hexagon. The slices demo cuts the middle plane of any design at any odd side, draws the hexagon triangle by triangle, and reports the mesh census beside its closed forms, the carpet-and-net partition, and the pieces and holes over k = 1..16.

One boundary first. The cuts page also slices a solid along x + y + z = s and also finds a hexagon, but everything else differs: there the solid is bang dim 3, code 126 at base 2, fractal from the start, and the result - every slice a Sierpinski gasket, scheduled by the binary digits of the height - comes from a digit argument, with no mesh and no census. Here the solid is the plain filled cube of odd side, the cut is the single middle plane, and the result is a triangular mesh counted directly. The two pages share a plane and a hexagon and not one number.

The solid slice, counted

Take the plane x + y + z = 3*side/2 through the centre of [0,side]^3. It crosses the cells of the three middle diagonal layers, meeting each crossed cell in a triangle or a hexagon; cutting the hexagons into six from their centres leaves pieces that are all equilateral triangles of one size, tiling the section - a regular hexagon whose area is 6*side^2 triangles. The rim contributes 6*side triangle sides, and the piece vertices are exactly the triangular-lattice points of the closed hexagon, 3*side^2 + 3*side + 1 of them. Every interior side is shared by exactly two triangles, so counting sides two ways gives 3*F = 2*I + B at F the triangles, I the interior edges and B the boundary edges, which fixes the edge count, and the whole census follows. (Proved; each step is also asserted numerically in the build, not waved through.)

Every count in the table carries the factor 6*side except the last: the vertex count is odd and carries none. The carry demo runs the same cut at any base and dimension, where the count remembers only a carry, so ceil(D/2) past terms decide every term and the growth exponent misses the generic value, above it at odd dimensions and below it at even ones.

Prior art. The vertex count 12*k^2 - 6*k + 1 is OEIS A154105, printed there as 12n^2 + 18n + 7 under an index shift of one: substituting k = n + 1 gives 12(n+1)^2 - 6(n+1) + 1 = 12n^2 + 18n + 7, and both forms return 7, 37, 91, 169, 271 over the first five terms. This lane also cites A154105 in sequences's A395241 row. https://oeis.org/search?q=id:A154105&fmt=json returns the name a(n) = 12*n^2 + 18*n + 7., offset 0, and data 7, 37, 91, 169, 271, 397, ..., so the name, the offset and the five terms above are all Verified at source.

Two consequences are exact, not asymptotic. The slice's own surface-to-volume ratio - boundary edges over triangles - is 6*side / 6*side^2 = 1/side at every size, and V - E + F = 1 identically - the slice is a topological disk at every k. (Proved; Verified with the Euler characteristic counted directly, never from the algebra, at every size built.) The polynomials are confirmed by direct census at k = 1..8 (mrlymath::six::census test the_five_closed_forms_match_the_census_to_eight, which reads the Euler characteristic as 1 at every size it builds), by a blind quadratic fit through k = 1..3 that reproduces k = 4..10 with zero residual (test a_blind_quadratic_fit_reproduces_the_wider_slices), and by fresh builds at k = 12, 16, 20 - at side 39: 9126 triangles, 13806 edges, 4681 vertices, 234 boundary edges (test fresh_builds_at_the_wide_sides_hold_the_forms). (Verified.)

One lemma serves every fill census below. Lemma (Proved). For any set of triangles of the mesh, the adjacency graph on them - one node per triangle, one edge per shared side - has exactly E' - B' edges, where E' and B' are the edge and boundary-edge counts of the sub-mesh those triangles span: every interior edge of the sub-mesh joins exactly two of its triangles, every boundary edge one, and no lattice edge lies in three. Verified by direct enumeration on twenty meshes - the solid and four design fills at base-3 levels one to four, sides 3 to 81 - with the adjacency count and the interior-edge count collected from different maps; the carpet's fill at side 81 reads E' = 28188, B' = 6642, adjacencies 21546 (mrlymath::six::census test the_fill_adjacency_counts_the_sub_mesh_interior_edges). The sub-mesh is the load-bearing word: read against the full hexagon mesh the identity is false - a fill-void edge is interior to the hexagon but joins no two fill triangles - and at side 27 the carpet's adjacency count is 2880 against the full mesh's E - B = 6480 (test the_lemma_needs_the_sub_mesh_and_not_the_hexagon).

The profile identity

For a binary array X let P_X(t) = Sum over filled cells of t^(r+c) be the anti-diagonal slice profile. Then

P_{A (x) B}(t) = P_A(t^(side_B)) * P_B(t)

because r + c = side_B*(r_A + c_A) + (r_B + c_B) under the Kronecker product. Proved.

• Stationary corollary: P_level(t) = prod_{j=0}^{level-1} P(t^(base^j)), the substitution product at one base.
• Mixed corollary: for an ordered word, P_w(t) = prod_i P_i(t^(side_{i+1}...side_level)), the mixed-radix substitution product.
• One identity therefore covers fractal slices, mixed-product slices, and the dimension ladder in cuts and the base ladder in spectra. It is the generating function of a digit-restricted set, so every tool for those sets applies to it.
• Verified on all words of length 2 and 3 at base 2, dim 2, over the 15 nonempty 2x2 tiles, zero mismatches, lab/py/slice-ladder-controls.
• Take the whole profile and never one coefficient: the total and the centre coefficient are order-blind, and the profile as a whole is not.

Open, and cheap:

• Is the profile a rational series in the word, as components, boundary, Euler characteristic and holes all are? Its Hankel rank is the first thing to measure.
• Which slice functionals are order-blind? Total is, the centre coefficient is, peak and support are not. Classify them.
• Non-diagonal lattice planes: x + y = c decomposes because the coefficient vector is all ones, and a general primitive alpha.x = c does not. That is where the carry automaton enters, and nothing is known there.

The vertex count is centered hexagonal

The closed hexagon of side side holds CH(side+1) lattice points, where CH(m) = 3*m^2 - 3*m + 1 is the centered hexagonal sequence 1, 7, 19, 37, 61, ... (OEIS A003215). With side 2k - 1 that is CH(2k): the slice vertex count is the centered hexagonal number at even index. (Proved.)

The arithmetic attached is short and mostly classical. 12*k^2 and 6*k are both divisible by 3, so the vertex count is 1 mod 3 for every integer k. (Proved.) And CH(m) = m^3 - (m-1)^3 = m^2 + m*(m-1) + (m-1)^2, a difference of consecutive cubes, so the centered hexagonal numbers that are prime are exactly the cuban primes, OEIS A002407 - a classical family, not a discovery of this page. What the slice adds is only the specialisation: its vertex counts walk the even-index half of that sequence.

The same form places the counts in the Eisenstein integers Z[omega], the ring behind the hexagonal lattice on the bases page. A rational prime splits in Z[omega] exactly when it is 1 mod 3, stays inert when it is 2 mod 3, and 3 alone ramifies - standard algebraic number theory, cited rather than re-derived. Since every vertex count is 1 mod 3, a prime vertex count is always a splitting prime, never inert and never the ramified 3, and it arrives with its own witness: CH(m) is already a value of the norm form at (m, m-1). One convention note: the bases page writes the norm as N(a + b*omega) = a^2 - a*b + b^2; the two forms trade under b -> -b, so in that convention CH(m) = N(m + (1-m)*omega).

The list itself. For k = 1..20 the prime vertex counts are 7, 37, 271, 397, 547, 919, 1657, 1951, 2269, 4219, at k = 1, 2, 5, 6, 7, 9, 12, 13, 14, 19; the other ten values, 91, 169, 721, 1141, 1387, 2611, 2977, 3367, 3781, 4681, are composite. Continuing to k = 40 adds 5167, 6211, 7351, 9241, 12097, 13669. All sixteen primes are 1 mod 3. (Verified by mrlymath::formulas::six test the_prime_vertex_counts_are_cuban_with_norm_witnesses, with a norm-form witness exhibited for each - 4219 = 37^2 + 37*38 + 38^2; the CH(2k) form and the 1 mod 3 residue run to k = 40 in test the_slice_vertex_count_is_centered_hexagonal_at_even_index.) The entry k = 19, value 4219, is easy to skip and is prime.

Four fills on one hexagon

The rest of the slice story needs the four historical families, and their naming needs care. The slice statements below are about the level-1 fill at odd side: the design's parity rule applied to the side^3 grid directly. A triangle of the mesh belongs to a design when the cell it was cut from is filled.

Carpet and net are one self-complementary class - that is why the core page aliases both names to bang dim 3, code 23 - and bang dim 3, code 232 is its complement member, a different truncation of the same class, not a second design; its fill polynomial 4*k^3 - 9*k^2 + 6*k - 1 is the worked corollary on the method page. This page does not call the fourth family void, because the core page's alias void is the canonical bang dim 3, code 24, which fills 2*k^3 - 3*k^2 + k, while bang dim 3, code 129 fills 2*k^3 - 3*k^2 + 3*k - 1 = k^3 + (k-1)^3, the centered cube numbers (OEIS A005898) - same class, different truncation, different polynomial. Nor is it the checkerboard: fill where i + j + l is even is bang dim 3, code 105, a different design again. This page calls it the void, after its two corners (0,0,0) and (1,1,1). (Verified: orbit walks over the 48 signed permutations and cell-by-cell counts, mrlymath::six::topology test the_four_families_fill_the_slice_and_name_their_classes.)

Carpet and net partition the hexagon. Their corner rules partition {0,1}^3 outright - every parity vector has popcount at most 1 or at least 2 - so every cell of any grid is filled by exactly one of the two, and in particular every crossed cell hands its whole section to exactly one. The two slice fills therefore partition the solid hexagon cell for cell:

carpet(side) + net(side) = 6*side^2

(Proved by the corner partition; Verified at every odd side 1..31, with disjointness and covering tested triangle by triangle: 42 + 12, 72 + 78, 204 + 90, 210 + 276, 486 + 240 at side 3, 5, 7, 9, 11, up to 3696 + 2070 = 5766 at side 31, mrlymath::six::topology test carpet_and_net_partition_the_hexagon_triangle_by_triangle.) Two cautions. This does not follow from the 3D complement identity - a volume identity says nothing about how a section decomposes; the extra fact, tested directly, is that the section decomposes along the same cells. And the split is not by the parity of i + j + l: the crossed cells lie in three consecutive diagonal layers and every layer splits between the two families - at side 5 the three layers split 15/3, 7/12, 15/3 - so the partition is by popcount, not by parity. (Refuted; Verified.) Measured by section area instead of by triangle, the same identity extends to even side, where the split is exactly half and half. (Verified at side 1..16 by mrlymath::six::topology test the_layer_weighted_area_is_a_second_route_to_the_fill.)

Components and holes

The carpet's slice fill runs two regimes in alternation, both governed by the same centered hexagonal numbers as the vertex count. At odd k it falls into CH((k+1)/2) disjoint pieces with no holes; at even k it is one connected piece pierced by CH(k/2) holes. Over k = 1..14 the component counts run 1, 1, 7, 1, 19, 1, 37, 1, 61, 1, 91, 1, 127, 1 and the hole counts 0, 1, 0, 7, 0, 19, 0, 37, 0, 61, 0, 91, 0, 127. (Verified at k = 1..14 by mrlymath::six::topology test the_carpet_slice_counts_its_pieces_and_holes_two_ways, which counts pieces by triangle adjacency and takes the hole count twice by routes sharing nothing: the piece count less the Euler number of the filled sub-mesh, and again as void regions the rim never reaches. CH(6) = 91 and CH(7) = 127 at k = 11..14 are predicted by the law and come out right in both regimes, the law itself asserted in the same test.)

The other families sort cleanly. The tree's and the void's slices are hole-free at every k checked (1..10), and the net's carries the same sequence shifted by one in k - holes 1, 7, 19, 37 at odd k = 3, 5, 7, 9 - so carpet and net are the two families that puncture the hexagon, in opposite phase. (Verified, mrlymath::six::topology test the_other_families_puncture_in_opposite_phase, both hole routes agreeing at every size.) CH is a classical sequence; its double occurrence in this slice - once as the vertex count, once as the component-and-hole law - is the fact.

Surface: the face-count recurrence

Now the 3D families, no slice. Substitute a tile of side side to level i and count unit faces: V(i) visible, H(i) hidden between two filled cubes. Cells are multiplicative, cells(i) = fc^i with fc the tile's fill, so V(i) + H(i) = 6*fc^i at every level.

The recurrence (Proved). Substitution creates no new contacts inside a copy; the only new hidden faces arise across an interface where two filled cells of the tile meet face to face. Opposite faces of a copy carry the same fill pattern - checked as arrays, not merely as counts - and face fills are multiplicative, so an interface along axis a contributes p_a^i contacts at level i, with p_a the tile's face fill on that axis, and each contact hides two faces. Writing adj_a for the tile's face-adjacent filled pairs along axis a:

V(i+1) = fc*V(i) - 2 * sum_a adj_a * p_a^i
H(i+1) = fc*H(i) + 2 * sum_a adj_a * p_a^i

When every adjacency-carrying axis shares one face fill l2, the source term collapses to 2*W*l2^i with W = sum_a adj_a, and the system closes: H(i) = 2*W*(fc^i - l2^i)/(fc - l2), V(i) = 6*fc^i - H(i). Both counts then live in the span of fc^i and l2^i - exactly the statement that (V, H) evolves by one fixed 2x2 matrix with eigenvalues (fc, l2), unique whenever W > 0 and l2 != fc.

Prior art. The carpet row is not new. 2*20^i + 4*8^i is OEIS A332705 verbatim, "Number of unit square faces (or surface area) of a stage-n Menger sponge", formula contributed by Allan Bickle, Nov 2022. Its printed terms 72, 1056, 18048, 336384 are A332705(1..4), recomputed here from the closed form and matching exactly. The identification is recorded in REFS.md and belongs here, at the point of derivation. A direct request to https://oeis.org/search?q=id:A332705&fmt=json returns the name verbatim as quoted above, offset 0, data 6, 72, 1056, 18048, 336384, ..., and the formula block "From Allan Bickle, Nov 28 2022: a(n) = 2*20^n + 4*8^n". So the closed form, the attribution and the term alignment 72, 1056, 18048, 336384 = A332705(1..4) are all Verified at source.

(Verified against brute-force face counts at side 3 to level 4, side 5 to level 3, side 7 to level 2, mrlymath::three::census tests the_face_ledger_prints_the_family_closed_forms and the_face_matrix_fits_its_eigenvalues_and_predicts; the carpet at side 3 reads visible 72, 1056, 18048, 336384 against hidden 48, 1344, 29952, 623616. Matrices fitted exactly over the rationals from levels 1..3 predict the next level, and trace and determinant give (fc, l2) in all six fitted cases - e.g. carpet at side 3: [[12, 4], [8, 16]], trace 28 = 20 + 8, determinant 160 = 20 * 8.) Across bases the second eigenvalue is a face count of the family: side^2 - floor(side/2)^2 for the carpet, floor(side/2)^2 for the net, ceil(side/2)^2 for the tree - 8, 1, 4 at side 3, 21, 4, 9 at side 5, 40, 9, 16 at side 7, nine of nine. (Verified.) The tree is the case that makes the rule bite: its copies touch only along one axis, and it is that axis's face fill, 4 rather than 6, that drives the recurrence.

The void row is the degenerate case, and a second eigenvalue of 0 for it is wrong. (Refuted.) With no face-adjacent pair, W = 0, the hidden channel is never fed, and the substitution step is literally fc times the identity - eigenvalues (fc, fc) - while the face data sits on a single ray, so no 2x2 matrix is determined by it at all. The law for that family is the next section's, with no second eigenvalue to quote.

This matrix shares nothing but a name with the digit-borrow transfer matrices this lane uses elsewhere: those are automata on the binary digits of a position, this one is a 2x2 face ledger. Two arguments, one word.

No hidden faces

(Proved.) Two cells share a unit face exactly when they differ by 1 in one coordinate - which flips exactly one bit of the parity vector. So a design ever hides a face if and only if two of its filled corners sit at Hamming distance 1. The void's corners are at distance 3, so no two of its filled cubes ever meet face to face, at any side and any level: every face is exposed, and

surface(k) = 6 * cells(k), cells(k) = 2*k^3 - 3*k^2 + 3*k - 1 = k^3 + (k-1)^3

with surface(level) = 6 * 9^level across fractal levels at base 3 - no hidden-face correction term, ever. (Proved; Verified by direct face counts at k = 1..12, in 2D as edges = perimeter, and at levels 1..4, mrlymath::three::census test the_void_buries_no_face.) Hamming distance is preserved by cube symmetry, so total exposure is a class property. The test is also exhaustive: sweeping all 256 designs at side 3, 5, 7, surface = 6 * cells holds for exactly the 35 designs whose filled corners are pairwise at Hamming distance at least 2 - the independent sets of the cube graph - and they form whole classes: bang dim 3, code 0, bang dim 3, code 1, bang dim 3, code 6, bang dim 3, code 22, bang dim 3, code 24, bang dim 3, code 105. (Verified by mrlymath::three::census test total_exposure_holds_for_the_independent_corner_sets and mrlymath::bang::universe test total_exposure_names_the_independent_corner_sets.)

Corners

A last piece of 3D background, because it explains an asymmetry the complement identity leaves open. (Proved.) At odd side 2k - 1 a grid-corner coordinate a sees the cell indices a-1 and a, clipped to the grid: the boundary values 0 and side see only an even index, the 2k-2 interior values see one of each parity. So a design touches every one of the (side+1)^3 grid corners if and only if its rule contains the all-even corner - the grid corner at the origin can be touched by no other cell. Carpet, tree and the void all contain it, so their solids touch the whole grid, 8*k^3 corners. The net does not - it needs two odd coordinates - so a grid corner is touched exactly when at least two of its coordinates are interior:

net vertices = m^3 + 6*m^2 = 8*k^3 - 24*k + 16 = 8*(k-1)^2*(k+2), m = 2k-2

short of the full grid by 24*k - 16. (Proved; Verified by brute force at k = 1..20 in 3D and k = 1..24 in 2D, where the same argument gives 4*k^2 - 4, and the all-even criterion swept over all 256 designs at k = 1..3: exactly the 128 rules containing the all-even corner touch every corner, mrlymath::three::census tests the_net_falls_short_of_the_grid_corners and the_all_even_rule_touches_every_grid_corner, the 128 rules named again by mrlymath::bang::universe test half_the_rules_hold_the_all_even_corner.) Two footnotes. Odd side is load-bearing: at even side the boundary coordinate sees an odd index and every family loses corners. And this is the geometric half of the complement story on the method page: the carpet and its complement trade cells exactly, but not vertices, because complementing the rule loses the all-even corner that every boundary grid corner depends on.

The slice ink, proved order by order

The Walsh spectrometer - the exact quasipolynomial that reads a 3D parity design's slice ink off its Walsh spectrum, level by level, ink_dim(side) = S0 - (1/2) S3 s + [(2/3) S1 - (1/3) S2 s]/side + [(2/3) S2 - ((1/3) S1 + (1/2) S3) s]/side^2 at s = (-1)^((3*side-1)/2), with S_j the design's level-j Walsh sum, proved in the walsh-spectrometer lane - has all three orders proved: the constant terms and the 1/n and 1/n^2 terms alike. (Proved.)

Fix odd side and let

P_side = { (x,y,z) in {0..4*side-1}^3 : x + y + z = 6*side - 2, z even }

be the plane the slice lives on. For a macro-parity triple e in {0,1}^3 let N_e(side) count the points of P_side whose macro coordinates (floor(x/4), floor(y/4), floor(z/4)) are congruent to e mod 2. Then N_e(side) depends only on the weight k = |e|, and with side 2h + 1,

N_k(2h+1) = [t^(3h+1)] (1 + 6t + t^2) E_h(t)^(3-k) O_h(t)^k,
E_h(t) = 1 + t^2 + ... + t^(2h), O_h(t) = t + t^3 + ... + t^(2h-1).

The three ingredients are all elementary. Writing x = 4u + r_x and likewise for y and z, the residue sum r_x + r_y + r_z can only be 0, 4 or 8, with multiplicities 1, 6, 1 read off (1 + t + t^2 + t^3)^2 (1 + t^2) - the same 1:6:1 law as the constant terms, now kept at the lower orders instead of averaged away. Each macro variable ranges over 0..2h, so a parity triple of weight k contributes E_h^(3-k) O_h^k. And 6*side - 2 = 0 mod 4 because the side is odd. The generating function proves on sight that only the weight matters and never which entries are odd.

Extraction is a finite binomial calculation, not an asymptotic: substituting the two geometric series and using [t^m](1-t^2)^(-3) = C(m/2 + 2, 2) for even m >= 0 and 0 otherwise gives the four exact quadratic quasipolynomials, (9*side^2 + 18*side + 21)/8 at k = 0 and side 1 mod 4 down to (9*side^2 - 18*side + 21)/8 at k = 3 and side 3 mod 4. Since slice ink is linear in the eight N_e(side), dividing by the slice normalization can produce only constant, 1/side and 1/side^2 terms, with the two residue classes carried by s = (-1)^((3*side-1)/2). No approximation enters anywhere. (mrlydemo::walsh_spectrum is the generator: it reads a code's four level sums and prints the law's fill at every odd side, and the mrlydemo fixture holds those fills against slice_census, the hexagon counted triangle by triangle, at codes 23 and 11 over side 1..11, zero mismatches. (Verified.) Neither test code carries a weight-3 corner, so N_3(side) is untouched by the crate; the eight-triple split, the popcount aggregate against C(3,k) N_k(side) and the 28 layers to side 55 are the walsh-spectrometer lane's own check.) The spectrometer demo points the slice at any of the 256 designs, bars its four level sums, and plots this law against the ink counted triangle by triangle at every odd side, with a mystery mode that hides the code and asks you to read the recipe back.

Scope guard. This page is the base-3 census at dim 3 and nothing on it speaks to the slice DIMENSION at other dimensions. The sign law sgn(dim_slice - (log(fill) / log(base) - 1)) = (-1)^(dim+1) at dim_slice the dimension of the slice, Conjecture S, is on no note - spectra names the lane that carries it, under WHERE THE REST LIVES - and the transfer matrix both it and Conjecture R run on is defined once in cuts. Nothing here should be read as evidence for or against either.

Coprimality on the slice

One pointer. The arithmetic of the slice - which of a design's slice points are visible from the origin - lives on the coprimality spine under "Coprimality on the slices", with the generator lab/py/slice-coprimality. The one-line summary: a slice point's gcd divides its height, so slice coprimality is finite arithmetic per height, the base prime peels the central slice one step off-centre, each foreign prime costs 1/p^2 rather than the solid's 1/p^3, and the central slice's bill is the factorization of the repunit in the base, so its density never converges.

Where the numbers live

Everything above is computed by ../crates/mrlymath: six::census carries the mesh census, its extrapolations and the adjacency lemma, six::topology the fills, the partition, the pieces and the holes, formulas::six the closed forms and the prime vertex counts, and three::census the face recurrence, total exposure and corners, with bang::universe naming the corner-set classes; mrlydemo's slice_census and slice_series put them in the browser behind the slices demo. The profile identity is checked by lab/py/slice-ladder-controls, the slice pointer is owned by lab/py/slice-coprimality, and the slice ink is rebuilt by mrlydemo::walsh_spectrum, whose law fills the mrlydemo fixture holds against slice_census triangle by triangle. Sequences that come out of this construction are held to the standard in the sequence ledger.