Spectra

Spectra

The diagonal slice of a cube fractal is a hexagon with a tile grammar. Everything published about that grammar stops at base 3; this page asks what happens at every other odd base.

  • The base axis is the open thread. The dimension axis - the same d - 1 comparison run along D at fixed base 3 - is settled and lives on the shelf.
  • The spectra demo draws the Laplacian of a design, its degenerate families and the slope that reads the spectral dimension.
  • Every number on this page is regenerated by lab/odd-base-slice-grammar, a raster-free digit recursion cross-checked against a direct layer census. What stays Conjecture is the grammar itself: a 2x2 map reproducing cell counts is weaker than a substitution acting on tiles, and only base 3 has the latter from a source.

WHAT IS KNOWN

Cited facts only.

  • The cut is Perez-Duarte's. "Slice of Menger", showing "a very interesting pattern of stars and hexagons" (Flickr), credited directly by Abel.
  • Hart popularised it in Mathematical Impressions, with a Scientific American mirror.
  • Cook published working code, applying the base-3 digit predicate to a plane through the cube's centre (Cook 2011). Its prose says the normal runs to (1, 1, 1) while the listing sets normal = (1, 1, 0.5), so the published plane is not the centroid diagonal.
  • The substitution rule is Abel's, verbatim: "replace each hexagon with 6 hexagons and 6 triangles, and replace each triangle with 1 hexagon and 3 triangles". As a matrix on (hexagons, triangles) that is [[6,1],[6,3]], trace 9, determinant 12.
  • The dimension is Abel's, and he states it as a computation rather than a proof: 3^d = (9+sqrt(33))/2, so d = log_3((9+sqrt(33))/2) = 1.8184, with the hedge "it takes a bit more work to turn the above computation into a full proof".
  • A299916 counts holes, not tiles. Its name is a(n) = A299914(2n+1), offset 0, terms 1, 6, 42, 306, 2250, 16578, 122202, signature (9,-12); the Menger reading is a comment on the entry, not the definition. That the two are the same object is Proved, by the hexagram bijection in cuts, and the index shift is load-bearing: the mesh-triangle census is A299916(n+1).
  • The two-tile move is published at base 3. Hocking's Bridges paper, resolved in REFS, treats the base-3 slice as a closed fractal family on a hexagon and a triangle, which is a directed-graph iterated function system under another name. What is unpublished is every other base.
  • The nearest real theory looks elsewhere. Slice dimension is solved for almost every plane (Marstrand 1954, Mattila 1975); the centroid diagonal is a single maximally arithmetic plane, exactly the case those theorems exclude. So a plane landing off d - 1 contradicts nothing and dodges no theorem.
  • The upstream is grey literature. A photograph, a video, three blog posts and an OEIS comment. Adjacent work does not close the gap: one generalisation runs along dimension, another changes the solid, and none touches base 5, 7 or 9.

THE CLAIM

  • Conjecture. For every odd base b, the centroid diagonal slice of the parity solid mrly_bang_d3_23 is a graph-directed set on exactly two tiles, whose 2x2 integer substitution matrix is a fixed rational function of b within each class of b mod 4, and whose slice dimension sits above d - 1 when b = 3 mod 4 and below it when b = 1 mod 4.
  • The method builds no raster. The plane x + y + z = 3*b^L/2 meets three diagonal layers and the coordinate sum splits as sum_k b^k * sigma_k over independently chosen digit triples, so the tile census is a memoised digit recursion with nothing allocated.
  • The four rules. Verified by lab/odd-base-slice-grammar; that they are tile grammars is the claim above. Two-term rules x9 -12 at b = 3, x11 +62 at 5, x42 -288 at 7, x28 +693 at 9, with dimensions 1.8184 / 1.6869 / 1.8026 / 1.7204 against d - 1 = 1.7268 / 1.7304 / 1.7430 / 1.7544.
  • Their arithmetic is only self-consistent: each printed dimension is log_b of the dominant root of its own printed rule, so a wrong rule and its own wrong dimension agree by construction.
  • The base-3 rung is the one external target, and it passes. Verified. A cold census returns Abel's [[6,1],[6,3]] and hexagons 1, 6, 42, 306, 2250, 16578, 122202; the grammar cannot be fitted to that.
  • The matrices. Verified by lab/odd-base-slice-grammar. [[7,3],[30,4]] at b = 5, [[30,3],[24,12]] at 7, [[19,9],[96,9]] at 9, all non-negative, reproducing cell counts level after level. Counts obeying a fixed 2x2 map is weaker than a substitution acting on tiles, and only b = 3 has the latter from a source.
  • The closed form. Verified at b = 3..21 by lab/odd-base-slice-grammar; every odd b is the claim above. On (hexagons, triangles): [[3(b+1)(3b-1)/16, (b+1)(b+5)/32], [3(b+1)^2/8, 3(b+1)^2/16]] when b = 3 mod 4, and [[(3b^2+6b+7)/16, 3(b-1)(b+3)/32], [3(b-1)(3b+5)/8, (b+3)^2/16]] when b = 1 mod 4. Both reproduce every census matrix at b = 3..21.
  • What would falsify it. Any odd base whose slice needs a third tile symbol; any census breaking its own two-term recurrence past the fitted levels; any base where the mod-4 side is wrong.
  • What a result would be worth. No candidate row returns an OEIS hit, and no paper treats the diagonal cross-section at any base but 3. A confirmed row at b = 5 is new; an unconfirmed one is a number in a file.

OPEN QUESTIONS

  • Which sponge is the base-b sponge? Cook's predicate is "at most one coordinate in the middle third"; this page inherits "at most one odd coordinate" from mrly_bang_d3_23. The two agree at b = 3 and nowhere else, and there is no canonical base-5 Menger sponge, so the whole generalisation rests on an unstated choice.
  • The choice is not cosmetic, and the two rules land on opposite sides. Verified. At b = 5, D = 3 the middle-digit rule fills 112 of 125 and its central diagonal slice sits ABOVE d - 1, excess +2.888e-02 with dim_slice = 1.960651 against 1.931768; the odd-coordinate rule fills 81 of 125 and sits BELOW, 1.6869 against 1.7304. No d - 1 claim at b = 5 can be quoted without naming the rule.
  • Does the mod-4 split have a mechanism, or is it numerology? Conjecture. It holds at every odd b up to 401 by exact rational comparison in lab/odd-base-slice-grammar, so it is not four data points. The candidate mechanism - the middle diagonal layer sits at coordinate sum 3*(b-1)/2, odd exactly when b = 3 mod 4, forcing a different parity of cell into the middle layer in each class - is unwritten and unchecked.
  • The split is a property of the rule as well as the base. The mod-4 statement is about the odd-coordinate solid, and the middle-digit solid contradicts it at b = 5. Any statement of it must pin base, dimension and digit rule before it means anything.
  • Is the two-tile grammar geometric, or only arithmetic? Reproducing cell counts is weaker than a substitution acting on tiles, and no source supplies the latter past b = 3.
  • Does the base axis factor at all? cuts walks the dimension axis at fixed base 3 and has a theorem there: the order law, from a factorisation of the digit polynomial. The same three-step argument - digit polynomial, carry contraction, palindromic symmetry - is what the two-tile claim has never been given, and it is exactly the kind of statement that would decide the mod-4 split.

WHERE THE REST LIVES

  • The dimension axis at fixed base 3: the ceil(D/2) order law, the product formula over 3-adic angle towers, and the unconditional pinning |rho_D - fill/3| <= 2(D-1)/3 are the carlomitchener/research/slice-recurrence-order lane.
  • The sign law in every even dimension at bases 3 and 5, the certificate machines, the transient constant ln(R)/4, the tent rank law and the layer-2 window law are the carlomitchener/research/slice-sign-even-half lane.
  • The layer-2 window itself - its generator g_D, its ceiling C_D, the family shift law and the corrector law behind them - is regenerated by lab/smith-window.
  • So are the two statements that close the corrector law. Law E's slot has length N = J(e) - J(e-1) and a position p measured from the octave centre; the tent identity is min(p, N - 1 - p) = C_D - deg g_D, and the reach law is reach = R - jmax = 3 min(p, N - 1 - p) + 2 [e even] + [k odd](1 + p mod 2), with one row per odd octave escaping it at D = 4^m + 3.
  • Off those escaping rows floor(reach/3) = C_D - deg g_D + [k odd and e even]; on them it reads 1 against C_D - deg g_D = K - deg g_D = 0, so min(K - deg g_D, floor(reach/3)) = C_D - deg g_D at every row and the corrector law's statement reads off (b, e, k, R) with no span test in it. The deduction behind it still carries one.
  • The carry matrix M_even is defined once, in cuts, and is not redefined here.
  • The hexagram bijection and the mesh census: cuts. The hexagon mesh itself: slices. The fill polynomial of mrly_bang_d3_23: method.
  • Every finding on a tagged line: DISCOVERIES. Every source resolved: REFS.