radix-designs

• Dials the place slot of a design: a word is placed by a base in a ring of the plane rather than by base place value on Z^dim, and the classical self-similar curves become codes in that list.
• Objects: a ring R is Z[i] or Z[omega] with omega^2 = -1 - omega; a base is an element base of R of norm N(base) = q >= 2; the canonical residue system mod base is the q representatives of least norm with ties broken by argument in [0, 2 pi); a digit code names a subset of that system, one bit per residue in that order; a digit set is one representative per named class; a twist is a unit u_d of R per digit; the place map is phi_d(x) = (u_d x + d) / base and the word d_1 ... d_level lands on phi_(d_1)(... phi_(d_level)(0)).
• A design is the quintuple ring, base, digit code, representative vector, twist vector; a code alone names only the classes, because phi_(d + base m)(x) = phi_d(x) + m moves the attractor.
• The exact point of a word is carried as an integer of the ring: base^level times the point is sum_(j=1..level) (prod_(i<j) u_(d_i)) d_j base^(level-j), so every count below is exact and the plane is entered once, at printing.
• The generator is mrlynum::radix: Base with residues, class, mirrored and group; Radix with words, plane, distinct, fill, dimension, code and canonical; and the designs koch, gasket, twindragon, terdragon, flowsnake and tile.
• koch: the design base 3 on Z[omega] with digits 0, 1, 2+w, 2 and twists 1, 1+w, -w, 1 at levels 1 to 5, printing the point count, the fill, the distinct-point count and the largest plane distance to a float evaluation of the four maps z/3, e^(i pi/3) z/3 + 1/3, e^(-i pi/3) z/3 + 1/2 + i sqrt(3)/6, z/3 + 2/3.
• Those four maps are phi_d coefficient for coefficient, because e^(i pi/3) = 1 + w, e^(-i pi/3) = -w and (2 + w)/3 = 1/2 + i sqrt(3)/6, so (u_d x + d)/3 at (d, u) = (0, 1), (1, 1+w), (2+w, -w), (2, 1) is that list in order. The identity is exact and needs no run; the printed 1.241e-16 is the float self-check of the crate's ring arithmetic and measures nothing else.
• compare: each object against an independent iterated function system written in f64 from the maps the page states, iterated on the seed 0, word for word against Radix::plane at the same level, printing the largest distance and the motion allowed.
• The gasket is compared against the three similarities of ratio 1/2 fixing the vertices of an equilateral triangle, placed at (1, 1), (3, 1), (2, 1 + sqrt 3), which fixes the gasket only up to similarity, so a translation and a positive scaling are allowed: they are pinned by the two corresponding words 0^level and 2^level and then measured at every word, and the turn residual printed beside them is the check that no rotation was needed. The fitted translation is (1, 1)(1 - 2^(-level)) and not (1, 1), because both systems are seeded at 0 and 0 is not the image of 0 under the motion; it converges to (1, 1).
• The Koch curve, the twindragon and the norm-seven tile are compared with no motion allowed, and each is a float self-check rather than an independent identification: z/(1+i) and (z+1)/(1+i) are the radix maps of base 1+i on its two residues by definition, the seven maps (z + d)/(3+w) over the canonical residues are the radix maps of that base by definition, and the four Koch maps are phi_d by the identity above.
• compare also settles the terdragon. Reading the L-system F -> F + F - F at 120 degrees as a turtle, three segments to a level, and normalising by the endpoint gives base 2+w, digits 0, 1, 1+w and twists 1, w, 1: the 3^level segment starts match plane(level) word for word at level 8 to 7.5e-16, while the untwisted code 7 misses by 1.06. The reading is the terdragon's own L-system and is carried by no source here.
• named: five codes with their ring, base, q, the size |F| of its digit set F, the fill law, the similarity dimension 2 log|F| / log q, and the distinct-point counts to the largest level under a budget of 200000 points; no name is tested here, only in compare. It closes on the twisted glue witness.
• The glue witness is Z[i], base 2, F = {0, 1}, twists 1, -1, whose distinct-point count is 2^(level-1) + 1 at every level, printed and checked to level 16. Proof: the scaled point of the word whose 1s sit at positions j_1 < ... < j_t is sum_(k=1..t) (-1)^(k-1) 2^(level - j_k), an alternating sum of strictly decreasing powers of two with top exponent at most level-1; such a sum is 0 or lies in [1, 2^(level-1)], since the alternating tail is smaller than the leading term; and every integer n of [1, 2^(level-1)] is reached by exactly one choice, the greedy one, taking 2^a for the least a with 2^a >= n and recursing on n - 2^a, whose modulus is below 2^(a-1). So the count is 2^(level-1) + 1 at every level.
• today: every plane code at the rational bases m = 2 and m = 3, of norm q = m^2, at level 2, the radix word set against the mrlymath::bang cells of the same code, through the pixel map bit r m + c of the code is the cell at row r and column c, the column the real part and the row the imaginary part; it prints the count of codes checked and the count of mismatches. The plane code is read in box row-major order and not in the canonical residue order, which differs from it at m = 3.
• census: the classes of digit CODES under the residue action at Z[i] bases 2, 1+i, 2+i and Z[omega] bases 2, 2+w, 3, 3+w, by Burnside over the group and by a direct orbit walk over all 2^q codes, which must agree; it prints the canonical residue system of each base and the twist count at base 3 on Z[omega].
• The group of a base is the units acting on residues by multiplication, together with conjugation when the conjugate of the base is an associate of the base. Multiplying every digit by a unit v carries the attractor to v times the attractor over the same base, since v phi_d (v^(-1) x) = phi_(v d)(x), so the units act; conjugation carries base base to base conj(base), which is the same radix system exactly when conj(base) is an associate of base. That holds at 2 and 1+i on Z[i] and at 2, 2+w and 3 on Z[omega], and fails at 2+i and 3+w, so the abstract group R^* semidirect <conj> has order 2|R^*| at the first five bases and |R^*| at the last two.
• That abstract order is not the order of the permutation group it induces. The action on the residues is not faithful: the image has order 2, 1, 4, 6, 2, 12, 6 at the seven bases against the abstract 8, 8, 4, 12, 12, 12, 6, and at 1+i every element acts as the identity. Both orders are printed. Burnside over the abstract list stays correct, because the list is the image of one abstract group with each element once.
• The census counts digit CODES up to the group and never designs up to similarity or up to conjugacy, and the three quotients are incomparable: affine prints all three counts and the witnesses.
• affine: at every base of the census and every |F| from 0 to q, the classes of the untwisted canonical digit sets under the similarity group and under the full conjugacy group, in exact arithmetic, printed beside the code orbits of the same cell. Conjugating the place maps by an invertible real affine h(x) = H x + s gives (y + H d + s(base - 1))/base, again an untwisted place map of the same base exactly when H commutes with multiplication by 1/base, and s(base - 1) sweeps the plane because N(base) >= 2 forces base != 1. So the conjugacy group is the centraliser of 1/base extended by translations, and the similarity classes are the finer quotient in which H is restricted to v in Q(i)^* or Q(w)^* joined by the mirror.
• At a non-real base the centraliser of 1/base in the two by two real matrices is C, so the conjugacy group IS the similarity group and the two columns agree; at a real base 1/base is the scalar (1/base) I, it commutes with every H, and the conjugacy group is the whole real affine group GL_2 semidirect R^2, so the affine column is computed over GL_2(Q) semidirect Q^2 at 2 on Z[i] and at 2 and 3 on Z[omega].
• The mirror x -> v conj(x) + t preserves the untwisted family of the same base exactly when conj(base) = base, since a direct conjugacy keeps the derivative 1/base and a mirror one sends it to 1/conj(base); being an associate is not enough. At a real base the mirror is one element of the full affine group and not the only new one, and it is load-bearing for the similarity column alone.
• Cost and controls: a similarity test anchors two digits and an affine one anchors three, so a pair of codes of digit count n costs n^4 with a linear membership scan, and the affine column is counted by an exact normal form over Q instead, one key per code. The affine key is checked against two explicit conjugacies, H = [[0, 2], [1, -1]] carrying code 131 to code 137 and H = [[1, 1], [0, 1]] carrying code 7 to code 131; the similarity classes are asserted to refine the affine classes in every cell; the f64 rerun of the similarity normal form checks the exact arithmetic and not the group; and the per-|F| orbit counts summing to the census class count constrains the CODE column alone.
• Domain: all 2^q codes per base split by |F| for census and for affine, all 16 + 512 plane codes at the two rational bases for today, levels 1 to 5 for koch, levels 9, 7, 14, 5 and 8 for the gasket, the Koch curve, the twindragon, the norm-seven tile and the terdragon in compare, levels 1 to 11, 17, 11, 6, 8 for the five codes of named in the order printed, and levels 1 to 16 for the glue witness.

RUN

bash scripts/cargo.sh cargo run --release -p radix-designs -- koch compare named today census affine

• Run from mrlyprod/. Verbs may be given in any combination and run in order in one process; no argument runs all six.
• koch, compare, today and census each run in under 0.01 s, affine in 0.08 s and named in 0.08 s; the whole study including the build is 1.3 s of wall clock.
• bash scripts/cargo.sh cargo test -p mrlynum radix pins the residue systems, the Koch levels 1 and 2, the real-base cell and the fill law inside the crate.

WITNESSES

• beneath, The radix dial - the closed form sum_(j=1..level) (prod_(i<j) u_(d_i)) d_j b^(-j) and the fill law |F|^level: koch and named.
• beneath, The radix dial - today's designs as the untwisted real-base row, 528 plane codes at m = 2, 3 and level 2 with 0 mismatches, the code read in box row-major order: today.
• beneath, The radix dial - the canonical residue system of a real base m is the box {x + y i : 0 <= x, y < m} at m = 2 and at no larger m, since the box holds m-1 of norm (m-1)^2 >= 4 where its class holds -1 of norm 1; at m = 3 the canonical system is 0, 1, i, -1, -i, 1+i, -1+i, -1-i, 1-i: today.
• beneath, The radix dial - the twisted glue witness Z[i], base = 2, F = {0, 1}, twists 1, -1, where level 2 has fill 4 and 3 distinct points and the distinct count is 2^(level-1) + 1 at every level: named.
• beneath, The radix dial - the gasket is code 7 at base 2 on Z[omega] up to the printed translation and scaling, the terdragon is code 7 at base 2+w twisted by 1, w, 1 and not untwisted, and the Koch curve is code 147 at base 3 twisted by 1, 1+w, -w, 1: compare.
• beneath, The radix dial - the class counts 12, 4, 12, 8, 6, 84, 28 over 16, 4, 32, 16, 8, 512, 128 codes at the seven bases, Burnside and orbit walk agreeing at every base, and the acting image orders 2, 1, 4, 6, 2, 12, 6: census.
• beneath, The radix dial - the code census is neither the similarity census nor the conjugacy census, and no two of the three quotients are comparable: over the seven bases the similarity classes total 5, 3, 8, 6, 4, 117, 22 and the affine classes 5, 3, 8, 5, 4, 88, 22 against the code classes 12, 4, 12, 8, 6, 84, 28, the count of untwisted canonical digit sets exceeding the code count at base 3 on Z[omega] under either name, 88 or 117 over 84; there the 84 three-digit codes fall in 13 orbits, 9 similarity classes and 2 affine classes, codes 131 and 137 splitting an orbit and codes 7 and 42 merging two under similarity, while 131 and 137 are affinely conjugate by H = [[0, 2], [1, -1]] and 7 and 131 by H = [[1, 1], [0, 1]]: affine.

SOURCES

• Hutchinson 1981, 3.3(2) - the Koch curve is the attractor of four similitudes each carrying a_1 a_5 to a_i a_(i+1) with positive determinant; the vertex positions are read from its Figure 3.2 and not from its text, so this study states the four maps itself.
• Gilbert 1986 - the radix representation in base -n + i tiles the plane. None of the bases here is of that form, so nothing in this study rests on it.