|
|
| (24 intermediate revisions by 2 users not shown) |
| Line 3: |
Line 3: |
| * Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.) | | * Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.) |
| * Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice. | | * Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice. |
| * A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink] | | * A scale imprint is that of a Fokker block if and only if it is the [[product word|product]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink] |
| * If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s | | * If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s |
| * Any convex object on the lattice can be converted into a hexagon. | | * Any convex object on the lattice can be converted into a hexagon. |
| * Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps. | | * Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps. |
| | | * An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]]. |
| == Unproven Conjectures == | | == Conjectures == |
| * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. | | * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. |
| == MV3 proofs ==
| |
| Under construction
| |
| === Definitions and theorems ===
| |
| Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used).
| |
| ==== Definition: Unconditionally MV3 ====
| |
| ''S'' is ''unconditionally MV3'' or ''intrinsically MV3'' if ''S'' is MV3 for all possible choices of step ratio x:y:z.
| |
|
| |
| ==== Definition: EMOS ====
| |
| ''S'' is ''elimination-MOS'' (EMOS) if the result of removing (all instances of) any one of the step sizes is a MOS.
| |
| ==== Definition: PMOS ====
| |
| ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
| |
| ==== Definition: AG ====
| |
| ''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties:
| |
| # ''S'' can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
| |
| # ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.
| |
|
| |
| These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.
| |
|
| |
| For theorems relating to the AG property, see [[AG]].
| |
| ==== Definitions: Billiard scale ====
| |
| Let n = a_1 + ... + a_r be the scale size, w a scale word with signature a_1 X_1, ..., a_r X_r, let L be a line of the form L(t) = (a_1, ..., a_r)t + v_0, where v_0 is a constant vector in R^r. We say that L is ''in generic position'' if L contains a point (0, α_1, α_2, ... α_{r-1}) where α_i and α_i/α_j for i != j are irrational.
| |
|
| |
| Say that an r-step scale ''S'' is a ''billiard scale'' if any appropriate line in generic position, (a_1, ..., a_r)t + v_0, has intersections with coordinate level planes x_i = k that spell out the scale as you move in the positive t direction.
| |
| <!--===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====
| |
| Assume wlog there are more L's than s's.
| |
|
| |
| Take the graph of the brightest mode of the mos, M_b(x) (right = L, up = s). We claim that this is the required graph of F(x) = floor(b/a*x).
| |
|
| |
| M_b <= F: Prove that F(x) describes a mos.
| |
|
| |
| Say F has #s s's and #L L's across interval [m, m']. Say there is #s+t small steps and #L-t large steps on some k step [r, r'], t >= 2. This implies that the slope of the line b/a* x itself satisfies
| |
|
| |
| (F(m')-F(m)-1)/(m'-m) <= b/a <= (F(m')-F(m)+1)/(m'-m).
| |
|
| |
| (bounded by "floor minus ceiling" and "ceiling minus floor" slopes; this is because x-x' <= x-floor(x') <= floor(x)+1-floor(x').)
| |
|
| |
| Rearranging,
| |
| F(m') - F(m) - 1 <= b/a(m'-m) <= F(m') - F(m) + 1
| |
|
| |
| But F(m') -F(m) = #s and m'-m = #L. So #s -1 <= b/a*#L <= #s+1. Do the same thing for the "bad" interval [r', r] and you get #s+t-1 <= b/a(#L-t) <= #s+t+1.
| |
|
| |
| Thus b/a#L <= b/a(#L-t), a contradiction.
| |
|
| |
| M_b >= F: (bc it's a mos) Suppose there is an x-value n_0 where M_b(n_0) <= F(n_0) - 1. n_0 > 1 since otherwise, M_b(1) < 0. Let k = min(n_0, n-n_0), n = scale size. Then find three different k-mossteps/average slopes by taking the interval [n_0-k, n_0] before n_0, one interval containing n_0 and one interval after n_0. (We already know that mosses are slope-LQ.)
| |
|
| |
| Since M_b is a mos mode, there is a k-step within [0, n_0] that has the slope which is just smaller than (F(n_0)-1)/n_0 (1). Similarly, there is a k-step within [n_0, n] that has the slope which is just bigger than (F(n_0)+1)/(n-n_0). These slopes are "two or more steps away" from each other, which is a contradiction. (State this more formally)
| |
|
| |
| <!--==== MV3 Theorem 1 (WIP) ====
| |
| ''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:''
| |
| # ''S is unconditionally MV3.''
| |
| # ''(a) S is PMOS, or (b) S is of the form x'y'z'y'x', or (c) S has signature nx ny nz, n ≥ 2.''ppp
| |
| # ''(a) S is AG *and* is of the form ax by bz, or (b) S is of the form x'y'x'z'x'y'x', or (c) S is one of the exceptions to PMOS in statement 2.''
| |
| ====== MV3 implies LQ except in the case "xyzyx" (WIP) ======
| |
|
| |
| ====== MV3 + LQ implies EMOS (WIP) ======
| |
| Proof sketch:
| |
|
| |
| Let L = L(t) = (a, b, c)t + (0, α, β) be a line in generic position corresponding to the signature aX bY cZ. The projection matrices
| |
|
| |
| <math>
| |
| P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \
| |
| P_2 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \
| |
| P_3 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},
| |
| </math>
| |
|
| |
| (which map (x, y, z) to (x, y), (y, z) and (z, x), respectively) map L to lines in R^2 that are in generic position (i.e. they intersect the x- and y-axes at irrational points). The projections record intersections with two of the planes to intersections with x- and y- axes, and these intersections must spell out the result of removing one of the step sizes; hence the resulting scales must be mosses.
| |
|
| |
| (MV3 has not been used yet)
| |
|
| |
| This in particular implies that xyzyx is not LQ.
| |
| ====== MV3 + EMOS implies PMOS (WIP) ======
| |
| Suppose w_{YZ} is a word made by identifying Y and Z into one letter, say Q.
| |
|
| |
| Suppose w_{YZ} has a k-step, 1 < k <= n/2, which comes in three sizes
| |
|
| |
| * v_1 = A_1 X + B_1 Q,
| |
| * v_2 = A_2 X + B_2 Q,
| |
| * v_3 = A_3 X + B_3 Q.
| |
|
| |
| If some A_i and A_j differed by more than 2, a contradiction would result, as all intermediate combinations must be attained (since scooting over by one step changes the numbers of Q's and X's by <= 1). So we assume we have sizes
| |
| * v_1 = v(s_1) = AX + BQ,
| |
| * v_2 = v(s_2) = (A-1)X + (B+1)Q,
| |
| * v_3 = v(s_3) = (A+1)X + (B-1)Q.
| |
|
| |
| Plan: v_2 and v_3 are sizes that are problematic when they occur together.
| |
|
| |
| Let EX(w), EY(w), EZ(w) be mosses that result from eliminating X, Y and Z. MV3 implies that for any possible choices of s_i, EX(s_1), EX(s_2), EX(s_3) each only comes in one possible size as B-, (B+1)- and (B-1)-steps in EX(w). [EX means eliminate X]
| |
|
| |
| ====== PMOS implies AG (except in the case xyxzxyx) (WIP) ======
| |
| -->
| |
|
| |
|
| [[Category:Fokker block]] | | [[Category:Fokker block]] |
| Line 105: |
Line 15: |
| [[Category:Rank 3]] | | [[Category:Rank 3]] |
| [[Category:Scale]] | | [[Category:Scale]] |
| [[Category:Theory]] | | [[Category:Pages with open problems]] |