Counting d.o.f.s in periodic frameworks. Louis Theran (Aalto University / AScI, CS)

Transcription

1 Counting d.o.f.s in periodic frameworks Louis Theran (Aalto University / AScI, CS)

2 Frameworks Graph G = (V,E); edge lengths l(ij); ambient dimension d Length eqns. pi - pj 2 = l(ij) 2 The p s are a placement of G / realization of (G, l)

3 Frameworksʹ Deformation space = local solutions to pi - pj 2 = l(ij) 2 mod rigid motions Degrees of freedom = dim (deformation space)

4 Rigidity, flexibility Rigidity question: is the deformation space zero dimensional? Rigid Flexible

5 Quiz!

6 Quiz! [Thorpe] wing a piece of bilayer of vitreous silica imaged in AFM and STM mode to show the O atoms as red dis ed discs FIX. The local covalent bonding leads to the yellow equilateral triangles that are freely jointed

7 Quiz!

8 Quiz!

9 Combinatorial rigidity Combinatorial rigidity question: which graphs are rigid? Deformation space is a finite-dimensional algebraic set, well-def d dimension

10 D.o.f. heuristic ( Maxwell counting ) m d n - d(d + 1)/2 Each point contributes d variables Each edge contributes 1 equation Always d (d + 1) / 2 rigid motions Don t waste any

11 Geometry to combinatorics m d #V(G ) - d(d + 1)/2 Theorem (Laman): Generically, in d = 2, this implies independence of length equations. (Rigidity if m = 2n 3.)

12 Genericity Genericity is loosely no special geometry Almost all placements are generic Non-generic set is algebraic Non-generically frameworks exhibit universality [Kapovich-Millson] Most general problem very hard

13 Why combinatorial rigidity? Generic frameworks can be general enough Can check Laman 2n 3 in O(n 2 ) time simple pebble game algorithms [Hendrickson- Jacobs, Berg-Jordán, Lee-Streinu] Useful to know if your problem is non-generic

14 Hypothetical zeolite Graph is infinite how to compute with it Structure is symmetric any symmetric structure satisfies lots of extra equations very non-generic looking [Rivin-Treacy-Randall]

15 Periodic frameworks [Borcea-Streinu] A periodic framework (G, l, Γ) is an infinite framework with Γ < Aut(G) l(γ(ij)) = l(ij) Γ free abelian, rank d finite quotient A realization G(p,Λ) is a realization periodic with respect to a lattice of translations Λ, which realizes Γ Motions preserve the Γ-symmetry

16 [Whiteley]

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19 1 vertex orbit 2 edge orbits

20 Not allowed Not one vertex orbit!

21 Colored graphs (0,1) (1,-1) (0,0) (0,0) (0,0) (-1,0)

22 Counting for periodic frameworks Each vertex orbit determined by one representative total dn variables from there Lattice representation is a d d matrix d 2 more variables For subgraphs, we will have to distinguish how much of the symmetry group they see

23 m 2n 3 (0,1) (1,-1) (0,0) (0,0) (0,0) (-1,0)

24 m 2n (0,1) (1,-1) (0,0) (0,0) (0,0) (-1,0)

25 m 2n (0,1) (1,-1) (0,0) (0,0) (0,0) (-1,0)

26 Generic periodic rigidity Theorem (Malestein-T): For dimension 2 Z 2 rank m 2(n + k) - 3-2(c - 1) connected comps. characterizes generic independence of length equations. Minimal rigidity if m = 2n + 1 Generic here is choice of vertex orbits Actually an instance of a more general counting heuristic Always necessary Known to be sufficient for more groups in dim 2 Similar results for fixed-area unit cell, fixed lattice, etc.

27 Allowed!

28 Ultrarigidity [Borcea] Let (G, p, L) be a realization of (G,l, Γ) (G, p, L) is (periodically) ultrarigid if it is rigid for any (finite-index) sub-lattice Λ < Γ, (G, p, L) is a rigid realization of (G, l, Λ) Related concept: ultra 1-d.o.f. (in 2d) e.g, 4-regular lattices

29 Challenges Generic rigidity characterized by the rank of one matrix (rigidity/compatibility/ matrix) here there is an infinite family of matrices Not completely clear finite ultrarigidity is a generic property Some evidence towards no We don t know a priori what extra bulk modes look like

30 Algebraic characterization [Connelly-Shen-Smith 14 + Power 13] A realization (G, p, L) is infinitesimally ultrarigid if and only if: It is infinitesimally periodically rigid The matrix with ijth row, ij E(G,φ) edge direction! vector! (.. dij.. dij {γij -1,ω}.) has rank dn for all ω 1 i j comp. wise mult {δ,ω} := (ζ1 δ 1,,ζd δ d ), ζi root of unity

31 Consequences Extra bulk modes fix the lattice [Connelly-Shen-Smith]: Nice geometric argument Direct derivation: Representation theory Can check ultrarigidity in finite time [Malestein-T] new a priori bound on order of ζ s

32 Counting (G,γ) a colored graph with Γ ( Z d ) colors ψ : Γ Δ, epimorphism to a finite cyclic Δ Ultra Maxwell Count for (G, ψ (γ)) m d n d T(G, ψ (γ)) # c.c. s w/ Δ rank > 0 for all ψ. Finitely many suffice. Sufficient in 2d if (G,γ) is independent as a periodic framework

33 Algorithms and combinatorics For m = 2n + 1, have a combinatorial algorithm polynomial in m (but not γ) for generic infinitesimal periodic ultra rigidity Useful for small colors For m = 2n, have a polynomial time algorithm for fixedarea periodic ultrarigidity Via some combinatorial equivalences Uses the pebble game, still only O(n 4 )

34 Questions Finite vs. infinitesimal ultra-rigidity Irrational points on the RUMS Faster algorithms

35 Thanks!