A Game of Life on Penrose tilings
AA GAME OF LIFE ON PENROSE TILINGS
DUANE A. BAILEY AND KATHRYN A. LINDSEY
Abstract.
We define rules for cellular automata played on quasiperiodic tilings of the plane arising fromthe multigrid method in such a way that these cellular automata are isomorphic to Conway’s Game ofLife. Although these tilings are nonperiodic, determining the next state of each tile is a local computation,requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. Asan example, we show a version of a “glider” moving through a region of a Penrose tiling. This constitutesa potential theoretical framework for a method of executing computations in non-periodically structuredsubstrates such as quasicrystals. Introduction
Inspired by Conway’s Game of Life ([3]), various researchers have investigated properties of similarly-defined cellular automata played on Penrose tilings (e.g. [4, 7, 8, 10, 11]). Since Penrose tilings are notperiodic ([6]), the challenge is to define a finite state cellular automaton which is locally computable –meaning that the next state of a tile depends only on the current states of finitely many “nearby” tiles –and yet has interesting emergent global properties. In particular, the existence of “gliders” and the abilityto support universal computation is of interest.In this note, we present a natural method for embedding the original Game of Life in any tiling of theplane that arises via the multigrid method, a class of tilings that include Penrose rhomb tilings. The resultingcellular automata, defined on quasiperiodic tilings, are locally computable and precisely mimic the behaviorof the original Game of Life. In particular, they admit gliders, signal delivery, universal computation, andreproduction.Quasiperiodic tilings may be thought of as “toy models” of quasicrystals - physical substances which,like quasiperiodic tilings, exhibit order but not periodicity. Penrose tilings, in particular, are a geometricmodel for icosahedral quasicrystals [9]. This investigation is motivated by the promise of new media forhosting computation – a topic which gives rise to the question of how to “compute” in non-periodicallystructured substrates. 2.
Statement of Results
Quasiperiodic tilings generated by the “multigrid” method admit ribbons of tiles. A ribbon consistsof a bi-infinite sequence of sequentially adjacent tiles that share an edge that is parallel to some fixed vector.In a tiling generated by an n -multigrid, every edge of the tiling is parallel to one of n distinct vectors, say v , . . . , v n ∈ R . Define a tile to be of type ( i, j ) if has two edges parallel to v i and two edges parallel to v j .A regular tiling has (cid:0) n (cid:1) types of tiles, each of which is a parallelogram and has a unique type. A tile withmore than four edges, which by definition belongs to a singular tiling, may be of more than one type (a tileof type ( i, j ) may have additional edges not parallel to v i or v j .) For any choice of ( i, j ) , i (cid:54) = j , in any tilingarising from a multigrid construction, the set of all ribbons passing through all tiles of type ( i, j ) forms asquare grid graph (i.e. the pattern formed by the ribbons is topologically the same as the pattern of lines ona piece of graph paper). The “vertices” of the square grid graph are the type ( i, j ) -tiles. See Figure 1.This square grid graph structure determines a natural isomorphism between the set of type ( i, j ) tilesin a regular quasiperiodic tiling arising from a multigrid construction and the grid of squares in Conway’soriginal Game of Life. This isomorphism preserves the property of adjacency of tiles within the lattice, andis unique up to choosing which tile represents the origin in the square grid Z and which directions of ribbons Date : August 29, 2017.2010
Mathematics Subject Classification. primary 37B15, 53C23; secondary 68Q80.
Key words and phrases. cellular automata, quasiperiodic tiling, quasicrystal, Penrose tiling, Game of Life.Second author supported by an N.S.F. Mathematical Sciences Research Postdoctoral Fellowship. a r X i v : . [ n li n . C G ] A ug D. BAILEY AND K. LINDSEY correspond to the positive x and y directions in Z . Since discerning ribbons only entails evaluating theparallelism of edges of adjacent tiles, picking out the natural lattice structure of the set of type ( i, j ) tiles inany region requires only local knowledge of the tiling. Our fundamental observation is that the original Game of Life may be “played” on this lattice of type ( i, j ) tiles. That is, in a regular tiling, each tile of type ( i, j ) has precisely eight neighbors – the eight ( i, j )type tiles that are “neighboring” in the lattice structure determined by the square grid graph of ribbons.Equipped with this observation, several obvious variants of the “game” are possible:
Game Variants: (i) We pick a pair of grid indices ( i, j ), play the Game of Life on the lattice consisting of all ( i, j ) typetiles, and declare all tiles that are not type ( i, j ) to be “dead.”(ii) As in variant (i), we pick a pair of grid indices ( i, j ) and play the Game of Life on the latticeconsisting of all type ( i, j ) tiles. However, instead of declaring tiles of other types to be dead,we associate each tile that is not type ( i, j ) to a nearby type ( i, j ) tile. Section § i, j ) to a nearby type ( i, j ) tile. We call thetype ( i, j ) tiles dominant , and we call the set of non-dominant tiles associated to a dominant tilethe supporting tiles . We declare that a supporting tile is alive if and only if its associated dominanttile is alive. (This is equivalent to specifying that the neighbors of a supporting tile are the eightdominant tiles that neighbor its associated dominant tile.) (See Figure 2.)(iii) In a regular tiling arising from a n -multigrid, there are (cid:0) n (cid:1) types of tiles, and each tile is of a uniquetype. We simultaneously play a different copy of the Game of Life on each lattice consisting of alltiles of a given type. These copies do not interact.(iv) As in variant (iii), we play a different copy of the Game of Life on each lattice consisting of all tilesof a given type, but on a singular tiling. In this case, copies of the Game of Life played on differentsets of tiles interact. Tiles that belong to more than one lattice serve as channels of communicationbetween the copies of the Game of Life played of these lattices. (See Section § The original Game of Life
Let X be a tiling of the plane, and let T be the set of tiles in X . A cellular automaton played on X with a set of states A = { a , . . . , a n } is a map f : T A → T A such that(i) each tile t ∈ T has a “neighborhood” N ( t ) ⊂ T consisting of finitely many “nearby” tiles(ii) there is a “local rule” map, r , defined on the set of all configurations of all neighborhoods and takingvalues in A , so that for all configurations c ∈ T A and all t ∈ T , π t ( f ( c )) = r ( π N ( t ) ( c )), where π i isthe restriction to the i th coordinate.The space T A is the phase space, or set of all possible configurations for the cellular automaton; a point c ∈ T A is an assignment of one state in A to each tile t ∈ T . Condition (ii) says that, regardless of theglobal configuration c , the next state of each tile t is can be computed from the current states of the tiles inthe neighborhood N ( t ) using the local rule r .In Conway’s original Game of Life ([3]), T is the standard tiling of the plane by isometric squares, and A = { , } . Tiles in state 1 are said to be “alive” or “living,” and tiles in state 0 are said to be “nonliving” or“dead.” For each tile t , N ( t ) is the eight tiles which neighbor t (i.e. are vertically, horizontally, or diagonallyadjacent to t ). The local rule r is r ( t ) = t is living and precisely 2 or 3 tiles in N ( t ) are alive1 if t is dead and precisely 3 tiles in N ( t ) are alive0 otherwiseInvented in 1970 by John Conway and popularized by Martin Gardner in an article in ScientificAmerican ([3]), the Game of Life exhibits many interesting local configurations, including gliders, gliderguns, spaceships, and replicators (cite). It was shown to support universal computation, with the proofutilizing streams of gliders to transmit signals ([1]). Dave Greene found the first replicator in the Game ofLife in 2013 ([5]). GAME OF LIFE ON PENROSE TILINGS 3
Figure 1.
Eight ribbons of a “square grid graph” of ribbons in a Penrose tiling are coloredin. The two chosen vectors, v i and v j , are the vectors parallel to the edges of the purpletiles. The purple ( i, j )-type tiles are the intersections of the grid of pink ribbons and the gridof blue ribbons. Only four ribbons of each ribbon grid are colored, so that the reader maymore clearly see the geometric structure, but one could complete the coloring by making allof the ( i, j )-type tiles purple and extending pink and blue ribbons out from these tiles.4. The multigrid method
In this section, we review the multigrid method (introduced by de Bruijn in [2]) and use it to justifyour assertion that the set of all ribbons passing through all type ( i, j ) tiles forms a square grid graph.4.1.
General multigrids.
The multigrid method is a method of constructing quasiperiodic tilings. A grid is an infinite family of equally spaced, parallel lines in the plane. (Note that his use of the word “grid”differs from that of the “square grid graph” of Section § n -multigrid , n ≥
2, sometimescalled just a multigrid , is the union of n grids in the plane, with no two grids parallel to the same vector. Amultigrid (or the associated tiling) is said to be regular if no point is the intersection of more than 2 lines;otherwise it is singular .A multigrid determines a tiling of the plane (see § v ,the tile corresponding to v is a parallelogram whose edges are perpendicular to the lines intersecting at v .In the case of a singular multigrid, if k ≥ v , the associated tile has oneedge for each of the 2 k half-lines incident to v , and the edges are perpendicular to the associated half-linesand arranged in the same order.Consequently, all tiles that correspond to intersections of a given line, say L , with other lines in amultigrid have opposite edges that are perpendicular to L (and hence parallel). The line L corresponds inthe tiling to a ribbon of tiles that all have a pair of edges that are perpendicular to L . Two grids in themultigraph correspond in the tiling to a “square grid graph” of ribbons, and the intersection points of thesetwo grids correspond to the set of tiles that have edges perpendicular to the lines in both of these grids (seeFigure 1). D. BAILEY AND K. LINDSEY
Figure 2.
Sixteen generations of a “glider” as it passes over a Penrose tiling, read fromleft-to-right, top-to-bottom. Here we are using Game Variant (ii); the living dominanttiles are black, and the supporting tiles associated to living dominant tiles are gray. Theconfigurations in each column of the figure represent the glider in the same position (relativeto the grid of dominant tiles). The different configurations that represent the glider in thesame position are due to the local structure of the tiling; a glider will assume identicalconfigurations at times, but that timing is, itself, nonperiodic.4.2.
Constructing Penrose tilings from pentagrids.
For concreteness, we describe how to constructthe tiling associated to a 5-multigrid (called pentagrid) in which the grids are parallel to the vectors1 , ζ , ζ , ζ , ζ , where ζ = e πi . This is the case described by de Bruijn and which gives rise to Penrosetilings. An analogous process defines the tiling associated to a general n -multigrid (see, e.g. [12, 2]).First, we define the 5-multigrid, which in this case is also called a pentagrid . Fix real numbers γ , . . . , γ . For j = 0 , ...,
4, define the j th grid to be the set { z ∈ C | Re( zζ − j ) + γ j ∈ Z } , which consists of of equally spaced parallel lines. The pentagrid determined by γ , ..., γ is the union of grids0 to 4.We now define a map K : C → Z that describes the coordinate of points in the plane relative to eachof the five grids. Namely, K associates to each point z ∈ C the point K ( z ) ∈ Z whose j th coordinate isgiven by K j ( z ) = (cid:100) Re( zζ − j ) + γ j (cid:101) . GAME OF LIFE ON PENROSE TILINGS 5
Define the map φ : Z → C by φ ( (cid:126)k ) = (cid:126)k · (1 , ζ , ζ , ζ , ζ ) . The set of vertices of the associated tiling is the image of the plane C under φ ◦ K . As z travels in a smallcircle around a point mentioned by n ≥ φ ◦ K ( z ) picks out a sequence of 2 n points in C ; straight line segments connecting these 2 n vertices (in cyclic order) are the edges of a tile in thetiling. Each tile of the tiling comes from such an intersection of grid lines. De Bruijn proved [2] that if thepentagrid determined by γ , . . . , γ is regular, then the associated tiling is a Penrose tiling, and conversely,every Penrose tiling arises from such a pentagrid.A tile of type ( i, j ), which comes from an intersection point of a line in the i th grid and a line in the j th grid, has edges parallel to ζ i and ζ j . If this intersection point is mentioned by precisely n ≥ n edges, of which two are parallel to ζ i and two are parallel to ζ j . If the tile is aquadrilateral, whether it is a thick rhomb or a thin rhomb depends on the parity of i − j . A Penrose tilingexhibits ten different tile orientations (5 rotations of both thin and thick tiles); these correspond to the (cid:0) (cid:1) ways that two of the five grids can intersect. The intersection of a fixed line of grid i with the other lines ofthe multigrid corresponds to a ribbon of tiles that all have edges parallel to ζ i . Example.
Figure 3 shows a small section of a pentagrid. Consider the intersection of line 0 of grid 0 andline − z . In a small neighborhood of z , K takes on four distinctvalues: K ( z ) = (0 , , , , − K ( z ) = (1 , , , , − K ( z ) = (1 , , , , K ( z ) = (0 , , , , z corresponds to the tile (a “thick” rhomb) whose four vertices have coordinates φ ( K ( z )) = ζ + ζ − ζ ,φ ( K ( z )) = 1 + ζ + ζ − ζ ,φ ( K ( z )) = 1 + ζ + ζ , and φ ( K ( z )) = ζ + ζ . Since ζ j has unit magnitude, we can see that the edges of the tile all have unit length. Since the coordinatesof the four points differ by a single multiple of 1 and/or ζ , the edges of the tile are parallel to the vectors 1or ζ in C . 5. Notes on Game Variants
Variant (ii): dominant and supporting tiles.
Section § G ) and the set of points in the plane that are mentionedby more than one grid of G . For each pair of grid indices ( i, j ), the union of grids i and j determines adecomposition of the plane into parallelograms. To be a true decomposition of the plane (i.e. each pointbelongs to precisely one parallelgram), view each parallelogram to be a product of left-closed, right-open (orright-open, left-closed) intervals. If a point v is in the intersection of two (or more) grids and belongs toparallelogram P , define the tile corresponding to v to be in the support of the type-( i, j ) tile associated tothe top left (or any other consistent choice) corner of the parallelogram P .5.2. Variant (iv): singular tilings and interacting copies of the Game of Life.
Suppose precisely3 lines of a multigrid intersect at a point z ; let i, j, k be the indices of the three grids. The tile associatedto z has six sides, is of types ( i, j ), ( j, k ), and ( i, k ), and belongs to the three corresponding lattices of tiles.Thus, this one tile may have as many as 24 neighbors – eight neighbors in each of three lattices. We maywish to impose a different local rule for tiles with more than eight neighbors. The state of this tile impactsthe evolution of each of the three “copies” of the Game of Life.In general, a tile that represents the intersection of n ≥ (cid:0) n (cid:1) types andthus belongs to (cid:0) n (cid:1) lattices, each of which hosts a copy of the Game of Life. If each of its neighboring tilesarise from the intersection of precisely two multigrid lines (i.e. is a parallelogram), the tile will have 8 · (cid:0) n (cid:1) neighbors. Understanding the qualitative effects of different possible local rules for tiles of more than onetype is beyond the scope of this work.In the case of Penrose tilings, given any 3 distinct edge directions, there is at most one tile that hasedges in all 3 directions (as well as, possibly, other directions). Thus, communication between the gamesplayed on the three associated lattices of tiles can occur only at this one location. Such a tile results from D. BAILEY AND K. LINDSEY -2 -1 0 1 2 - - - - - - - -
10 2
Grid 0 G r i d G r i d G r i d G r i d z z z z τ Figure 3.
The intersection, z , of line 0 of grid i = 0 and line -1 of grid j = 4. The nearbypoints z , z , z , and z are mapped by K to four different points in Z . The images ofthese points under φ are the vertices of a tile in the associated tiling.the intersection of lines in 3 distinct grids of the pentagrid. To see that there can be at most one such tile,we may assume without loss of generality that two of the grids are grid 0 and grid 1, corresponding to thevectors ζ and ζ . Let M be an affine deformation of the complex plane that maps these two vectors to1 and i in C . Denote by u the image under M of ζ j , for any fixed j ∈ { , , } . Since both the real andcomplex coordinates of u are irrational, the three vectors 1, i , and u are rationally independent. Therefore,at most one point is in the intersection of grids 0, 1, and j ; this proves the assertion that a (nonregular)Penrose tiling has most one tile of type (0 , , j ).6. Discussion
In some ways, our approach may be seen to have skirted the problem of how to “compute” in nonpe-riodic substrates. However, the fundamental observation of this work is that nonperiodic tilings may haveperiodic features that can be harnessed for computation. The collection of all tiles of type ( i, j ) naturallyforms a lattice, but tiles that occupy adjacent positions in the lattice may not be adjacent in the tiling asa whole (they are, however, a bounded distance apart). Game variant (ii) may be interpreted as a way toaddress this issue; one may consider the union of a dominant tile together with its supporting tiles as a singlegiant tile, eliminating the dead space between dominant tiles. While interaction between copies of the gameof life played on different lattices can occur at tiles with more than four sides, (nonregular) Penrose tilingshave at most one tile that belongs to any three distinct lattices; generalizations of multigrids in which linesof the grid are not required to be spaced equally could give rise to more tiles that are loci for communcationbetween the lattices. It remains to be seen to what extent our approach can inform implementation ofcomputation in real-world nonperiodic substrates.
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Department of Mathematics, Williams College, Williamstown, MA 01267, U.S.A.
E-mail address : [email protected] Department of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A.
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