The Significance of Multiway Methods

It’s all about methods the place there can in impact be many doable paths of historical past. In a typical commonplace computational system like a mobile automaton, there’s all the time only one path, outlined by evolution from one state to the subsequent. However in a multiway system, there will be many doable subsequent states—and thus many doable paths of historical past. Multiway methods have a central position in our Physics Undertaking, notably in reference to quantum mechanics. However what’s now rising is that multiway methods the truth is function a fairly basic basis for a complete new “multicomputational” paradigm for modeling.

My goal right here is twofold. First, I need to use multiway methods as minimal fashions for development processes primarily based on aggregation and tiling. And second, I need to use this concrete utility as a approach to develop additional instinct about multiway methods usually. Elsewhere I’ve explored multiway methods for strings, multiway methods primarily based on numbers, multiway Turing machines, multiway combinators, multiway expression analysis and multiway methods primarily based on video games and puzzles. However in learning multiway methods for aggregation and tiling, we’ll be coping with one thing that’s instantly extra bodily and tangible.

After we consider “development by aggregation” we sometimes think about a “random course of” by which new items get added “at random” to one thing. However every of those “random prospects” in impact defines a distinct path of historical past. And the idea of a multiway system is to seize all these prospects collectively. In a typical random (or “stochastic”) mannequin one’s simply tracing a single path of historical past, and one imagines one doesn’t have sufficient data to say which path it is going to be. However in a multiway system one’s all of the paths. And in doing so, one’s in a way making a mannequin for the “entire story” of what can occur.

The selection of a single path will be “nondeterministic”. However the entire multiway system is deterministic. And by learning that “deterministic entire” it’s usually doable to make helpful, fairly basic statements.

One can consider a selected second within the evolution of a multiway system as giving one thing like an ensemble of states of the sort studied in statistical mechanics. However the basic idea of a multiway system, with its discrete branching at discrete steps, will depend on a stage of elementary discreteness that’s fairly unfamiliar from conventional statistical mechanics—although is completely easy to outline in a computational, and even mathematical, method.

For aggregation it’s simple sufficient to arrange a minimal discrete mannequin—a minimum of if one permits express randomness within the mannequin. However a significant level of what we’ll do right here is to “go above” that randomness, establishing our mannequin by way of a complete, deterministic multiway system.

What can we study by this entire multiway system? Nicely, for instance, we will see whether or not there’ll all the time be development—regardless of the random decisions could also be—or whether or not the expansion will generally, and even all the time, cease. And in lots of sensible purposes (assume, for instance, tumors) it may be crucial to know whether or not development all the time stops—or by way of what paths it could possibly proceed.

A whole lot of what we’ll at first do right here entails seeing the impact of native constraints on development. Afterward, we’ll additionally take a look at results of geometry, and we’ll examine how objects of various shapes can combination, or finally tile.

The fashions we’ll introduce are in a way very minimal—combining the best multiway buildings with the best spatial buildings. And with this minimality it’s virtually inevitable that the fashions will present up as idealizations of all types of methods—and as foundations for good fashions of those methods.

At first, multiway methods can appear quite summary and tough to understand—and maybe that’s inevitable given our human tendency to assume sequentially. However by seeing how multiway methods play out within the concrete case of development processes, we get to construct our instinct and develop a extra grounded view—that may stand us in good stead in exploring different purposes of multiway methods, and usually in coming to phrases with the entire multicomputational paradigm.

The Easiest Case

It’s the final word minimal mannequin for random discrete development (usually known as the Eden mannequin). On a sq. grid, begin with one black cell, then at every step randomly connect a brand new black cell someplace onto the rising “cluster”:

After 10,000 steps we’d get:

However what are all of the doable issues that may occur? For that, we will assemble a multiway system:

A whole lot of these clusters differ solely by a trivial translation; canonicalizing by translation we get

or after one other step:

If we additionally cut back out rotations and reflections we get

or after one other step:

The set of doable clusters after t steps are simply the doable polyominoes (or “sq. lattice animals”) with t cells. The variety of these for successive t is

rising roughly like okay^t for giant t, with okay just a little bigger than 4:

By the way in which, canonicalization by translation all the time reduces the variety of doable clusters by an element of t. Canonicalization by rotation and reflection can cut back the quantity by an element of 8 if the cluster has no symmetry (which for giant clusters turns into more and more possible), and by a smaller issue the extra symmetry the cluster has, as in:

With canonicalization, the multiway graph after 7 steps has the shape

and it doesn’t look any easier with various rendering:

If we think about that at every step, cells are added with equal chance at each doable place on the cluster, or equivalently that every one outgoing edges from a given cluster within the uncanonicalized multiway graph are adopted with equal chance, then we will get a distribution of chances for the distinct canonical clusters obtained—right here proven after 7 steps:

One function of the massive random cluster we noticed initially is that it has some holes in it. Clusters with holes begin growing after 7 steps, with the smallest being:

This cluster will be reached by way of a subset of the multiway system:

And in reality within the restrict of huge clusters, the chance for there to be a gap appears to method 1—although the overall fraction of space lined by holes approaches 0.

One approach to characterize the “house of doable clusters” is to create a branchial graph by connecting each pair of clusters which have a typical ancestor one step again within the multiway graph:

The connectedness of all these graphs displays the truth that with the rule we’re utilizing, it’s all the time doable at any step to go from one cluster to a different by a sequence of delete-one-cell/add-one-cell adjustments.

The branchial graphs right here additionally present a 4-fold symmetry ensuing from the symmetry of the underlying lattice. Canonicalizing the states, we get smaller branchial graphs that not present any such symmetry:

Totalistically Constrained Development (4-Cell Neighborhoods)

With the rule we’ve been discussing to this point, a brand new cell to be connected will be anyplace on a cluster. However what if we restrict development, by requiring that new cells will need to have sure numbers of current cells round them? Particularly, let’s contemplate guidelines that take a look at the neighbors round any given place, and permit a brand new cell there provided that there are specified numbers of current cells within the neighborhood.

Beginning with a cross of black cells, listed here are some examples of random clusters one will get after 20 steps with all doable guidelines of this kind (the preliminary “4” designates that these are 4-neighbor guidelines):

Guidelines that don’t enable new cells to finish up with only one current neighbor can solely fill in corners of their preliminary situations, and might’t develop any additional. However any rule that enables development with just one current neighbor produces clusters that continue to grow perpetually. And listed here are some random examples of what one can get after 10,000 steps:

The final of those is the unconstrained (Eden mannequin) rule we already mentioned above. However let’s look extra rigorously on the first case—the place there’s development provided that a brand new cell will find yourself with precisely one neighbor. The canonicalized multiway graph on this case is:

The doable clusters right here correspond to polyominoes which can be “all the time one cell extensive” (i.e. haven’t any 2×2 blocks), or, equivalently, have perimeter 2t + 2 at step t. The variety of such canonicalized clusters grows like:

That is an growing fraction of the overall variety of polyominoes—implying that the majority massive polyominoes take this “spindly” kind.

A brand new function of a rule with constraints is that not all places round a cluster might enable development. Here’s a model of the multiway system above, with cells round every cluster annotated with inexperienced if new development is allowed there, and purple if it by no means will be:

In a bigger random cluster, we will see that with this rule, a lot of the inside is “lifeless” within the sense that the constraint of the rule permits no additional development there:

By the way in which, the clusters generated by this rule can all the time be straight represented by their “skeleton graphs”:

random clusters for all of the (grow-with-1-neighbor) guidelines above, we see totally different patterns of holes in every case:

There are altogether 5 kinds of cells being distinguished right here, reflecting totally different neighbor configurations:

Right here’s a pattern cluster generated with the 4:{1,3} rule:

Cells indicated with have already got too many neighbors, and so can by no means be added to the cluster. Cells indicated with have precisely the precise variety of neighbors to be added instantly. Cells indicated with don’t at present have the precise variety of neighbors to develop, but when neighbors are stuffed in, they may be capable to be added. Typically it should end up that when neighbors of cells get stuffed in, they’ll really forestall the cell from being added (in order that it turns into )—and within the specific case proven right here that occurs with the two×2 blocks of cells.

The multiway graphs from the foundations proven listed here are all qualitatively comparable, however there are detailed variations. Particularly, a minimum of for lots of the guidelines, an growing variety of states are “lacking” relative to what one will get with the grow-in-all-cases 4:{1,2,3,4} rule—or, in different phrases, there are an growing variety of polyominoes that may’t be generated given the constraints:

The primary polyomino that may’t be reached (which happens at step 4) is:

At step 6 the polyominoes that may’t be reached for guidelines 4:{1,3} and 4:{1,3,4} are

whereas for 4:{1} and 4:{1,4} the extra polyomino

may also not be reached.

At step 8, the polyomino

is reachable with 4:{1} and 4:{1,3} however not with 4:{1,4} and 4:{1,3,4}.

Of some observe is that not one of the guidelines that exclude polyominoes can attain:

Totalistically Constrained Development (8-Cell Neighborhoods)

What occurs if one considers diagonal as effectively orthogonal neighbors, giving a complete of 8 neighbors round a cell? There are 256 doable guidelines on this case, comparable to the doable subsets of Vary[8]. Listed here are samples of what they do after 200 steps, ranging from an preliminary cluster:

Two instances that a minimum of initially present development listed here are (the “8” designates that these are 8-neighbor guidelines):

Within the {2} case, the multiway graph begins with:

One may assume that each department on this graph would proceed perpetually, and that development would by no means “get caught”. Nevertheless it seems that after 9 steps the next cluster is generated:

And with this cluster, no additional development is feasible: no positions across the boundary have precisely 2 neighbors. Within the multiway graph as much as 10 steps, it seems that is the one “terminal cluster” that may be generated—out of a complete of 1115 doable clusters:

So how is that terminal cluster reached? Right here’s the fragment of multiway graph that results in it:

If we don’t prune off all of the methods to “go astray”, the fragment seems as half of a bigger multiway graph:

And if one follows all paths within the unpruned (and uncanonicalized) multiway graph at random (i.e. at every step, one chooses every department with equal chance), it seems that the chance of ever reaching this specific terminal cluster is simply:

(And the truth that this quantity is pretty small implies that the system is way from confluent; there are lots of paths that, for instance, don’t converge to the fastened level comparable to this terminal cluster.)

If we preserve going within the evolution of the multiway system, we’ll attain different terminal clusters; after 12 steps the next have appeared:

For the {3} rule above, the multiway system takes just a little longer to “get going”:

As soon as once more there are terminal clusters the place the system will get caught; the primary of them seems at step 14:

And in addition as soon as once more the terminal cluster seems as an remoted node in the entire multiway system:

The fragment of multiway graph that results in it’s:

To this point we’ve been discovering terminal clusters by ready for them to seem within the evolution of the multiway system. However there’s one other method, much like what one may use in filling in one thing like a tiling. The thought is that each cell in a terminal cluster will need to have neighbors that don’t enable additional development. In different phrases, the terminal cluster should encompass sure “native tiles” for which the constraints don’t enable development. However what configurations of native tiles are doable? To find out this, we flip the matching situations for the tiles into logical expressions whose variables are True and False relying on whether or not specific positions within the template do or don’t comprise cells within the cluster. By fixing the satisfiability downside for the mixture of those logical expressions, one finds configurations of cells that would conceivably correspond to terminal clusters.

Following this process for the {2} guidelines with areas of as much as 6×6 cells we discover:

However now there’s an extra constraint. Assuming one begins from a related preliminary cluster, any subsequent cluster generated should even be related. Eradicating the non-connected instances we get:

So given these terminal clusters, what preliminary situations can result in them? To find out this we successfully should invert the aggregation course of—giving ultimately a multiway graph that features all preliminary situations that may generate a given terminal cluster. For the smallest terminal cluster we get:

Our 4-cell “T” preliminary situation seems right here—however we see that there are additionally even smaller 2-cell preliminary situations that result in the identical terminal cluster.

For all of the terminal clusters we confirmed earlier than, we will assemble the multiway graphs beginning with the minimal preliminary clusters that result in them:

For terminal clusters like

there’s no nontrivial multiway system to indicate, since these clusters can solely seem as preliminary situations; they will by no means be generated within the evolution.

There are fairly a couple of small clusters that may solely seem as preliminary situations, and don’t have preimages underneath the aggregation rule. Listed here are the instances that slot in a 3×3 area:

The case of the {3} rule is pretty much like the {2} rule. The doable terminal clusters as much as 5×5 are:

Nevertheless, most of those have solely a reasonably restricted set of doable preimages:

For instance we’ve got:

And certainly past the (size-17) instance we already confirmed above, no different terminal clusters that may be generated from a T preliminary situation seem right here. Sampling additional, nevertheless, extra terminal clusters seem (starting at measurement 25):

The fragments of multiway graphs for the primary few of those are:

Random Evolution

We’ve seen above that for the foundations we’ve been investigating, terminal clusters are fairly uncommon amongst doable states within the multiway system. However what occurs if we simply evolve at random? How usually will we wind up with a terminal cluster? After we say “evolve at random”, what we imply is that at every step we’re going to take a look at all doable positions the place a brand new cell may very well be added to the cluster that exists to this point, after which we’re going to select with equal chance at which of those to really add the brand new cell.

For the 8:{3} rule one thing shocking occurs. Regardless that terminal clusters are uncommon in its multiway graph, it seems that no matter its preliminary situations, it all the time ultimately reaches a terminal cluster—although it usually takes some time. And right here, for instance, are a few doable terminal clusters, annotated with the variety of steps it took to achieve them (which can also be equal to the variety of cells they comprise):

The distribution of the variety of steps to termination appears to be very roughly exponential (right here primarily based on a pattern of 10,000 random instances)—with imply lifetime round 2300 and half-life round 7400:

Right here’s an instance of a giant terminal cluster—that takes 21,912 steps to generate:

And right here’s a map exhibiting when development in numerous elements of this cluster occurred (with blue being earliest and purple being newest):

This image means that totally different elements of the cluster “actively develop” at totally different occasions, and if we take a look at a “spacetime” plot of the place development happens as a perform of time, we will affirm this:

And certainly what this means is that what’s occurring is that totally different elements of the cluster are at first “fertile”, however later inevitably “burn out”—in order that ultimately there are not any doable positions left the place development can happen.

However what shapes can the ultimate terminal clusters kind? We will get some thought by a “compactness measure” (of the sort usually used to review gerrymandering) that roughly provides the usual deviation of the distances from the middle of every cluster to every of the cells in it. Each “very stringy” and “roughly round” clusters are pretty uncommon; most clusters lie someplace in between:

If we glance not on the 8:{3} however as an alternative on the 8:{2} rule, issues are very totally different. As soon as once more, it’s doable to achieve a terminal cluster, because the multiway graph exhibits. However now random evolution virtually by no means reaches a terminal cluster, and as an alternative virtually all the time “runs away” to generate an infinite cluster. The clusters generated on this case are sometimes far more “compact” than within the 8:{3} case

and that is additionally mirrored within the “spacetime” model:

Parallel Development and Causal Graphs

In increase our clusters to this point, we’ve all the time been assuming that cells are added sequentially, one by one. But when two cells are far sufficient aside, we will really add them “concurrently”, in parallel, and find yourself constructing the identical cluster. We will consider the addition of every cell as being an “occasion” that updates the state of the cluster. Then—identical to in our Physics Undertaking, and different purposes of multicomputation—we will outline a causal graph that represents the causal dependencies between these occasions, after which foliations of this causal graph inform us doable total sequences of updates, together with parallel.

For example, contemplate this sequence of states within the “all the time develop” 4:{1,2,3,4} rule—the place at every step the cell that’s new is coloured purple (and we’re together with the “nothing” state initially):

Each transition between successive states defines an occasion:

There’s then causal dependence of 1 occasion on one other if the cell added within the second occasion is adjoining to the one added within the first occasion. So, for instance, there are causal dependencies like

and

the place within the second case extra “spatially separated” cells have been added that aren’t concerned within the causal dependence. Placing all of the causal dependencies collectively, we get the whole causal graph for this evolution:

We will recuperate our authentic sequence of states by selecting a selected ordering of those occasions (right here indicated by the positions of the cells they add):

This path has the property that it all the time follows the course of causal edges—and we will make that extra apparent through the use of a distinct structure for the causal graph:

However usually we will use any ordering of occasions in step with the causal graph. One other ordering (out of a complete of 40,320 prospects on this case) is

which provides the sequence of states

with the identical closing cluster configuration, however totally different intermediate states.

However now the purpose is that the constraints implied by the causal graph don’t require all occasions to be utilized sequentially. Some occasions will be thought of “spacelike separated” and so will be utilized concurrently. And in reality, any foliation of the causal graph defines a sure sequence for making use of occasions—both sequentially or in parallel. So, for instance, right here is one specific foliation of the causal graph (proven with two totally different renderings for the causal graph):

And right here is the corresponding sequence of states obtained:

And since in some slices of this foliation a number of occasions occur “in parallel”, it’s “quicker” to get to the ultimate configuration. (Because it occurs, this foliation is sort of a “cosmological relaxation body foliation” in our Physics Undertaking, and entails the utmost doable variety of occasions occurring on every slice.)

Totally different foliations (and there are a complete of 678,972 prospects on this case) will give totally different sequences of states, however all the time the identical closing state:

Notice that nothing we’ve carried out right here will depend on the actual rule we’ve used. So, for instance, for the 8:{2} rule with sequence of states

the causal graph is:

It’s price commenting that every little thing we’ve carried out right here has been for specific sequences of states, i.e. specific paths within the multiway graph. And in impact what we’re doing is the analog of classical spacetime physics—tracing out causal dependencies particularly evolution histories. However usually we may take a look at the entire multiway causal graph, with occasions that aren’t solely timelike or spacelike separated, but additionally branchlike separated. And if we make foliations of this graph, we’ll find yourself not solely with “classical” spacetime states, but additionally “quantum” superposition states that will have to be represented by one thing like multispace (by which at every spatial place, there’s a “branchial stack” of doable cell values).

The One-Dimensional Case

To this point we’ve been contemplating aggregation processes in two dimensions. However what about one dimension? In 1D, a “cluster” simply consists of a sequence of cells. The only rule permits a cell to be added every time it’s adjoining to a cell that’s already there. Ranging from a single cell, right here’s a doable random evolution in line with such a rule, proven evolving down the web page:

We will additionally assemble the multiway system for this rule:

Canonicalizing the states provides the trivial multiway graph:

However identical to within the 2D case issues get much less trivial if there are constraints on development. For instance, assume that earlier than inserting a brand new cell we depend the variety of cells that lie both distance 1 or distance 2 away. If the variety of allowed cells can solely be precisely 1 we get habits like:

The corresponding multiway system is

or after canonicalization:

The variety of distinct sequences after t steps right here is given by

which will be expressed by way of Fibonacci numbers, and for giant t is about .

The rule in impact generates all doable Morse-code-like sequences, consisting of runs of both 2-cell (“lengthy”) black blocks or 1-cell (“quick”) black blocks, interspersed by “gaps” of single white cells.

The branchial graphs for this technique have the shape:

random evolutions for all doable guidelines of this kind we get:

The corresponding canonicalized multiway graphs are:

The principles we’ve checked out to this point are purely totalistic: whether or not a brand new cell will be added relies upon solely on the overall variety of cells in its neighborhood. However (very similar to, for instance, in mobile automata) it’s additionally doable to have guidelines the place whether or not one can add a brand new cell will depend on the whole configuration of cells in a neighborhood. Principally, nevertheless, such guidelines appear to behave very very similar to totalistic ones.

Different generalizations embody, for instance, guidelines with a number of “colours” of cells, and guidelines that rely both on the overall variety of cells of various colours, or their detailed configurations.

The Three-Dimensional Case

The sort of evaluation we’ve carried out for 2D and 1D aggregation methods can readily be prolonged to 3D. As a primary instance, contemplate a rule by which cells will be added alongside every of the 6 coordinate instructions in a 3D grid every time they’re adjoining to an current cell. Listed here are some typical examples of random clusters fashioned on this case:

Taking successive slices by way of the primary of those (and coloring by “age”) we get:

If we enable a cell to be added solely when it’s adjoining to only one current cell (comparable to the rule 6:{1}) we get clusters that from the surface look virtually indistinguishable

however which have an “airier” inner construction:

Very like in 2D, with 6 neighbors, there can’t be unbounded development until cells will be added when there is only one cell within the neighborhood. However in analogy to what occurs in 2D, issues get extra difficult once we enable “nook adjacency” and have a 26-cell neighborhood.

If cells will be added every time there’s a minimum of one adjoining cell, the outcomes are much like the 6-neighbor case, besides that now there will be “corner-adjacent outgrowths”

and the entire construction is “nonetheless airier”:

Little qualitatively adjustments for a rule like 26:{2} the place development can happen solely with precisely 2 neighbors (right here beginning with a 3D dimer):

However the basic query of when there may be development, and when not, is sort of difficult and refined. Particularly, even with a particular rule, there are sometimes some preliminary situations that may result in unbounded development, and others that can’t.

Typically there may be development for some time, however then it stops. For instance, with the rule 26:{9}, one doable path of evolution from a 3×3×3 block is:

The complete multiway graph on this case terminates, confirming that no unbounded development is ever doable:

With different preliminary situations, nevertheless, this rule can develop for longer (right here proven each 10 steps):

And from what one can inform, all guidelines 26:{n} result in unbounded development for , and don’t for .

Polygonal Shapes

To this point, we’ve been “filling in cells” in grids—in 2D, 1D and 3D. However we will additionally take a look at simply “inserting tiles” with out a grid, with every new tile attaching edge to edge to an current tile.

For sq. tiles, there isn’t actually a distinction:

And the multiway system is simply the identical as for our authentic “develop anyplace” rule on a 2D grid:

Right here’s now what occurs for triangular tiles:

The multiway graph now generates all polyiamonds (triangular polyforms):

And since equilateral triangles can tessellate in a daily lattice, we will consider this—just like the sq. case—as “filling in cells in a lattice” quite than simply “inserting tiles”. Listed here are some bigger examples of random clusters on this case:

Basically the identical occurs with common hexagons:

The multiway graph generates all polyhexes:

Listed here are some examples of bigger clusters—exhibiting considerably extra “tendrils” than the triangular case:

And in an “successfully lattice” case like this we may additionally go on and impose constraints on neighborhood configurations, a lot as we did in earlier sections above.

However what occurs if we contemplate shapes that don’t tessellate the airplane—like common pentagons? We will nonetheless “sequentially place tiles” with the constraint that any new tile can’t overlap an current one. And with this rule we get for instance:

Listed here are some “randomly grown” bigger clusters—exhibiting all types of irregularly formed interstices inside:

(And, sure, producing such footage appropriately is way from trivial. Within the “successfully lattice” case, coincidences between polygons are pretty simple to find out precisely. However in one thing just like the pentagon case, doing so requires fixing equations in a high-degree algebraic quantity discipline.)

The multiway graph, nevertheless, doesn’t present any instantly apparent variations from those for “successfully lattice” instances:

It makes it barely simpler to see what’s happening if we riffle the outcomes on the final step we present:

The branchial graphs on this case have the shape:

Right here’s a bigger cluster fashioned from pentagons:

And keep in mind that the way in which that is constructed is sequentially so as to add one pentagon at every step by testing each “uncovered edge” and seeing by which instances a pentagon will “match”. As in all our different examples, there isn’t any choice given to “exterior” versus “inner” edges.

Notice that whereas “successfully lattice” clusters all the time ultimately fill in all their holes, this isn’t true for one thing just like the pentagon case. And on this case it seems that within the restrict, about 28% of the general space is taken up by holes. And, by the way in which, there’s a particular “zoo” of a minimum of small doable holes, right here plotted with their (logarithmic) chances:

So what occurs with different common polygons? Right here’s an instance with octagons (and on this case the limiting whole space taken up by holes is about 35%):

And, by the way in which, right here’s the “zoo of holes” on this case:

With pentagons, it’s fairly clear that difficult-to-resolve geometrical conditions will come up. And one might need thought that octagons would keep away from these. However there are nonetheless loads of unusual “mismatches” like

that aren’t simple to characterize or analyze. By the way in which, one ought to observe that any time a “closed gap” is fashioned, the vectors comparable to the perimeters that kind its boundary should sum to zero—in impact defining an equation.

When the variety of sides within the common polygon will get massive, our clusters will approximate circle packings. Right here’s an instance with 12-gons:

However in fact as a result of we’re insisting on including one polygon at a time, the ensuing construction is way “airier” than a real circle packing—of the sort that will be obtained (a minimum of in 2D) by “pushing on the perimeters” of the cluster.

Polyomino Tilings

Within the earlier part we thought of “sequential tilings” constructed from common polygons. However the strategies we used are fairly basic, and will be utilized to sequential tilings fashioned from any form—or shapes (or, a minimum of, any shapes for which “attachment edges” will be recognized).

As a primary instance, contemplate a domino or dimer form—which we assume will be oriented each vertically and horizontally:

Right here’s a considerably bigger cluster fashioned from dimers:

Right here’s the canonicalized multiway graph on this case:

And listed here are the branchial graphs:

So what about different polyomino shapes? What occurs once we attempt to sequentially tile with these—successfully making “polypolyominoes”?

Right here’s an instance primarily based on an L-shaped polyomino:

Right here’s a bigger cluster

and right here’s the canonicalized multiway graph after simply 1 step

and after 2 steps:

The one different 3-cell polyomino is the tromino:

(For dimers, the limiting fraction of space lined by holes appears to be about 17%, whereas for L and tromino polyominoes, it’s about 27%.)

Going to 4 cells, there are 5 doable polyominoes—and listed here are samples of random clusters that may be constructed with them (observe that within the final case proven, we require solely that “subcells” of the two×2 polyomino should align):

The corresponding multiway graphs are:

Persevering with for extra steps in a couple of instances:

Some polyominoes are “extra awkward” to suit collectively than others—so these sometimes give clusters of “decrease density”:

To this point, we’ve all the time thought of including new polyominoes in order that they “connect” on any “uncovered edge”. And the result’s that we will usually get lengthy “tendrils” in our clusters of polyominoes. However another technique is to attempt to add polyominoes as “compactly” as doable, in impact by including successive “rings” of polyominoes (with “older” rings right here coloured bluer):

Normally there are lots of methods so as to add these rings, and ultimately one will usually get caught, unable so as to add polyominoes with out leaving holes—as indicated by the purple annotation right here:

After all, that doesn’t imply that if one was ready to “backtrack and take a look at once more”, one couldn’t discover a approach to prolong the cluster with out leaving holes. And certainly for the polyomino we’re right here it’s completely doable to finish up with “excellent tilings” by which no holes are left:

Normally, we may contemplate all types of various methods for rising clusters by including polyominoes “in parallel”—identical to in our dialogue of causal graphs above. And if we add polyominoes “a hoop at a time” we’re successfully making a selected alternative of foliation—by which the successive “ring states” end up be straight analogous to what we name “generational states” in our Physics Undertaking.

If we enable holes (and don’t impose different constraints), then it’s inevitable that—simply with unusual, sequential aggregation—we will develop an unboundedly massive cluster of polyominoes of any form, simply by all the time attaching one edge of every new polyomino to an “uncovered” fringe of the prevailing cluster. But when we don’t enable holes, it’s a distinct story—and we’re speaking a few conventional tiling downside, the place there are finally instances the place tiling is inconceivable, and solely limited-size clusters will be generated.

Because it occurs, all polyominoes with 6 or fewer cells do enable infinite tilings. However with 7 cells the next don’t:

It’s completely doable to develop random clusters with these polyominoes—however they have an inclination to not be in any respect compact, and to have numerous holes and tendrils:

So what occurs if we attempt to develop clusters in rings? Listed here are all of the doable methods to “encompass” the primary of those polyominoes with a “single ring”:

And it turns that in each single case, there are edges (indicated right here in purple) the place the cluster can’t be prolonged—thereby demonstrating that no infinite tiling is feasible with this specific polyomino.

By the way in which, very similar to we noticed with constrained development on a grid, it’s doable to have “tiling areas” that may prolong solely a sure restricted distance, then all the time get caught.

It’s price mentioning that we’ve thought of right here the case of single polyominoes. It’s additionally doable to think about with the ability to add a entire set of doable polyominoes—“Tetris type”.

Nonperiodic Tilings

We’ve checked out polyominoes—and shapes like pentagons—that don’t tile the airplane. However what about shapes that may tile the airplane, however solely nonperiodically? For example, let’s contemplate Penrose tiles. The essential shapes of those tiles are

although there are extra matching situations (implicitly indicated by the arrows on every tile), which will be enforced both by placing notches within the tiles or by adorning the tiles:

Beginning with these particular person tiles, we will construct up a multiway system by attaching tiles wherever the matching guidelines are happy (observe that every one edges of each tiles are the identical size):

So how can we inform that these tiles can kind a nonperiodic tiling? One method is to generate a multiway system by which at successive steps we encompass clusters with rings in all doable methods:

Persevering with for an additional step we get:

Discover that right here a number of the branches have died out. However the query is what branches exist that may proceed perpetually, and thus result in an infinite tiling? To reply this we’ve got to do a bit of study.

Step one is to see what doable “rings” can have fashioned across the authentic tile. And we will learn all of those off from the multiway graph:

However now it’s handy to look not at doable rings round a tile, however as an alternative at doable configurations of tiles that may encompass a single vertex. There seems to be the next restricted set:

The final two of those configurations have the function that they will’t be prolonged: no tile will be added on the middle of their “blue sides”. Nevertheless it seems that every one the opposite configurations will be prolonged—although solely to make a nested tiling, not a periodic one.

And a primary indication of that is that bigger copies of tiles (“supertiles”) will be drawn on prime of the primary three configurations we simply recognized, in such a method that the vertices of the supertiles coincide with vertices of the unique tiles:

And now we will use this to assemble guidelines for a substitution system:

Making use of this substitution system builds up a nested tiling that may be continued perpetually:

However is such a nested tiling the one one that’s doable with our authentic tiles? We will show that it’s by exhibiting that each tile in each doable configuration happens inside a supertile. We will pull out doable configurations from the multiway system—after which in every case it seems that we will certainly discover a supertile by which the unique tile happens:

And what this all means is that the one infinite paths that may happen within the multiway system are ones that correspond to nested tilings; all different paths should ultimately die out.

The Penrose tiling entails two distinct tiles. However in 2022 it was found that—if one’s allowed to flip the tile over—only a single (“hat”) tile is adequate to pressure a nonperiodic tiling:

The complete multiway graph obtained from this tile (and its flip-over) is difficult, however many paths in it lead (a minimum of ultimately) to “lifeless ends” which can’t be additional prolonged. Thus, for instance, the next configurations—which seem early within the multiway graph—all have the property that they will’t happen in an infinite tiling:

Within the first case right here, we will successively add a couple of rings of tiles:

However after 7 rings, there’s a “contradiction” on the boundary, and no additional development is feasible (as indicated by the purple annotations):

Having eradicated instances that all the time result in “lifeless ends” the ensuing simplified multiway graph successfully consists of all joins between hat tiles that may finally result in surviving configurations:

As soon as once more we will outline a supertile transformation

the place the area outlined in purple can probably overlap one other supertile. Now we will assemble a multiway graph for the supertile (in its “bitten out” and full variant)

and might see that there’s a (one-to-one) map from the multiway graph for the unique tiles and for these supertiles:

And now from this we will inform that there will be arbitrarily massive nested tilings utilizing the hat tile:

Private Notes

Tucked away on web page 979 of my 2002 ebook A New Type of Science is a observe (written in 1995) on “Generalized aggregation fashions”:

And in some ways the present piece is a three-decade-later followup to that observe—utilizing a brand new method primarily based on multiway methods.

In A New Type of Science I did focus on multiway methods (each abstractly, and in reference to elementary physics). However what I stated about aggregation was principally in a bit known as “The Phenomenon of Continuity” which mentioned how randomness may on a big scale result in obvious continuity. That part started by speaking about issues like random walks, however went on to debate the identical minimal (“Eden mannequin”) instance of “random aggregation” that I give right here. After which, in an try and “spruce up” my dialogue of aggregation, I began “aggregation with constraints”. In the principle textual content of the ebook I gave simply two examples:

However then for the footnote I studied a wider vary of constraints (enumerating them a lot as I had mobile automata)—and observed the shocking phenomenon that with some constraints the aggregation course of may find yourself getting caught, and never with the ability to proceed.

For years I carried across the thought of investigating that phenomenon additional. And it was usually on my checklist as a doable undertaking for a pupil to discover on the Wolfram Summer season College. Often it was picked, and progress was made in varied instructions. After which a couple of years in the past, with our Physics Undertaking within the offing, the concept arose of investigating it utilizing multiway methods—and there have been Summer season College tasks that made progress on this. In the meantime, as our Physics Undertaking progressed, our instruments for working with multiway methods tremendously improved—finally making doable what we’ve carried out right here.

By the way in which, again within the Nineties, one of many many matters I studied for A New Type of Science was tilings. And in an effort to find out what tilings had been doable, I investigated what quantities to aggregation underneath tiling constraints—which is the truth is even a generalization of what I contemplate right here:

Thanks

Firstly, I’d wish to thank Brad Klee for in depth assist with this piece, in addition to Nik Murzin for extra assist. (Thanks additionally to Catherine Wolfram, Christopher Wolfram and Ed Pegg for particular pointers.) I’d wish to thank varied Wolfram Summer season College college students (and their mentors) who’ve labored on aggregation methods and their multiway interpretation in recent times: Kabir Khanna 2019 (mentors: Christopher Wolfram & Jonathan Gorard), Lina M. Ruiz 2021 (mentors: Jesse Galef & Xerxes Arsiwalla), Pietro Pepe 2023 (mentor: Bob Nachbar). (Additionally associated are the Summer season College tasks on tilings by Bowen Ping 2023 and Johannes Martin 2023.)

See Additionally

Video games and Puzzles as Multicomputational Methods

The Physicalization of Metamathematics and Its Implications for the Foundations of Arithmetic

Multicomputation with Numbers: The Case of Easy Multiway Methods

Multicomputation: A Fourth Paradigm for Theoretical Science

Multiway Turing Machines

Combinators: A Centennial View—Updating Schemes and Multiway Methods

The Updating Course of for String Substitution Methods