Penrose Tilings and Periodicity

Transcription

1 Perose Tiligs ad Periodicity Christopher McMurdie Mathematics Departmet, Uiversity of Arizoa 67 N. Sata Rita Ave., Box 0089 Tucso, AZ Project: This is the first part of a cotiuig project o o-commutative geometry. The space of all perose tiligs offers a iterestig example of a NCG; work doe this semester covers properties of tiligs i geeral as well as specific properties of perose tiligs. This project is supported by a UofA udergraduate research assistatship. Work doe was assisted ad supervised by Arlo Caie, a graduate studet, ad his thesis advisor, Dr. Doug Pickrell.

2 We begi our discussio of Perose Tiligs with a discussio of the more geeral cocept of tiligs. Two-dimesioal tiligs are a plae fillig arragemet of a fiite umber of prototiles, such that the etire plae is covered with a prototile ad o tiles overlap. The most familiar example might be a ifiite grid of squares. I this case, the prototile is the square. This is also oe of the relatively few tiligs that ca be costructed usig idetical, regular -gos. I fact, the oly regular -gos to tile the plae are the equilateral triagle, the square, ad the hexago. Proof: Cosider a tilig of the plae by regular -gos for some!. A tilig iduces a graph o the plae. Thus, at ay particular vertex, there is some k! such that k ( it ) = 360. It is well kow that the iterior agle of a regular ( ) 80 k ( ) 80 -go is give by it =. Thus, = 360, so = k. ( I + ) 4 Let I =. The = I +, so = k!. Thus, + = k. Sice I I k!, k # so 4 k I = #. If 4!, the I is a factor of 4, i.e. I {-4,-,- I,,,4}. Sice = I +, ad!, {3,4,6}. Thus, the set of regular -gos satisfyig the ecessary coditios for a edge-to-edge tilig are the regular 3-go, 4-go, ad 6-go, or the equilateral triagle, square, ad regular hexago. It turs out that geeralizig the coditios for whether prototiles ca tile the plae or ot is much more complicated as the umber of prototiles icreases. As a example, we have foud that two differet prototiles, a regular -go ad -go, ca oly tile the plae if the figures cosidered are the petago ad decago, the square ad octago, or the triagle ad hexago. Note the icreased complexity of the proof, foud i the appedix. Cosiderig more geeral polygos, ad more geeral umbers of prototiles would require differet methods of aalysis. Periodicity. A importat distictio to make betwee tiligs of the plae is whether they exhibit traslatioal symmetry. For example, the tilig cosistig of a grid of squares of uit legth is periodic because traslatio up a uits ad across b uits yields the same patter ab, #. Choose a vertex. The, the vector a,b begiig at this vertex is a vector of traslatio symmetry. It is easy to thik of other, more complex tiligs, which also exhibit traslatioal symmetry. Cosider the tilig of regular decagos ad regular petagos. A vector betwee the ceter of ay petago ad the ceter of ay other petago describes traslatioal symmetry. As oe costructs tiligs, it seems difficult to coceive of costructig aperiodic tiligs of the plae. Perose Tiligs, as will be discussed later, are examples of such aperiodic tiligs of the plae.

3 Perose Tiligs. Prototiles are fudametal shapes which ca be arraged o a plae to form a tilig. Perose tiligs cosist of two isosceles triagles, oe with a acute vertex agle ad oe with a obtuse vertex agle, such that the diagram i Figure is valid. These prototiles have a iterestig scalig relatioship that oe ca thik of as aalogous to the golde rectagle. The followig proof is a costructio of a double decompositio, ad demostrates the relatioship of the successive sizes of prototiles. The fudametal tiles of a Perose Tilig decompose twice with the ratio :τ. Proof: Figure Figure We assume Figure. We the costruct a lie from D to some poit E o $%%& BC such that EDC = BDA. Sice ' ABC is isosceles, BAC = BCA. Thus, ' CED ~ ' ABD by AA similarity. Note that ' BDC is isosceles, so that ACB = CBD. Call this agle α. The, cosiderig ' ABC, ad lettig

4 ABD = β, we fid that 3α + β = π. Now cosider ' CDE. It must also be isosceles sice it is similar with a isosceles triagle. Thus, we fid from π π ' DEC thatα + β = π. Combiig these expressios, α = ad β =. We the 5 5 $%%& costruct a lie from D to some poit F o AB such that FDA = α. The, FDA = FDB. Now costruct a lie from F to some poit G o $%%& AD such that FGD = β. Clearly, ' GDE ' FDB by AAS cogruece. It is also clear that these triagles are cogruet to ' DCE by AAS. Cosider ' DCB. Sice it is isosceles, DCB = DBC ad by trasitivity, we have that ' DBC ~ ' ABC by AA cogruece. Thus, DC = BC. But, AC = AD + DC ad AD = BC. Thus, BC AC DC BC = BC BC + DC. Arbitrarily, let DC =, BC = x. The x = x+ + 5 x x = 0 x= = τ. It is a simple exercise to fiish ad show that ' AGF ' DEB ad that the ratio of similarity betwee ' AGF, ' CDB isτ. Ifiite Plae. This decompositio process is perhaps the most fudametally importat aspect of Perose Tiligs, because it is what isures that Perose Tiligs exist, which are arbitrarily large. By successively repeatig this double decompositio to each prototile, ad the elargig the dimesios of the graph to retai the prototiles origial size, essetially we have expaded the graph by a factor ofτ. I this limitig process, we ca ask aother importat questio. What is the ratio of tiles with a acute vertex agle, to tiles with a obtuse vertex agle? We fid that the ratio is τ! Proof: Let λn be the ratio of prototiles with acute vertex agles to those with obtuse agles after N double decompositios. We ote that after every double decompositio, the obtuse prototiles each split ito two ew obtuse prototiles ad a acute prototile, while each origial acute prototile splits ito a ew obtuse prototile ad a ew acute oe. I symbols represetig the umber of obtuse ad acute prototiles at each stage N of double decompositio, ON+ = ON + AN, AN+ = ON + PN. Thus, ON + ON+ AN λn + λn + = = =. A O N+ N + λn + A What is the limit of λ as N? Cosider the Fiboacci umbers defied by the sequece UN = UN + UN. Defie the fuctio Å() :!! to be the fuctio such that Å(0) =, Å () =, ad >, Å( ) = Å( ) + Å ( ). N

5 The the value λ = N ( Å( N) + Å(N ) ) λ0 + ( Å( N) + Å(N ) ) ( Å(N ) + Å(N ) ) λ + ( Å(N ) + Å(N ) ) 0, as ca be show iductively. Sice Å is a icreasig fuctio, it is easily see that Å( N) + Å(N ) lim ( λn ) = lim. But by properties of Å, this reduces to N N Å ( N ) + Å ( N ) Å( N) + Å(N + ) Å(N + ) Å( a+ ) lim = lim = lim. Sice N (N ) + ( N) N (N + ) a Å Å Å Å( a) Å( a+ ) Å( a ) Å( a + ) Å( a+ ) = Å( a) + Å ( a ), = +. The lim exists, so let Å( a) Å( a) a Å( a) Å( a + ) Å( a ) lim = L. The lim =. So we fid that a Å( a) a Å( a) L L= + L = L+ L= τ. Thus, lim ( λn ) = τ. L N Aperiodicity. The followig argumet is preseted i Tiligs, Chaotic Dyamical Systems ad Algebraic K-Theory (see Resources) ad is paraphrased below. We ow cosider why Perose Tiligs are aperiodic. Suppose to the cotrary that a Perose tilig T was periodic, the there would be some vector v of traslatioal symmetry. Let T be the tilig after a process of deflatio (where edges of the same type are erased, ad the tilig is scaled dow by a factor of, leavig the prototiles τ the same size as the origial tilig). The T still has the same symmetry vector v, though it has bee reduced by. Repeatig this process sufficietly may times τ leads to a cotradictio, because the vector v caot be smaller tha the size of the prototiles. Thus, o symmetry vector v is permissible, ad the tilig is aperiodic. Ufortuately, this argumet is isufficiet because it applies to periodic tiligs that have decompositio. Cosider the periodic tilig of equilateral triagles ad regular hexagos. There exists a decompositio that erases edges ad preserves sufficietly large symmetry vectors. However, give ay vector of symmetry, the decompositio could be performed eough times to make the vector smaller tha the legth of either prototile. A more rigorous argumet for the aperiodicity of Perose Tiligs is required that utilizes the properties of Perose Tiligs that are uique from those of periodic tiligs.

6 Appedix Which two regular polygos of the form ad will tile the plai edge-to-edge? Petago ad Decago, Triagle ad Hexago, ad Square ad Octago. Proof: Sice this is a tilig of two prototiles, there must exist at least oe vertex where at least oe of each prototile is preset. We model the relatioship of this vertex. ( ) ( ) k 80 k 80 k, k,! such that + = 360 () The first term i () is the cotributio from the -go(s) ad the secod term is the cotributio from the -go(s). Now, we seek to limit the values of k, k. Sice the first term i () is goig to be a positive umber, we have k ( ) 80 < < k <. As oted i a previous result, K =! K { 3, 4, 6}. Sice k < K, k {,,3,4,5}. A similar argumet limits k, ad we fid that k {, }. We retur to k, k,,3,4,5 such that for >3, (), ad rewrite as { } { } ( ) ( ) k k + = 4 This equatio ca be re-writte to solve for i terms of k, k. k+ k = k+ k. We hold k fixed to determie what values of k make the expressio itegral. A exhaustive check of the possible values of k yields these results: + k k=: = k + k k=: k! k {, 4} =! k { }, Note that we are oly iterested i the cases whe >. The the fial set of k,, k, that satisfies equatio () is ( ) {(,0,,5 ),(,8,,4 ),(,6,4,3 ),(,6,,3 )}

7 This correspods to the petago ad decago, the square ad octago, ad two refereces to the equilateral triagle ad hexago. Ay edge-to-edge tilig of regular ad gos must be a subset of this set. Resources Tasadi, Tamas. Perose Tiligs, Chaotic Dyamical Systems ad Algebraic K-Theory. April 09, 00. < B. Grubaum ad G.C. Shephard. Tiligs ad Patters. Freema, New York, 989.