TILES FROM COVERINGS. a b c

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1 TILES FROM COVERINGS Previn: You re plying ll the wrong notes! Morecme: No, I m plying ll the right notes ut not necessrily in the right orer. Dernge Zongons Throughout this rticle we shll e concerne with zongons: polygons with n even numer of equl sies n opposite sies prllel, i.e. polygons with equl sies so rrnge tht the resulting figure hs hlf-turn symmetry. (The letters ientify orienttions. A repete letter therefore inictes prllel sie.) e c f f c e Specificlly we shll e concerne with the most symmetricl exmple: the evensie regulr polygon. If we strt with zongon n mix up the orer of sies the lw of ition of vectors tells us tht we shll still en up with polygon:

2 Our oject in this piece is to otin monoherl tilings, tilings using just one shpe. The lst exmple is not very promising. How cn we improve our chnces? Some Semlnce of Orer: Escherifiction One principle employe y Escher in his own tilings, n pplie in schools throughout the ln to ecorte the wlls of the mths eprtment, is to tke prllelogrm n complicte ech pir of opposite sies in the sme wy, thus mintining the trnsltionl symmetry of the tile. The ctivity is populr ecuse the chilren mke the exciting iscovery tht simple mthemticl ie cn mit rtistic complexity. We shll use it now to moify our regulr 2 n -gons. In this figure we tke our regulr polygon n reflect prt in chor:

3 c c M2 M1 c The chilren shoul prove tht the ot-shpe piece is zongon. (It hs 2 symmetry xes, necessrily perpeniculr, implying hlf-turn symmetry. By contriction or otherwise oler stuents shoul justify my phrse necessrily perpeniculr.) We see tht the prt where the originl letter orer is reverse my e trnslte to mtch corresponing prt on the other sie of the polygon. This property is unique to even-sie regulr polygons. It rises from the fct tht they hve 2n symmetry xes n tht the prouct of reflections in two prllel mirrors ( M1, M 2 ) is trnsltion. If we flip jcent sections of p n q sies, p + q = n, we shll hve performe the Escher trick se on rectngle. We cn suivie section: p + (q 1 + q 2 ) = n. The resulting Escher figure is hexgon. It hs opposite sies prllel ut not necessrily equl, ( more generl zongon if you like). Note how Escher rectngles persist s figures inscrie in the originl polygon:

4 The chilren shoul convince themselves etter still, their peers - tht this hexgon tiles the plne. Escher Tiles from Coverings This rings us to coverings. (We shll see why in moment.) Grünum n Shephr (1987) open their monumentl work with efinition which this Venn igrm expresses: Wys to fill the plne Pckings - no overlps Tilings Coverings - no gps From this, tilings emerge s the Golilocks plne-filling: no gps, no overlps, just right. The tiles on roof constitute covering. But if there were no shows to revel tht the sltes overlppe you coul think you were looking t tiling. Roof tiles (usully) form perioic tiling, one tht cn e generte y trnslting prt. My oservtion my e expresse in the theorem, offere without proof: perioic coverings imply perioic tilings. Our interest in coverings is tht we cn tke the ite out of one sie of our polygon y lpping congruent one. Lpping n Pliting As mthemticl roofer I cn either lp my sltes or plit them. (If you fin sltepliting emning crft, rememer I m mthemticl roofer, nxious only to mintin the istinction etween slte n tile.) In the first rrngement I cn lift off sltes in certin orer without isturing ny others (though, if the roof is infinite in extent, I won t know where to strt). In the secon, I cn lift no slte

5 without isturing others. (Agin, tht pplies to n infinite expnse or the interior of finite one.) These two lppings illustrte the two cses mentione erlier. The first uses 10-gons n is se on the rectngle; the secon uses 12-gons n is se on the hexgon. In the secon cse we further issect tile using pttern locks to show the rtistic possiilities in these irregulr polygons. In oth those cses ump comes opposite ite. But this nee not e so. The wy we ve lppe 10-gons in the next exmple is n interesting cse.

6 c e c e e c e 10-gons re use, ut the resulting tile is n 8-gon. The prllelogrm use for the Escher trnsltion is not rectngle. Mrke in white re the vertices which our lpping scheme requires to coincie. Mrke in white too re the fce orienttions which result. Ly out Polyron Frmework 10-gons on the floor in the ove wy n sk the chilren to show tht vertices t the lck points coincie n the sie etween them hs the orienttion shown. They shoul reprouce the figure using LOGO or Geometer s Sketchp, useful exercise in itself. Let us see wht hs hppene to our originl 10-gon s permuttion of the sequence of sie orienttions:! cece$ " # ceec% & We then lose the two es. We coul hve strte with the zongon forme from flipping the segment c in our originl 10-gon, then trnsposing c n :! cc$ " # cc% &

7 Becuse we re eling with single tiles me from regulr polygons the sequence of sies efines the tiling completely. Tke, for exmple: cgfehicefgih How mny sies hs tile? 18. Where o the vertices fll? Certinly where there is rek in the letter sequence re cycliclly. We fin 6 here so we know we hve in fct foun them ll: c! gfe! hi! c! efg! ih! We cn lso work out the orienttions of the sies concurrent t vertex with the two either sie of the rrow. Tken in pirs, the sequences eh,e,h shre letter ut revel thir: h e e! h! e h! Likewise for g,i,gi. Notice too tht eh n gi vertices lternte roun tile. Reers my like to use LOGO or Geometer s Sketchp to rw prt of this tiling. How out: cefcfe? How mny sies hs tile? 12. Where o the vertices fll? Certinly t the 3 points mrke y the soli rrows ut we know there must e 4 or 6. Since stretches of corresponing length must occupy corresponing positions in the first 6 n the lst 6 letters, we cn locte 4 th t the position shown y the oule-hee rrow: c! ef " c " fe " A wor out lpping n pliting. Two sltes cn only utt or lp n we cn trce lines of sltes where left lps right or vice vers. Pliting escries wht hppens in two imensions. Here we plit regulr 12-gons to prouce fmilir tiling. A vertex is point where 3 or more sltes (or tiles) meet. Notice tht the sense of superposition reverses when we move from one vertex to the next:

8 Notice too tht the symmetry of the regulr 12-gon llows trnsltions symmetriclly in 3 irections se on the regulr hexgon shown. So fr ll our tiles hve fce the sme wy. In the next tiling we plit 10-gons to prouce esign with tiles in two orienttions. Note however tht the rrngement of sltes without regr to lpping or pliting llows ifferent tiles. In the next picture we use Polyron Frmework 10-gons to show this neutrl strting point, efine y the prllelogrm unit cell. Three ifferent tiles re mrke. Ech necessrily hs the re of the unit cell. 1 n 2 oth hve umps corresponing to ites, result from lpping, n prouce tilings with tiles in the sme orienttion. (Pirs of sies in which the sltes simply utt ginst ech other contriute the single eges to the Escher hexgons, shown in white.) But we hve to plit the sltes to prouce 3, in which ite comes opposite ite, n ump opposite ump. The two orienttions re colour-coe in the igrm eneth.

9 1 2 c e 3 e c m2 m1 g2 m2 g1 g1 g2 m1 The tiles re zongons: the permuttion mintins repete sequence of (5) symols:

10 ! cece$ " # cece% & They hve two symmetry xes ( m1, m2 ). They fit together to give tiling with xes of glie reflection running in 2 irections ( g1, g2 ). Notice how those ox trnsltion cell. The tiling hs centres of 2-fol rottion symmetry site t the mi-points of sies where tiles in the sme orienttion ut. The next picture shows how the sltes re plite. To see the lternting pttern shown y the 12-gon pliting ove we hve to fuse vertices joine y utte eges.! " " We hve mrke 3 ngles roun prticulr vertex.! is the interior ngle of the regulr 10-gon.! is lesser ngle resulting from the flippe vertex. So tht they my pply the result elsewhere, oler chilren my seek n expression for! where k vertices of regulr m -gon hve een flippe. [m " 2(k + 1)]# Reers my like to confirm tht! =. Ressuringly, the sustitution m k = 0 gives the correct expression for!. In the present cse m = 10, k = 1 n we hve! = 108. Check:! lso 2! " # 360 " 144 = =

11 For our polygons m = 2n n the ientity simplifies to:! = (n " k " 1)# n. Go ck to the rin-shpe tile we otine y lpping 12-gons n o the rithmetic for this vertex: m=12 k=1 k=2 k=0 Wht vlues k 1,k 2,k 3 re possile in generl t vertex where 3 tiles/sltes meet? Reers my like to confirm tht k 1 + k 2 + k 3 = m! 6 = n! 3. 2 Lstly, consier slte where two jcent sections hve een flippe, i.e. the chors in which the reflections occur shre vertex. Confirm tht we my use the sme eqution, summing the numers of flippe vertices either sie of the common vertex. A nice cse occurs when! = 0 so tht two originl slte eges come together. Our formul tells us tht k = n! 1. But we cn see tht stright wy y foling slte in hlf. Here n = 7, so k = 6. In the ltter, one chor ecomes sie of the originl polygon, proviing trivil reflection xis, rwing our ttention to the fct tht! is hlf the interior ngle,! 2, (though we coul hve inferre tht fct from the lefthn figure too):

12 1! 5! 6 You cn imgine n nimte sequence eginning with the right-hn figure, which we cn cll (0,6), n proceeing (1,5), (2,4), (3,3), (4,2), (5,1), (6,0). As the numer of sies increses, the polygon pproches circle n! pproches hlf of 180 = 90, s require y the ngle in semicircle theorem. Tht in turn is specil cse of the sme segment theorem. Since! = " + #,! is constnt of the 2 polygon,! is function only of k, n k is constnt throughout given segment, we hve stisfie tht too. Screen or Floor? One woul like the or to e inclusive. One woul like the pupils to strt with mnipultives then migrte to the keyor where chnges cn e rung without limit. But tht is counsel of perfection. This whole exercise is 2-imensionl n tht my swing it for you. When I ws lying out my Polyron ecgons on the floor the chilren roun me wtche in incomprehension s they put their own pieces to the purpose for which they were intene: uiling polyher. Reers who nevertheless wish to know where to get the mterils mentione in the text shoul e-mil me. Note tht there re interctive computer environments for oth regulr polygons n the prticulr shpes mking up the set of pttern locks. References Grünum, B. n Shephr, G. C. 1987, Tilings n Ptterns, W. H. Freemn n Compny, ch.1, p.16. Keywors: Angle; Trnsltion; Zongon. Author Pul Stephenson, The Mgic Mthworks Trvelling Circus, Ol Coch House, Pen y Pylle, Holywell CH8 8HB e-mil: stephenson@mthcircus.emon.co.uk