Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 Plastc Number: Costructo ad Applcatos Lua Marohć Polytechc of Zagreb, 0000 Zagreb, Croata lua.marohc@tvz.hr Thaa Strmeč Polytechc of Zagreb, 0000 Zagreb, Croata matemata.tvz@gmal.com Abstract I ths artcle we wll costruct plastc umber a heurstc way, explag ts relato to huma percepto threedmesoal space through archtectural style of Dom Has va der Laa. Furthermore, t wll be show that the plastc umber ad the golde rato specal cases of more geeral defto. Fally, we wll expla how va der Laa s dscovery relates to percepto ptch space, how to defe ad tue Padova tervals ad, subsequetly, how to costruct chromatc scale temperamet usg the plastc umber. (Abstract) Keywordsplastc umber; Padova sequece; golde rato; musc terval; musc tug ; I. INTRODUCTION I 98, shortly after abadog hs archtectural studes ad becomg a ovce mo, Has va der Laa dscovered a ew, uque system of archtectural proportos. Its costructo s completely based o a sgle rratoal value whch he called the plastc umber (also ow as the plastc costat): 4.48.... () Ths umber was orgally studed by G. Cordoer 94. However, Has va der Laa was the frst who explaed how t relates to the huma percepto of dffereces sze betwee threedmesoal objects ad demostrated hs dscovery (archtectural) desg. Hs ma premse was that the plastc umber rato s truly aesthetc the orgal Gree sese,.e. that ts cocer s ot beauty but clarty of percepto (see []). I ths secto we wll expla how va der Laa bult hs ow system of proportos ad whch way t determes all basc elemets (buldg blocs) of hs archtectural style. I the followg secto we wll compute () exactly ad show that the plastc umber ad the golde rato are two cases of the same defto, obtaed by varyg the sgle parameter, space dmeso. I the fal secto plastc umber s dscussed wth the ptch space of musc. It wll be explaed how the plastc umber s related to our percepto of musc tervals ad to the Wester toe chromatc scale toato (temperamet). Dom Has va der Laa (90499), was a Dutch archtect ad a member of the Beedcte Order. The word plastc was ot teded to refer to a specfc substace, but rather ts adjectval sese, meag somethg that ca be gve a threedmesoal shape (see []). Gérard Cordoer (909), was a Frech egeer. He studed the plastc umber (whch he called the radat umber) whe he was just years old. Fgure. Twety dfferet cubes, from above A. Percepto of Proportos: a Expermetal Study Has va der Laa thought that, percevg a composto of objects space, t oly matters how ther dmesos relate to each other, wth the huma body as a referece pot. The followg example llustrates hs reasog. Image a ple of, say, 0 woode cubes. Let all cubes have a dstct sze, where edge legth of the smallest oe equals 5 cm ad edge legth of the largest oe 5 cm. Fg. shows a example of such a set, vewed from above. If cubes are close eough to each other, le they are the pcture, we automatcally terpret them terms of how they relate to each other. All cubes beg of smlar color, texture ad shape, we obvously relate them terms of sze. Our bra wll, furthermore, decde to relate edge legths, sce they represet, vsually, the most acceptable formato. Due to the lmtatos of our atural ablty to measure ad compute, these relatos are terpreted as ratos of small atural umbers. Ths s what maes our percepto so terestg. I our example wth set of cubes thgs are qute smple: we wll automatcally decde to relate dfferet cubes terms of ther edge legth. Oce relatos have bee establshed, process of orderg them a sequece aturally taes place. Ths s the most terestg part: how wll our bra decde to accomplsh such tas? Idexg cubes wth umbers 0 seems http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 5
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 too tedous to do at a glace. To costruct shorter sequece we wll frst cosder cubes of approxmately the same sze practcally equal, thus creatg groups ad the orderg them by sze. Va der Laa amed that groups types of sze. The reaso for hs scetfc approach through a expermetal study was to aswer the followg questo: what s the smallest proporto a : b of two legths, where a s loger tha b, by whch they belog to dfferet types of sze? Let us ow mage smlar stuato wth 0 dfferet cubes, but where edge legth of largest cube equals 60 cm. Now the largest cube seems too bg for the smallest oe, because our bra cosders relatg them terms of proportos too dffcult ad mmedately gores t. Such objects stay separated; whe oe s focus of vew, the other seems to dsappear. Ths proves that types of sze are related to each other up to some pot where they dffer too much. They are therefore grouped categores, whch va der Laa called orders of sze, wth whch they ca be easly related. The other questo he wated to aswer emprcally s: what s smallest proporto b : a of two legths, where a s loger tha b, by whch they belog to dfferet orders of sze? Aswers to these two questos precsely expla grouds of huma percepto, subsequetly provg ts objectvty. Has va der Laa foud these aswers by coductg smple expermets whch objects had to be sorted by sze. After statstcal aalyss, two umbers emerged: frst of them, 4/, aswers the frst questo, ad other oe, /, the secod. We ca express these mportat results form of a sgle defto: two objects belog to dfferet types of sze f quotet of ther szes s about 4/ (plastc umber), whle they belog to dfferet orders of sze f oe object s about seve tmes larger tha the other. Rato / s also used to determe whether two objects the same type of sze are each other s eghborhood. Va der Laa used t hs wor to relate buldg blocs to the opegs betwee them. He calculated the deal relato betwee the wdth of a space ad the thcess of the walls that form t: seve to oe o ceter. If t would exceed ths rato, the earess of the elemets as va der Laa called t would be lost. Fgure. Types of sze wth a order of sze (source: []) Fgure. Costructo of the Form Ba (source: [4]) System of proportos costructed from the plastc umber s llustrated Fg.. Legths of bars show the pcture crease by geometrc progresso, accordg to the plastc umber Ψ, from to.59.... Therefore, they represet all eght types of sze wth oe order of sze. I the rest of ths text, umber 0.698... wll be deoted by ψ. B. The Form Ba Has va der Laa thought that the art of archtecture les gvg spaces meagful dffereces, for whch the plastc umber s a uque tool. I hs wor, he dstgushed three ds of basc objects, or forms: blocs, bars, ad slabs. He called ther composto thematsmos: the ordered arragemet of dfferet forms. Usg ths, he created the Form Ba, a set cosstg tally of 6 shallow base blocs, show o the left sde of Fg.. Ther depth (heght) s equal, whle ther wdths ad legths crease accordg to the plastc umber rato, remag the same order of sze. Oe group of blocs (red) has lttle dfferece legth, wdth ad heght. Aother group, the bars (yellow), dstgush themselves wth a varato oe drecto. I the thrd group, the slabs (blue), measuremets are exteded two drectos relatve to the smallest sze. I the mddle remas a fourth group (lght gray) whch va der Laa calls the blac shapes. These last shapes have propertes of all the aforemetoed groups: bloc, bar ad slab. I each group oe core shape could be poted out from the mddle of each group (slghtly lghter color). These four powerful core shapes are the represetatves of ther group. O the rght sde of Fg. base blocs are the same maer vared the thrd dmeso, heght, thus formg a quastetrahedro, whch completes the costructo of the Form Ba. Its elemets are llustrated Fg. 4. Has Va der Laa s most famous wor s the Abbey of St. Beedctusberg Vaals (see Fg. 5), cosstg of a upper church, a crypt ad a atrum (956968). He bult several other moasteres (Abbey Rooseberg Waasmuster, Moastery Mary ssters of St. Fracs, Moastery church Tomellla) ad also a prvate house Hus Naalde. He ot oly desged buldgs, but also furture ad eve a typeface (see Fg. 6). He chose the heghts of a bech ad a char to be the startg pot ad the appled the system of measuremets of the plastc umber to create the other dmesos. The typeface he http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 54
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 veted, the Alphabet stoe, s based o the Roma carved stoe captals. The basc fgures of the letters are the square ad the rectagle rato four to three, whch the form the letters ad eve the spaces betwee them. Fgure. Breag the segmet AB ad parts (source: []) II. CONSTRUCTION OF THE PLASTIC NUMBER Fgure 4. Elemets of the Form Ba Fgure 5. The Abbey Vaals (Fras de la Couse) Fgure 6. Va der Laa's typeface ad furture desg A. Evoluto of the Golde Rato Has va der Laa cocluded that umber 4/ s the ratoal approxmato of some value whch should be precsely defable. The base of hs mathematcal research s oe possble costructo of umber Φ, called golde rato or dve proporto, a acet aesthetcal axom. The latter wors well plae (ts smplest represetato beg the golde rectagle), but fals to geerate harmoous relatos wth ad betwee threedmesoal objects. Va der Laa therefore elevates defto of the golde rectagle terms of space dmeso. Golde rato ca be calculated by sectog the segmet AB two parts AC ad BC such that AB BC, () BC AC as show o the left sde of Fg.. Segmets AB ad BC are sdes of the golde rectagle. Lettg AB t follows BC BC BC BC BC. Golde rato s obtaed by solvg last equato (): ().6804... (4) Va der Laa breas segmet AB the smlar maer, but three parts. If C ad D are pots of subdvso, plastc umber Ψ s defed wth AB AD BC AC CD, (5) AD BC AC CD BD as llustrated o the rght sde of Fg.. Lettg AB, from AC BC, BD AD ad (5) follows. (6) http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 55
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 Usg Cardao s formula, () s obtaed from (6) as the oly real soluto. Segmets AC, CD ad BD ca be terpreted as sdes of a cubod aalogous to the golde rectagle. That cubod s obvously cotaed the Form Ba (lght blue rectagle Fg. s ts base). B. Harmoous umbers Numbers Ψ ad Φ ca be obtaed from more geeral defto, whch s based o the followg two theorems. Theorem. For gve atural umber, let f ( x) x x. () The there exsts real umber (, ) such that f 0 ad for every root r C of the polyomal f the followg statemet holds: r r. (8) Proof setch. Codto mples that P ( ) 0 ad P ( ) 0, therefore exstece of umber follows from the termedate value theorem. The rest of the statemet follows maly from EeströmKaeya theorem (see [9]). For or, () matches () or (6), respectvely; therefore t s ad. Corollary. For ay atural umber s. Proof. Assume the cotrary,.e. that there exsts atural umber such that. The () mples (9) whch s mpossble. Therefore for all. Theorem. For gve atural umber, let ( ),,,...,, H ( ) ( ) (0) H H,. The umber from Theorem s defed wth ( ) H lm. () ( ) H Proof setch. It s easy to see that () s characterstc polyomal for recurrece relato (0). Hece, there exst umbers ) C j C, j,,...,, such that H C j jrj C, where r, r,..., r C are roots of () dfferet tha. Now t s ot dffcult to prove that (8) mples (). Sequece H ( ) s Fboacc sequece for (see []) ad Padova sequece for (see [6]). ( () Let ow K be a hyperrectagle dmesoal space,, defed as Cartesa product of tervals: Fgure 8. Harmoous umber rato the plae (left) ad space (rght) j0 Legth of the logest sde of shortest sdes of () K ( ) K [0, H ]. () ( ) ( ) j K equals the sum of two. Now, () ca be terpreted as the lm ( ) () t of quotet of logest sdes of K ad K whe. For large these two adjacet hyperrectagles are cosdered almost smlar shape whle rato of ther szes equals approxmately. The latter s called harmoous umber (or clarty rato) for dmesoal Eucldea space. That s because up to the lmt of our percepto of space ( ), harmoous umber represets smallest (greater tha ) rato of szes of two smlar dmesoal objects such that a) ther dfferece sze s mmedately otceable, b) relato betwee ther szes s perceved as clear, pleasat, harmoous, atural ad stable. Statemets a) ad b), demostrated for ad Fg. 8, are fact osescal for because of percepto lmts 4 ; cocept of those harmoous umbers s just a terestg ductve hypothess. Approxmate values of the frst seve harmoous umbers are gve the followg table. 4 5 6 8 χ.680.4.0.6.4.8.090 Frst two harmoous umbers stad apart beg oly oes we ca experece. Moreover, they dffer from others pure mathematcal, abstract sese. Oe could easly prove that for x or x there exst atural umbers p ad q such that the followg s satsfed: p q x x, x x. () Real umbers x that satsfy these codtos are called morphc umbers. It ca be show (see [5]) that the plastc umber ad the golde rato are the oly two such umbers. Because for every, these are oly harmoous morphc umbers. 4 Percevg the sze of a object by vewg mples relatg legths of all ts sdes (to determe the logest oe). Hece oe must perceve these frst, what s possble, obvously, oly threedmesoal space ad ts subspaces. http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 56
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 III. PLASTIC NUMBER IN MUSIC A. Musc Itervals I ths secto we wll establsh coecto betwee the plastc umber ad the stadard Wester toe musc system (chromatc scale). Let us defe tervals wth the chromatc scale, spag oe octave, as mathematcal objects. Sce tervals re measured semtoes ad there are semtoes betwee root toes of two adjacet scale staces, we typcally represet them wth elemets of cyclc group Z. It cotas whole umbers from 0 to, whch are mapped to set of basc tervals wth a octave (uso, mor secod, major secod, mor thrd, major thrd, perfect fourth, trtoe, perfect ffth, mor sxth, major sxth, mor seveth ad major seveth) oetooe fasho, beg ther wdth semtoes. Furthermore, stadard addto of musc tervals, defed as sum of ther wdths, matches addto Z sce sum of tervals whch exceeds octave represets toe ext scale stace, so t ca be reduced to basc terval by subtractg semtoes. We wll deote group of basc tervals wth I = Z. To eep further otato smple, let us observe that for every whole umber there s a uque terval m I such that s cogruet to m modulo. We deote m, thus defg map B. The Ptch Space Z I. (4) Whe two toes T ad T are soudg smultaeously, we automatcally relate ther fudametal frequeces ad ( further text frequeces). We therefore observe them ptch space, whch comprses all fudametal frequeces. Frequecy rato : s called a terval from T (lower toe) to T (hgher toe). Our bra terprets frequeces as ther logarthms, so T ad T are perceved as log ad log, whle terval betwee them s log : log log. It seems to us that terval betwee T ad T s dfferetatot T. The followg pheomeo occurs: two frequecy ratos r ad r are perceved as the same terval dfferet octaves f r r for some whole umber. Ths s a equvalece relato: we wll deote r ~ r. Every frequecy rato r therefore has ts uque represetat r [,) such that r ~ r, defed wth r log r. r (5) To materalze the toe chromatc scale ptch space,.e. to traslate wrtte musc to soud, we must defe frequecy rato [,), whch we call tug or toato, for every basc terval I. That s called scale temperamet. Example s the equal temperamet, I. (6) Gve the toe of frequecy φ ad the whole umber, the set of frequeces { : I} s toe represetato of the chromatc scale th octave of the ptch space. Observe that j l, where, j ad l are basc tervals, mples that t has to be at least, otherwse the basc musc theory j l would cease to mae sese 5. How precse must the latter approxmato be? Perceved dfferece (terval) betwee toes of frequeces ad s typcally defed wth d,, where d, 00log [cets]. () Oe octave s therefore 00 cets wde. Smallest audble dfferece betwee two frequeces s about 0 cets. Mor secod, smallest terval wth the chromatc scale, s 00 cets wde. Therefore, two tugs ad of the same terval, belogg to dfferet temperamets, should dffer by less tha quartertoe,.e. satsfy d, 50. C. Types ad Orders of Sze Ptch Space Whle expermetg wth moochord, Pythagoras cocluded that tervals wth smplest frequecy ratos soud most harmoous 6. These are ratos :, : ad 4 :, whch represet oly three perfect tervals: octave, ffth ad fourth, respectvely. At the tme whe polyphoc musc of Europe started to develop these were oly cosoat tervals; all others were cosdered dssoat (ustable). Because Ψ s slghtly smaller tha 4/, more precsely d 4,. cets, t ca be terpreted as the hghest lower boud for tugs of perfect tervals. I other words, tug Ψ resembles frst perfect terval after trval uso. Therefore, for the gve toe of fre quecy φ, set of frequeces { :,,...,8} s a order of sze, whle ts elemets are types of sze ptch space. Its wdth s. 59, whch s slghtly wder tha two octaves plus mor seveth: d (, t 0 ). 6 cets. Therefore, Δ embraces frst elemets of harmoc seres (fudametal toe ad frst 6 overtoes). If the dstace betwee two toes belogg to staces of chromatc scale s less tha Δ, t s easy to relate them, because spectra of both toes are audble eough o the tersecto of ther domas. Whe the dstace s larger tha Δ, relatg becomes dffcult, especally for ds 5 Ths s the ma problem of scale temperato. For chromatc toe scale t has several solutos, whch were gve the course of cetures. Fally, the equal temperamet was chose as the least compromsg oe. 6 A terval betwee two toes seems as harmoous as audble tersecto of ther spectra s,.e. how soo ther hgher harmocs (overtoes) start to cocde. http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 5
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 soat tervals; whe t spas more tha several octaves, qualty of cosoace, or harmoy, s also lost all such tervals soud more or less dssoat. Perfect octave, or frequecy rato :, satsfes codtos a) ad b). It s hece harmoous umber for the ptch space, whch s oedmesoal by the defto. Lettg :, prevous defto of harmoous umbers exteds to the whole category of Eucldea spaces { E : N }. D. Plastc Number Temperamet Fally, t wll be show how va der Laa s types of sze S ( ), where s atural umber, geerate possble tugs for basc tervals. Multplyg () by yelds S( ) S( ) S( ) (8) for every atural umber ; hece types of sze satsfy Padova recurrece. Members of the sequece p ) ( P ), where ( 4 P s th Padova umber, obvously have the same property. They form the largest subsequece of the Padova sequece such that all ts members are mutually dfferet:,,4,5,,9,,6,,8,,49,... p (9) Iterval whch spas p semtoes s called Padova terval ad s deoted smply by p (frst te of them are show Fg. 9). Now, t s atural to assume that S () s proportoal to the logarthm of frequecy rato r of terval log ( r p,.e. ) S( ). (0) Because p, r should equal ; hece / S(). TABLE. PLASTIC NUMBER TEMPERAMENT Iterval Rato Devato Iterval Rato Devato Uso.0000 0 Trtoe.985 9. M. d.056 4.49 Perf. 5 th.5 +4.8 Maj. d.6 8.9 M. 6 th.5969 +0.4 M. rd.85 5.85 Maj. 6 th.685 +5.85 Maj. rd.54 0.4 M. th.90 +8.9 Perf. 4 th.5 4.8 Maj. th.89 +4.49 Solvg (0) r s gve wth r r s obtaed. Its represetat tug log { } () where { x} x x deotes fractoal part of some real umber x 0. These tugs ca be used to buld orgal plastc umber temperamet show Table, as follows. Fg. 0 shows devato of frequecy ratos r from the latter beg equally tempered tervals p / p, as they are the moder pao. Whe 5, that dfferece starts to grow rapdly. Therefore, oly r for 5 are usable. Idexes of the most sutable tugs for frst seve basc tervals are chose from the followg table. p d(, ) r p p d(, ) r p p d(, ) +.04 6 9 +5.85 4.49 5.85 0 0 4.49 4 0.4 8 4 0.4 5 4.8 4 5 +6.9 9 9 +5.85 4 8.9 5 6.9 0 4 0.4 5 6 9. Frst seve frequecy ratos (from uso to trtoe) Table are hece obtaed by computg r for,,4,,, ad 5, respectvely. Others are computed usg the prcple of the verse terval,.e. dvdg by tug for mor secod yelds tug for major seveth ad so forth. r p, Fgure 9. Frst te Padova tervals, bult o the mddle C Fgure 0. Devato of plastc umber tugs from equal temperamet REFERENCES [] Rchard Padova, Dom Has Va Der Laa ad the Plastc Number, pp. 89 Nexus IV: Archtecture ad Mathematcs, eds. Km Wllams ad Jose Fracsco Rodrgues, Fuceccho (Florece): Km Wllams Boos, 00. [] Rchard Padova: Dom Has va der Laa: Moder Prmtve, Archtectura & Natura Press, 994. [] H. va der Laa, Le Nombre Plastque: quze Leços sur l'ordoace archtectoque, Lede: Brll, 960. [4] http://www.vaderlaastchtg.l/e(november, 0) [5] J. Aarts, R. J. Fo, ad G. Krujtzer, Morphc Numbers. Neuw Arch. Ws. 5, pp. 5658, 00. [6] I. Stewart, Tales of a Neglected Number. Sc. Amer. 4, 00, Jue 996. [] S. R. Fch, Mathematcal Costats, Cambrdge Uversty Press, pp. 5, 00. [8] James Murray Barbour: Tug ad Temperamet: A Hstorcal Survey, Courer Dover Publcatos, 95. [9] G. Sgh, W. M. Shah, O the EeströmKaeya theorem, Appled Mathematcs, 00,, 555560. http://www.arsacof.com 0. Natural scece (mathematcs, chemstry, bology, physcs) 58