globally rigid symmetric tensegrities by connelly, terrell, 1995

8/9/2019 Globally Rigid Symmetric Tensegrities by Connelly, Terrell, 1995

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TopologiestructuraieRevue in terdisciplinaire deg&om&rie appliqutie auxprob&mes de structure et demorphologie en design, enarchitecture et en g6nie.

StructuralTopologyInterdisciplinary journal ongeometry applied to problems ofstructure and morphology indesign, architecture andengineering.


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1 Topologie structurale l 21 l Structural Topology l 19951La revue est publiee par le groupe derecherche ((Tbpologie structurale H, sousla direction de Janos Baracs, avec lacollaboration de 1Association mathema-tique du Quebec et de Wniversite duQuebec h Montrkal.The journal is published by the Structuv-al76pology research group, under the direc-tion of Janos Baracs, with the collaborationof the Association mathkmatique duQuebec and the Universite du Quebec tiMontreal.

Cornit de directionManagement committee

Janos BaracsDirecteur, representant de son ecoleDirectol; representative of his schoolEcole darchitecture, Universite de MontrealWalter WhiteleyRepresentant de son ecoleRepresentative of his schoolDepartment of Mathematics and Statistics,York University, North York, OntarioHenry Crap0Representant de son ecoleRepresentative of his schoolEcole darchitecture, Universite de Montreal;INDIA, Rocquencourt, FranceRichard PallascioRepresentant de son departementRepresentative of his schoolDepartement de mathematiques etdinformatique,Universite du Quebec a MontrealVincent PapillonRepresentant de 1Association mathematiquedu QuebecRepresentative of the Association mathkmatiquedu QuebecCollege Jean-de-Breb euf, Montreal, Q uebec

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Cornit consultatif

-Adiisoryboad

Laszlo Fejes-T&h, mathematicianProfessor, Hungarian Academy o f Sciences,Budapest, HungaryBranko Griinbaum, mathematicianProfessor, University of Washington,Seattle, Washington, U.S.A.Lionel March, architectDirector, Centre for ConfigurationalStudiesProfessor, The Open University, MiltonKeynes, EnglandCedric Marsh, engineerProfessor, Centre for Building Studies,Concordia University, Montreal, QuebecGert Sabidussi, mathematicienProfesseur, Universite de Montreal,Montreal, QuebecMoshe Safdie, architectDirector, Urban Design Program, GraduateSchool of Design, Harva rd University,Cambridge, Massachusetts, U.S.A.Geoffrey Shephard, mathematicianProfessor, University of East Anglia,Norwich, England

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R&dactionEditorship

Janos BaracsRedacteur en chef/editor in chiefEcole darchitecture, Universite deMontrealHenry Crap0%dacteur/editovIkole darchitecture, Universitk deMontreal ;INRIA, Rocquencourt, FranceHaresh Lalvani%dacteur/editovSchool of Architecture, Pratt Institute,Brooklyn, New York, USARichard PallascioRkdacteur/editorDepartement de mathematiques etdinformatique,Universite du Quebec a MontrealJean-Luc Raymond%dacteur/editorDepartement de mathematiques etdinformatique,Universite du Quebec a MontrealWalter HlhiteleyRkdacteur/editorDepartment of Mathematics and Statistics,York University, North York, Ontario

Jll.


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Topologie structurale l 21 l Structural Topology l 1995 1

Parotian deux uis par ann6eiPartet embaftageunt nck4 s ans e prixdabunnemenr.Abonnement nstWtionnel(2 num&os]Canada 60 $ Can Autrespays 60$ USAbonnementndividuet2 num6rus) :Canada 35 $ Cm - Autrm pays : 35 $ USPublicatim IWO imesa yeaf:Subscriptionees ndudepostageo anypartof theworld.Subscriptionor institutim (2 ssuesj):

Contents

Singled-sidedMobius Modules- Michael YanoffSpatial geometriccompetencies development- P Mongeau, R. Pallascio and R. Allaire

Stellations of therhombic triacontahedronand beyond- Peter W. Messer

Families of Multi-directionalPeriodic Space Labyrinths- Haresh Lalvani

Globally RigidSymmetric Tensegrities- R. Connelly and M. Terrell

Sommaire

Module de Mobiusunilateres- Michael YanoffLe developpement des competencesspatiales geometriques- P. Mongeau, R. Pallascio and R. Allaire

Les etoilementsdu rhombitriacontaedreet plus- Peter W. Messer

->. .

Familles de labyrinthes spatiauxmultidirectionnels et periodiques- Haresh Lalvani

Tens&grit& symetriquesglobalement rigides- R. Connelly and M. Terre11


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Topologie structurale l 21 l Stmctural Topology l 1995

IForewordi The journal S~TUC~UYU~o@gy is published by the Structural Topology re-search group, with the collaboration ofthe Association Mathematique duQuebec and the Universite du Quebec aMontreal. The res earch group is aninterdisciplinary team bringing togethermathematicians, engineers, architects,designers, and artists.The particular field of interest of theJournal is the application of classicaland contemporary mathematics (espe -cially geometry) to the solution of mor-phological a nd structural problems aris -

ing in architecture and design. The prin-cipal themes of this research are ques-tions relevant to construction, namely:polyhedra, juxtaposition, and the ri-gidity.The aim of the Journal:l to gather together and promote inter-change between researchers who areinterested in the problems of struc-tural topology, at a theoretical or prac-tical level;l to publish recent results, recent appli-cations and unsolved problems inthese areas ;l to encourage interdisciplinary com-munication, thereby making the re-sults available to a wide audienc e;

l to describe teaching projects an dmaterials which illustrate thosethemes and which use those results.With these objectives, the Journalcontains:

l longer articles describing recenttheoretical advances, current projectsand applications in the field ofstruc-tural topology;l brief reports on

La revue lbpologie struc turale est publieepar le groupe de recherche ((?bpologiestructurale H,avec la collaboration de PAS-sociation ma thematique du Quebec et de1Universite du Quebec a Montreal. Legroupe de recherche est une equipemultidisciplinaire composee a la fois demathematiciens, dingenieurs, darchitec-tes, de designers et dartistes.Le principal champ dinteret de la re-ecent work, includ-ing unsolved pro blems and reviews of / vue est lapplication des mathematiques

Avant-propos

work r elevant to the themes of struc-tural topology;l expository articles which describe

methods and results in such a waythat students and practitioners can ap-ply them;l popular articles which translate theresults into a visual form ac cessible toa broad audience not familiar withtechnical mathematics.

We invite our readers to send theircomments on the Journal and to submitarticles for publication in forthcomingissues. The editors will make every e f-fort to maintain a balance between theo-retical, applied, and expository articles,and to develop contacts betw een the di-verse groups who sha re an interest inthe field of structural topology..lll.

classiques et contemporaines (speciale-ment de la geometric) a la solution deproblemes morpholog iques et structu-raux qui se posent en architecture et endesign. Ses principaux themes ont trait alarchitecture et sont: les poly&dres, lajuxtaposition et la rigiditk.La revue vise al rassembler et mettre en communica-tion les chercheur s interesses aux pro-blemes de topologie structurale, a unniveau theorique et pratique.l publier les resultats recents, les appli-cations recentes et les problemes nonresolus dans ces domaines.l encourager la communication interdis-ciplinaire, par consequent mettre lesresultats a la disposition dun large pu-blic.l d&-ire les projets et le materieldenseignement qui illustrent ces the-

mes et se servent de ces resultats.Avec ces objectifs, la revue com porte :l des articles longs decrivant des pro-gres theoriques recents, des projets entours et des applications dans le do-maine de la topologie structurale.l de beefs rapports sur des travauxrecents, y compris des problemes nonresolus et des comptes rendus de tra-vaux lies aux themes de la topologiestructurale.l des articles dintmduction decrivantles methodes et les resultats de faGontelle que les etudiants et les gens du

metier p uissent les appliquer.l des articles d5nt&& g&&al quitraduisent les resultats en une formevisuelle accessible a un large publicpeu familier avec les mathematiquestechniques.Nos lecteurs sont invites a nous en-voyer leurs commentaires sur la revueet a nous soumettre des articles pour pu-blication.La redaction sefforcera de maintenir unequilibre entre les articles theoriques ,appliques et dintroduction, et de deve-lopper des contacts entre les divers grou-pes qui partagent un meme inter& pourla topologie structurale.

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Topologie structurale l 21 l Structural Topology l 1995

Michae I VanoffInformational Graphics ConsultantP.O. Box 25452Chicago, Illinois 60625USA

French translation:Traduction franpise :Jean-Luc Raymond

Singled-sidedMiibius Modules

Abstmct

Te mobius strip is simple to construct and yet embodies

a sophisticated mathematical concept which forces usto reconsider the way we think about surfaces because

the mobius strip has no distinct front and back. It has only asingle side. Like much in mathematics the ideas relating to themobius strip can be extended. The Klein bottle is one exampleof such an extension concerning the fourth dimension.

This article presents another example which is quite dif-ferent from the Klein bottle. Here modular systems are introduced in which e ach module like the mobius strip is single-sided. Yet, as one would expect from modular systems, thesecan all be extended to span any given volume of three di-mensional space. The individual modules are constructedfrom simple square tiles.

While these systems are intriguing in themselves, theycould possrbly present challenging ideas for mathematicalinquiry or offer innovative structures for technological usesin fields like computer design and chemical engineering. ~1.

Module de M6biusunilathes

R&urn6

Le ruban de Mobius est simple a construire; pourtant ilconcretise un concept mathematique sophistique quinous oblige a reconsiderer notre conception des sur-

faces, car le ruban de Mobius na ni devant ni derriere. 11 epossede quun seul cot& Com me beaucoup de concepts enmathematique s, les idles re levant du ruban de Mobiuspeuvent etre etendues. La bouteille de Klein es t un exempledune telle extension dans la quatrieme dimension.

Cet article pres ente un autre ex emple tout a fait diRerentde labouteille de Klein. On presente ici des systemes modu-laires dans lesquels chaque module ne possede, comme leruban de Mobius, quun s eul c&e. Po urtant, comme on pour-rait sattendre de systemes modulaires, ceux-ci peuvent tousetre deploy& pour englober nimporte quel volume de les-pace tridimensionnel. Les modules individuels sont cons-truits a partir de simples tuiles c ar&es.

Quoiquintrigants en eux-meme s, ces systemes peuventeventuellement se presenter comme elements stimulants de


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Singled-sidedMiibius Modules Module deunilathes

Traditional realization of R6alisation raditionnellemijbius strip. dun ruban de Mobius.la Flat band of paper la Bande de papierfor making mgbius strip. plane pour construire unlb Assembled miibius ruban de Mobius.Strip. 1b Ruban de Mijbiusassemble.

Ruban de Mobius danslequel la torsion estconfin6e SIune seuletuile.

IntroductionThe mobius strip challenges the way we normally conceiveof surfaces. It has but a single side bounded by a single edge.The consequences of single-sideness can be appreciatedthrough a n example pertaining to computer electronics. If acircuit boa rd were a mobius strip then any two of its elec-tronic components regardless of location could be connectedby a circuit w ithout that circuit leaving the boards surface,without ever feeding it through a hole and without wrappingit around the boards edge. This wo uld be poss ible because allelectronic components would be on the same side of thecircuit board.

This article introduces a significant departure from theusual conception of single-sideness. Typical realizations ofmobius s trips depict them as isolated models, whereas thestructures which will be described here are composed ofmodules w hich can be connected together to span any vol-ume of three dimensional space in a periodic m anner Yet,each module individua lly has the property of single-sideness.The manifold repetition of this single-sidedness insures arelatively short route to be found from any location on thesurface of such a structure to the corresponding location onwhat would be normally considered the opposite side, with-out having to go around an edge.Construction constraintsIn the traditional realization of the mobius strip, a narrowband of paper is twisted 180 about its longer dimension be-fore its two shorter edges are joined. A first band and an as-sembled mobius strip are both illustrated in Figu~ 1. As canbe seen in the illustration, the result of this type of twisting isa curvilinear surface which doe s not facilitates the creation ofsystematic intermodular connections.

The first step towards modular construction is to establisha basic building unit which, here, will be a square tile. We canthen m ake a tiled mobius st rip by constructing a square bandfrom several square tiles having one of them twisted 180before it is assembled with the rest which all remain flat.

la recherche m athematique ou ofkir des idees de structuresinnovatrices pour des utilisations technologiques dans deschamps comme larchitecture des ordinateurs et le geniechimique. .111.lntmductionLe ruban de Mobius defie notre faGon habituelle de concevoirles surfaces. I1 ne possede quun seul tote borne par uneseule a&e. On peut apprecier les consequences de lunilate-ralite a laide dun exemple se rapportant a lelectronique delordinateur. Si une carte de circuits etait un ruban de Mobiusalors nimporte quelle paire de ses composants electroniques,peu importe leu r emplacement, pourrait etre connectee parun circuit sans que ce circuit ne quitte la surface de la carte,sans quon nait a le faire passer par un trou et sans quil necroise larete de la carte. Cela serait possible car toutes lescomposantes electro niques se situeraient sur le meme c&ede la carte de circuits.

Cet article presente une nouvelle orientation par rapport ala conception habituelle de lunilateralite. Les realisationstypiques de rubans de Mobius s e depeignent comme desmodeles isoles, tandis que les structures qui seront d&ritesici sont composees de modules qui peuvent etre lies entreeux pour englober nimporte quel volume de lespace tridi-mensionnel de faGon periodique. Cependant, chaque modulepossede individuellement la propriete dunilateralite. La re-petition multiple de cette unilateralite assure quon puissetrouver un chemin relativement court entre un quelconqueemplacement sur la surface dune te lle structure a lemplace-ment correspo ndant sur ce quon considererait normalementcomme le ((tote oppose)+ sans avoir a franchir une a &e.Contraintes de constructionDans la realisation traditionnelle du ruban de Mobius, onimprime a une etroite bande de papier une rotation de 180selon son axe le plus grand avant de joindre ses deux pluspetites a&es. Lillustration de la figue 1 presente une bandeplane et un ruban de Mobius assemble. Comme on peut lob-server dans lillustration, le resultat de ce type de torsion estune surface curvilineaire qui ne simplifie pas la creation deconnections intermodulaires sy stematiques.La premiere &ape vers une construction modulaire estletablissement dune unite de base de construction qui, ici,sera une tuile car&e. On peut alors faire un ruban de Mobius


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UP / HAUT

W

N

+ SDOWN I BAS

E

Figure 2 shows the result of such a tiling which all mobiusSome results from Certains r6sultats de 1 strips is single-sided.applying the 5 con- Iapplication des cinqstruction constraints. contraintes de construc-

The square-band version of the mobius strip is useful in3a A tile may assume tion. helping us understand how tiles can be used in making sin-only one of three orien- 3a Une tuile ne posskde gle-sided structures, bu t it would be beneficial to eliminatetat&s. Tile edges may quune seule des troisonly be co incidental or orientations. Les a&esmeet orthogonally, if des tuiles ne peuventthey intersect3b Corresponding tilesin consecutive parallellayers align into squarecolumns. Dash linesindicate one suchcolumn. Cwrespondingsides of correspondingtiles align like rungs ofa ladder.

que cdincider ou serencontrer de faconorthogonale si elles ontune intersection.3b Les tuiles corres-pondantes dans descouches par-alleles on-secutives salignent encolonnes car&es. Leslignes pointillees indi-quent une telle colonne.les c&es conespondantde tuiles correspondan-tes salignent comme lesechelons dune echelle.

twisted element, firstlyment of a modular system , n order to further aid develop-and, secondly, for the practicalreason that it is easier to fabricate mo dels by keeping all basicbuilding units flat. In fact, beyond eliminating twisted tiles,inter-tile angles of connection can be limited to multiples of90 and this restriction will simplify the task of conceivingand creating modular components. For these reasons thefollowing five construction constraints are employed:1) Basic building units are flat square tiles.2) Only one tile may be connected to any one edge of an-other tile, but it is permissible to leave a tile edg e uncon-

nected (or vacant).3) The angular measurement between two adjacent faces of

directly connected unit tiles can only be 90) 180 or 270.The angle measured betw een two adjacen t faces of di-rectly conn ected is a dihed%l angle.

4) Tile edges of basic building units may only meet in one ofthe same three above mentioned angles or else they mustbe coincidental.

5) If two tiles are positioned parallel (parallel is taken in theusual Euclidean sense) to each other and if it possible toconnect them with a line perpendicular to both, then thetiles must align in a column whose perpendicular crosssection is exactly the square shape of a single tile.


carrele par la construction dune bande carree a partir d eplusieurs tuiles car&es dont lune a subi une torsion de 180avant detre assemblee aux autres tuiles qui, elles, demeurentplanes. On montre, a la figure 2, le resultat dun tel carrelagequi, comme tous les rubans de Mobius, est unilatere.

La version carrelee du ruban de Mobius est utile pournous aider a comprendre comment les tuiles peuvent etreutilisees pour faire des structures unilateres, mais il pourraitetre avantageux deliminer lelement tordu, premierement,pour favoriser le developpement dun systeme modulaire et,deuxiememen t, pour la raison technique de simplification dela fabrication de modeles e n conservant planes toutes lesunites de construction de base. En fait, audela de lelimina-tion de s tuiles tordues, langle de jonction entre les tuilespeut etre limite a des multiples de 90 et cette restrictionsimplifiera la &he de conception et de creation de compo-santes modulaires. Cest pour ces raisons quon emploiera lescinq contraintes de construction suivantes :1) Les unites de construction de base sont des tuiles carrees

planes.2) Une seule tuile peut etre jointe a une quelconque a&e

dune autre tuile, mais il est permis de laisser une arete detuile sans onction (ou vacante).

3) La mesure angulaire entre deux faces adjacentes de tuilesdirectement jointes ne peut etre que 90) 180 ou 270.Langle mesure entre deux faces adjacentes de tuiles direc-tement jointes est un an@e dihhe.

4) Les aretes dune tuile dune unite de construction de

5

base ne peuvent se rencontrer que selon lun ou lautredes angles mentionnes plus haut ou sinon elles doiventcoincider.

) Si deux tuiles sont placees de facon parallele (paxallele estconsidere selon le sens euclidien habituel) lune a lautreet sil est possrble de les joindre par une droite perpendicu-laire aux deux, les tuiles doivent alors etre alignees en unecolonne dont la coupe transversale perpen diculaire a exac-tement la forme carree dune simple tuile.La figure 3a illustre les restrictions sur lorientation.

On les met ici en rapport avec des termes geographiques : ilexiste une orientation horizontale qui fait en sorte que la facedune tuile peut etre orientee vers le haut o u vers le bas, etdeux orientations verticales qui fon t en sorte que la facedune tuile est orientee selon les directions nord/sud ouest/ouest.

7

Singled-sided Module de Miibius


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M6bius Modules unilatbres

Figulle 3a illustrates the restrictions on orientation. Hereit is related to geographic terms: there is one horizontal orien-tation w hich results in a tiles face facing up or down, andtwo vertical orientations which result in a tiles faces facing inone of either pair of opposing directions of north/south oreast/west.

Connected t iles form a layer if the only dihedral angle ofinter-tile connection is 180. The tiles of a layer can at thesame time be part of a larger collection of connected tilesthat includes more than the layer as well a s other layers.

?h;roparallel layers are said to be consecutive if they areseparated by a perpendicular distance the length of one tileedge (see Figum 3b).

If a mutually perpendicular line meets the interior of twoparallel layers we say that the layers overlap. When oneviews a series of overlapping parallel layers he sees tilesneatly aligned into square columns. In each of any two paral-lel layers, two tiles that so align are said to be coxxespond-ing. J ust as corresponding tiles neatly a lign, so do their edgesbut like the rungs of a simple s traight ladder. Wo edges eachfrom one of two corresponding tiles which align are referredto as corresponding edges.

Finally, two faces each from one of two tiles in the samelayer or parallel layers have like orientation if they face inthe same direction while two faces each from one of two tilesin the same layer or parallel layers have opposing orienta-tions if they face in opposite directions ; the two faces of thesame tile have opposing orientations. In a similar fashion theterms opposing orientations and like orientations also applyto multiple-tile sections and entire surfaces of parallel layers.Although completed tiles structures with which we aredealing have only one face, we can still unambiguously referto two opposite sides relative to an individual tile or layerby considering it independently of the complete structurefrom which it comes.System for constructing a single-sided tile structureHaving established construction restraints, we can now proteed to make a single-sided surface from flat square tilesalone. To do this, the required angular rotation associatedwith a twisted element will b e achieved by distributing therotation among some of the inter-tile connections. Routesbetween any two locations on these single-sided surfaces cantake us from tile to tile without travelling all the way around

Des tuiles jointes forment une couche si le seul diedre dejonction inter-tuile est 180. Les tuiles dune couche peuventen meme temps app artenir a un ensemble plus vaste detuiles jointes qui inclut la couche en question aussi bien quedautres couches.Deux couches paralleles sont dites conskutives si ellessont separees par une distance perpendicu laire egale a lalongueur de Par&e dune tuile (voir figure 3b).

Si une droite rencontre linterieur de deux couches paral-leles tout en leur etant mutuellement perpendiculaire, ondira que les couches se chevauchent. Lorsquon observe uneserie de couches paralleles se chevauchant, on voit des tuilesproprement alignees en colonnes car&es. Dans chaque pairede couches paralleles, deux tuiles ainsi alignees sont ditescorrespondantes. Comme les tuiles correspondantes sali-gnent propreme nt, il en est ainsi de leurs a&es mais a lafaGon des echelons dune simple echelle droite. On appe lleraa&es cortespondantes deux a&es salignant appartenanta deux diffierentes tuiles correspondantes.Enfin, deux faces appartenant a deux tuiles distinctesdans la meme couche ou dans des couches paralleles ontm&ne orientation si elles sont orientees selon la memedirection. Deux faces appartenant a deux tuiles distinctesdans la meme couche ou dans des couches paralleles ont desorientations opposkes si elles sont orientees selon des di-rections opposees ; les deux faces dune meme tuile ont desorientations opposees. Dune maniere similaire, les terrnes((orientations opposees ))et ((meme orientation ))sappliquentegalement a des sections de multiples tuiles e t aux surfacesentieres de couches paralleles.Quoique les structures completes de tuiles dont on traitene possedent quune seule face, on peut encore faire refe-rence sans ambiguite aux ((deux c&es opposes )) elativementa une tuile ou une couche particuliere en la considerant in-dependamment de la structure complete dont elle est issue.Systhme de constructiondune st~cture unilathe de tuilesApres avoir etabli les restrictions de construction, on peutmaintenant proceder a la fabrication dune surface unilatereen nutilisant que des tuiles carrees planes. Pour ce faire, oncreera la rotation angulaire requise as sociee a un elementtordu en distribuant la rotation parmi certaines des jonc-tions inter-tuiles. Les chemins entre deux emplacemen ts



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Figure 4Creating linkage be-tween faces of con-secutive pamllel layers.Dash lines representdistances covered onsurface facing awayfrom viewer.4a Linking faces ofopposing orientation.4b Linking faces of likeorientation.4c Establishing single-sidedness by linkingfaces of both opposingand Iike orientations.

Cn%ion de liens entreles faces de couchesparalleles consecutives.Les lignes pointilleesrepr-esentent es distan-ces parcourues sur lessurfaces qui ne sont pasa la vue de Iobservateur.4a Liens entre des facesdorientations opposees.4b Liens entre des facesde meme orientation.4c itablissement deIunilateralite par lajonction de faces dorien-tations opposees et dememe orientation.

the edge of a single tile. The specific procedure for achievingthis involves utilization of two types of linkage between con-secutive layers of tiles.

One type of linkage cre ates connections between faces oflike orientation and the other type of linkage creates connec-tions between faces of opposing orientation. Methods fo rcreating each type of linkage are outlined below. It should bekept in mind that these are but two of a number of possibili-ties. These two methods, involving only consecu tive layers(as represented in figure 3b), were chosen because they arestraightforward.

?I3 ink two faces of two cons ecutive parallel layers a nd ofopposing orientation, one must first either create or locate asituation where eac h of two corresponding tiles (one in eachof the consecutive parallel layers) have correspondingvacant edges (edges not connec ted to another tile). Such anedge can either be on the outer perimeter of a layer or bor-dering a hole in the layers interior. Since the consecutivelayers are separated by a distance of only one tile ed ge, asingle tile can be placed between and connected to each ofthe two corresponding vacant edges. This newly introducedtile then serves as a bridge for a route from either face of oneof the consecutive layers to the opposite face of the otherconsecutive layer. In Figure 4a this type of link is depictedas well as a route from location P on the bottom of the lowerlayer to location Qon the top of the upper layer. Note that inall of Figu~ 4, tiles serving as inks are represented asoblongs for illustrative purposes only. While oblong tileswould function in such a situation, square tiles are preferable

quelconques sur ces surfaces unilatkres peuvent se faire detuile en tuile s ans amais croiser larete dune simple tuile.La procedure specifique am enant c ette realisation impliquelutilisation de deux types de liens entre les couches conse-cutives de tuiles.

Un premier type de lien tree d es onctions entre des fa-ces de meme orientation et lautre typ e de lien tree des onc-tions entre des faces dorientations opposees. On d&it ci-dessous les methodes respectives pour realiser chaqu e typede lien. On ne doit pas oublier quil ne sagit que de deuxpossrbilites parmi un grand nombre. On a choisi ces deuxmethodes, nimpliquant que des couches consecutives (tellesque representees a la figure 3b), car elles sont directes.

Pour lier deux face s appartenant a deux couches parallelesconsecutives d orientations opposees, on doit dabord creer oureperer un emplacement ou chacune de deux tuiles corres-pondantes (une dans chacune des couches paralleles conse-cutives) possedent des a&es correspondante s vacantly (desa&es non jointes a une autre tuile). Une telle tuile peut soitse situer sur le perimetre exterieur dune couche ou etre a lafrontiere dun trou a linterieur dune couche. Puisque lescouches consecutives sont separees par une distance equiva-lente a une a&e de tuile, une seule tuile peut etre placeeentre les deux couches et se oindre a chacune des deux are-tes correspondante s vacan tes. Cette nouvelle tu ile sert alorsde pont pour un chemin qui irait de nimporte quelle face delune des couches consecutives vers la face opposee de lautrecouche consecutive. La figu~ 4a illustre ce type de lien dememe quun chemin du lieu P sur le dessous de la couche

Location Q Location Q Location QOn top On top On topPosition Q Position Q Position QAu dessus 1 Au dessus 1 Au dessus

4aLocation POn bottomPosition PEn dessous 4b

Location ROn topPosition RAu dessus

Location R ---I Location POn top On bottomPosition R Position PAu dessus En dessous

9

Singled-sided Module de Miibius


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MGbius Modules unilatitres

i-10

Figure 5~~~ -~ __~Example of single-sidedmobius module.5a Exploded view of asingle module.All tile edges are shown.5b Isometric view of asingle module.All visible tile edges areshown.5c Exploded view of4 connected modules.Only perimetric tileedges are shown.5d isometric view of4 connected modules.Only perimetric tileedges are shown.

Exemple de module deMobius unilatere.5a Vue eclatee dunmodule simp le. Toutesles a&es de tuiles sontdessinees.5b Vue sometrique dunmodule simp le. Toutesles at-&es visibles sontdessinees.5c Vue eclatee de lajonction de quatre mo-dules. Seules les a&esdes tuiles perimetriquessont dessinees.5d Vue isometrique dela onction de quatremodules. Seules lesaretes des tuiles perime-triques sont dessinees.

in order to maintain uniformity and structural simplicity.?ro ink two faces of like orientation of two consecutive

parallel lay ers a method similar to linking opposing faces isused but the configuration of tiles is not quite as simple. At-tention must first be directed towards two adjacent tile posi-tions, A and B in one of the consecutive layers and the corre-sponding adjacent tile positions A and B respectively in theother consec utive layer. In the first layer position A must befilled with a tile while position B is empty. In the secondlayer the reverse is true with tile position A empty and posi-tion B occupied. Linkage as previously described is accom-plished with a single tile which in this instance is connectedboth to the vacant edge of A which borders the empty tileposition B in the first layer and to the vacant edge of B whichborders empty tile position of A in the second layer. Therendering of such a link is offered in Figure 4b. In order tobetter describe the nature of this type of linkage, a route be -tween position Qon the top of the upper layer and position Ron the top to the bottom layer is shown.

Tb obtain a single-sided tile configuration, both types oflinkages are installed between two overlapping consecutiveparallel tile layers. Thus each face of each layer is linked to

inferieure vers le lieu Qsur le dessus de la couche sup&rieure. Notons q ue dans toutes les illustrations de la figurr= 4,les tuiles servant de liens ne sont representees comme desrectangles que pour des raisons graphiques . Meme si destuiles rectangulaires auraient pu jouer le meme role dans unetelle situation, il est preferable dutiliser des tuiles carrees afinde maintenir luniformite et la simplicite structurale.

Pour lier deux faces de meme orientation appartenant adeux couches paralleles consecutives, on utilise uric me-thode se mblable a celle utilisee pour lier des faces dorienta-tions opposees mais la configuration des tuiles nest pas touta fait aussi simple. On portera tout dabord notre attentionsur deux positions de tuiles adjacentes, A et B, dans lune descouches consecutives et sur les positions des tuiles adjaccn-tes correspon dantes, A et B, appartenant 2 lautre coucheconsecutive. Dans la premiere couche, la position A doit etrecomblee par une tuile tandis que la position B est vide. Dansla seconde couche, linverse est vrai, cest-a-dire que la posi-tion A est vide et la position B est occupee. Le lien decritprecedemment se realise a laide dune simple tuile qui, dansce cas, est jointe a la fois a larete vacante de A qui borde laposition vide de tuile B dans la premiere couche, et a larete

~ 5a 5b 5d



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Figure 6Example of single-sidedmijbius module.6a Exploded view ofsingle module. All tileedges are shown.6b Three modulesshowing the only 3allowable orientationsfor this modular system.All visible tile edges areshown.6c An assortment of 6interconnected modules.Only perimetric tileedges are shown.

Exemple de module deMijbius unilatere.6a Vue &latee dunmodule simp le. Toutesles at-&es des tuiles sontdessinees.6b Trois modules mon-trant les seules troisorientations admissiblespour ce systeme modu-laire. Toutes es at-&esvisibles des tuiles sontdessinees.6c Un assortiment desix modules joints.Seules les a&es destuiles p&imWiques sontdessinees.

both the face of like orientation and the face of opposing ori-entation in the other layer which yields one continuous tiledsurface that includes b oth faces of both lay ers as well as allthe faces of the tiles involved in creating th e linkage. This isillustrated in Figure 4c. For further clarification, a route isindicated from position P on the bottom of the lower layer toposition Qon the top of the upper layer to the lower layersposition R on the top face of the tile from whence thisjour-ney commenced. This demonstrates single-sidedness.

nects faces of like orienta tion and the other connects faces ofopposing orientation. All tiles both within the stacked layersand the ones serving as inks must be assembled according tothe five construction constraints given earlier.

Single-sided miibius modulesThe design of a single-sided miihius module is a matter ofarranging two or more consecutive parallel layers of tiles intoa stack and then linking all the layers accord ing to the twomethods outlined above. Moreover, at least two consecu tivelayers must be connected by two links where one link con-

vacante de B$ qui borde la position vide de tuile A@ dans laseconde couche. Le rksultat dun tel lien est illustri: 5 lafigure 4b. Afin de mieux decrire la nature de ce type de lien,on montre un chemin allant de la position Qsur le dessus dela couche sup&ieure & a position R sur le dessus de la cou-the infkrieure.Pour obtenir une configuration unilatere de tuiles, on doitinstaller les deux types de liens entre de ux couches de tuilesparalleles con&cutives se chevauchant . Ainsi, chaque face dechaque couche est li6e 5 la fois & a face de m6me orientationet 2 la face dorientation opposee dans lautre co uche, ce quidonne lieu 2 une surface continue de tuiles incluant les deuxfaces des deux couches de meme que toutes les faces destuiles impliquees dans la cr6ation du lien. Ceci est illustre 2la figu~ 4c. Pour plus de clarte, on y indique un cheminallant de la position P sur le dessous de la couche inferieure 2la position Qsur le dessus de la couche sup&ieure, puis versla position R de la couche infi5rieure sur le dessus de la tuilecorrespondan t 2 lorigine de ce chemin. Ceci dkmontrelunilateralit6.Tb this point, these guidelines can apply to any single-sided tiled structure wh ich follow the five construction re-

6a 6b 6c

11 II

S ingled-sidedM6bius Modules Module de Miibiusunilathes


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straints. What distinguishes modules is that, in addition tofollowing the construction constraints within the frameworkof each individual module, they m ust also follow the sameconstraints among tiles which form intermodular connec-tions. And, this added requirement for which I have no gen-eral method, creates a situation which is tantamount to solv-ing a three-dimensional jigsaw puzzle. The main problem isthat of inadvertently joining more that two tiles at a commonedge and thus create a set ofbranching surfaces instead of asingle complex surface with m ultiple holes. While no generalrule is given, two examples are: Figuxm 5 and 6 illustratetwo systems of single-sided mobius modules.

Figum 5 shows a modular system in which the modulesall have the same orientation. 5a and 5b depict intramodularassembly as four consecutive ho rizontal layers linked by ver-tical tiles of both north/south and east/west orientations.5c and 5d convey how ent ire modules are connected to oneanother. This particular pattern of intermodular connectionsturns out to have the same configuration as closely-packedsame-sized cubes having no face of one cube overlap an edgeof another cube; cubes which touch face to face align pre-cisely.

The result of the system in F&u= 5 is a series of expan-sive horizontal layers regularly disrupted by holes and byvertical linking tiles. Of course the whole configura tion couldbe rotated to place the expansive layers into a vertical posi-tion. In any case, we can say that this type of structure isbiased towards one set of parallel planes, those which formthe expansive layers. The second example of a single-sidedmobius module shows no such bias.Figum 6 depicts this second example. As with the exam-ple in Figuxe 5, intramodular assembly can be accomplishedby stacking horizontal layers of tiles and connecting themwith appropriate linking tiles; this process is show in 6a. Withrespect to relative intramodular assembly, all modules areidentical, however, in addition to linking horizontally stackedlayers in sets of three, we also link layers in sets of three withnorth/south orientation an d with east/west orientation. Allthree orientations are represented in 6b, all three must beutilized and all three must be rigidly maintained or the resultwill violate the five construction constraints.These modules connect to each other to form a three-dimensional checkerboard pattern where modules alternatewith empty cube-shaped spaces. An assortment of inter-

Modules de MGbius unilathesLe design dun module de Mobius un ilatere est une questiondarrangement de deux couches paralleles consecutive s ouplus en une pile, et de liens entre ces couches s elon les deuxmethodes d&rites precedemment. De plus, au moins deuxcouches consecutives doivent etre jointes par deux liens ouun des liens joint des faces de meme orientation et lautrejoint des faces dorientations opposees. mutes les tuiles, aussibien celles appartenant aux couches empilees que cellesjouant le role de liens, doivent etre assemblees en respectantles cinq contraintes de construction donnees plus haut.

A ce point, ces lignes directrices peuvent sappliquer animporte quelle structure unilatere de tuiles respectant lescinq contraintes de construction. Ce qui distingue les modu-les, cest quen plus de respecter les contraintes de construc-tion a linterieur des structures de chaque module individuel,ils doivent aussi respecter les memes contraintes pour lestuiles qui forment les jonctions intermodulaires. Et cette exi-gence supplementaire, pour laquelle je nai pas de methodeg&&ale, tree une situation qui est equivalente a la resolu-tion des casse-tete tridimensionnels. Le principal problemeest celui de joindre par inadvertance plus de tuiles en unea&e commune et de creer ainsi un ensemble ramifie desurfaces plutot quune seule surface complexe avec plusieurstrous. Bien quon ne donne pas de regle g&r&ale, les figures5 et 6 sont des exemples qui illustrent deux sy stemes de mo-dules de Mobius unilateres.

La figure 5 montre un systeme modulaire dans lequel lesmodules ont tous la meme orientation. 5a et 5b montrentlassociation intramodulaire composee de quatre coucheshorizontales consecutives liees par des tuiles verticalesdorientations nord/sud de meme quest/ouest. 5c et 5dillustrent la jonction entre des modules entiers. 11 avere quece motif particulier de jonctions intermodulaires possede lameme configuration quun ensemble etroitement tasse decubes de meme taille ou aucune face dun cube ne chevau-the une arete dun autre cube ; les cubes qui s e touchent facea face salignent de facon precise.

Le resultat du systeme de la figure 5 est une serie decouches horizontales expan sibles regulierement interrom-pues par des trous et par des tuiles verticales de liens.Lensemble de la configuration peut evidemment subir unerotation afin de placer les couches expansibles dans une posi-tion verticale. Dans tous les cas, on peut dire qu e ce type de



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modular connection are shown in 6c and F&urn 7 is a pho-tograph of a constructed model.

Together, the two examples given he re suggest that thereare many possible single-sided miibius modules even withinthe limitations of the five construction constraints. with therelaxation of some of the restrictions on tile shape and inter-tile connecting angles, the selection should be larger.Concluding remarksThe idea of the single-sided mijbius modules which offers asurface ro ute between any two points located on it is intrigu-ing in itselfboth as a theoretical concept and as a physicalmodel. It could be the source of some interesting mathemati-cal investigation such as what are the properties of the limitsurface as the number of connected modules spanning agiven finite volume becomes arbitrarily large and densewhile the size the modules tends towards zero? Or, can thereexist a system of modules that ha s a single edge as well as asingle side and if so what does it look like?

In the area of applications, as mentioned earlier, single-sided mijbius modules could serve as circuit bo ards for com-puters in order to enhance the configuration of electronic

Figure 7_____--hotograph of 12connected single-sided

mijbius modules.Photographie de lajonction de 12 modulesde Mobius uniIat&es.

structure est influcncee par lensemble des plans parallC:lesque constituent les couches exp ansibles. Le second exemplede module de Mijbius unilatere ne montre pas die telle in-fluence.

La figure 6 illustre ce second exemple. Comme danslexemple de la figure 5, lassociation intramodulaire peut serealiser e n empilant des couches horizontales de tuiles et enles joignant par des tuiles d e liens appropri@es; la figure 6amontre ce processus. Par rapport 2 lassociation intramodu-laire relative, tous les modules sent identiques; toutefois, enplus de lier horizontalement les couches empilCcs par en-sembles de trois, on lie aussi les couches par ensembles detrois selon les orientations nord/sL4d et est/ouest. La figure6b illustre les trois orientations; toutes les trois doivent etreutiliskes et doivent etre rigidement maintenues sans quoi leresultat violera les cinq contraintes de construction.

Ces modules sejoignent les uns aux autres pour formerun motif dechiquier tridimensionnel oti les modules alter-nent avec des espaces cubiques vides. On trouve 2 la figure6c un assortiment de joints intermodulaires. La figure 7 estLune photographic dun modele construit.

Ensemble, les deux exemples p resent& ici suggerent quily a plusieurs modules de Miibius unilateres possibles mGme

b3

2 lint&ieur des limites des cinqCl* 1 1 / * 1 contraintes de construction.1 * 8. 1 r 131 on iaisse tomDer certalnes ues restrictions sur la rorme aes 1tuiles et les angles de jonction inter-tuile, les possibilit& de-vraient etre encore lnlus nombreuses. IRemaques de conclusion ILidee des modules de Miibius unilateres offrant un chemin I~sur leur surface entre nimporte quelle p aire de I3oints situ&sur eux est fascinante en soi 2 la fois comme concept thtJori-que et comme modele physique. Ce pourrait &t-rc la sourcede certaines recherches math@ matiques interessantes tellesles questions suivantes: quelles sont les propri&s de la sur-face limite lorsque le nombre de modules joints r:nglobant WIvolume fini donne devient arbitrairement grand et densealors que la taille des modules tend vers z&o? 0~4, existe-t-i1un systkme de modules possedant une seule a&e et un seulcbti: et si oui, & quoi ressemblerait-il?

Dans le champs des applications, comme on 1 a men-tionne plus haut, les modules de Miihius unilati:rc,s pour-raicnt servir d e cartes de circuits pour les ordinatcurs cla~ls lebut dameliorer la configl4ration des composants elcctroni-

Singled -sidedMiibius Modules Module deunilathes


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components; specifically the modules could reduce the needfor drilling holes for electrical feed through since all compo-nents are on the same side of the board. In chemical engi-neering, single-sided mobius could serve as sites upon whichchemical reactions could take place.

But regardless of viewpoint, theoretical or practical, single-sided mobius modules offer us a fresh perspective on ex-pendable modular surface systems which can extend overvolumetric space. They allow us to readily think about suchsystems in a way that is more free from our usual conceptions of front v ersus back and top versus bottom in which wealmost instinctively favour front and top. The single-sidedmobius modules help us understand that any face of a sur-face is of use and worthy of consideration no matter howabstract or concrete the application involved may be.

ques; en particulier, les modules p ourraient reduire le besoinde perter des trous pour des connexions electriques ((passantau travers) puisque tous les composants sont sur le memec&5 de la carte. En genie chimique, les modules de Mobiusunilateres peuve nt servir com me lieux pou r les reactionschimiques.Mais quelque so it le point de vue, theorique ou pratique,les modules de Mobius unilateres n ous ofient une perspec-tive fraiche sur les systemes de surfaces modulaires expan-sablespouvant sete ndre audela dun es pace volumetrique.11s ous permettent daborder volontiers de tels systemesdans le sens ou on est plus libre par ra pport a nos concep-tions habituelles de devant/derriere et dessus/dessou s quinous portaient instinctivement a favoriser le devant et ledessus. Les modules de Mobius un ilateres nous amenent acomprendre que toute face dune surface est utilisable etdigne de consideration, peu importe que lapplication impli-quee soit abstraite ou concrete.

Topologie structurale l 21 l Structural Topology l 1 a95


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Spatial geometric Le d&eloppementcompetencies des comp&ences spatialesdevelopment gfZom&iques

Pierre MongeauUniversit6 du Quebec B RimouskiRichard PallascioRichard Allain?Universite du Quebec h MontrealC.I.R.A.D.E.C.P.8888, succursale AMont&a I (Qhbec)Canada H3C 3P8

English ranslation:Traduction anglaise :Hamut Kodjian

AbstractT e authors started with the hypothesis that spatialgeometric compe tencies development evolves, fromchildhood to the adult years, following the overlapping of geometric levels (topological, projective, affine, fol-lowed by metric). Since no geometrically complete and validresearch instrument has been identified, a new one had to bedeveloped. Grou ps of subjects at different ages-children (ll-12 years), a dolescents (15-l 6 years), young adults (18-24years), adults (30-40 years)-were tested to see how theyperformed at each geometric level. While tes t results do notconfirm the original hypothesis, they do show poor perform-antes related to projective representations. The authors at-tempt to explain these results in relation to the psychometric

~ factors of spatial relations and visualization.~ Introduction

For any representation of space and the objects within it to1 be accurate, it must progress according to the laws of geom-

R&wm&

Les auteurs ont emis lhypothese que le developpementdes competences spatiales geometiques se calque,de lenfance jusqua lage adulte, sur limbrication des

niveaux geometriques (top ologique, projectif affine puismetrique). Aucun instrument de recherche geometriquementcomplet et valide nayant ete recense, les auteurs en ont deve-loppe un no uveau. L es performances de groupes de sujets dedifferents ages, enfants (11-12 ans), adolescents (15-16 ans),jeunes adultes (18-24 ans), adultes (30-40 ans), ont ete testeesselon chacun des niveaux g eometriques. Les resultats ne per-mettent pas de confirmer lhypothese de depart. 11smettentcependant en evidence la farblesse des performances relativesaux representations projectives. Certaines explications sontavancees en lien avec les facteurs psychometriques de (( ela-tion ))spatiale et de (( isualisation ))spatiale.Introductionmute representation juste de lespace ou des objets quil con-

Spatial geometriccompetenciesdevelopmentLe dbeloppementdes comp&ences spatialesg45omWiques


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etry. Yet little is known about how spatial geometric compe-tencies development proceeds beyond childhood. The per-formance of groups of subjects at different ages has beenmeasured to better understan d how this development takesplace. A brief review of the major work done on spatial com-petencies is followed by the methodology used and the re-sults of the study.

Aside from the work done by Piaget, few studies havetried to measure how spatial competencies evolve, or exam-ined their geometric aspects. In fact, many re searchers(Cooper, Glaser, Kosslyn, Pellegrino, Shepard, Stemberg andothers: see [lOJ) seem more intere sted in better understand-ing the fundamental processes that underlie th e solving ofspatial competenc ies pro blems tha n they are in looking atthe intrinsic geometrical characteristics of the studied repre-sentations. The same holds true in psychometry , where astatistical analysis is made comparing results from a largenumber of subjects to specific items without any attentionbeing given to geometric characteristics. I nstead, an attemptis made to break results down into their principal compo-nents.

Thus, McGe e [8], Lohman [5,6], Eliot and Smith [2] andPellegrino, Alderton and Shute [lo] show, through exhaustiveand detailed analyses of the body of studies published in thefield, that the notion of spatial competencies as measured bythe psychometric tests generally used in research can bebroken down into two main factors: spatial relations andspatial visualization.

Spatial relations refers to the ability to define and under-stand the relationships that connect various elements to-gether into a spatial unity and to compare unities. The sub-ject must rec ognize a figure or shape seen from a differentpoint of view, for example.

Visualization refers to the ability to manipulate three-dimensional shapes mentally. Visualization is usually meas-ured using two-dimensional representations, or projections.Through a mental operation, the subject transforms one representation so that it correspon ds to another, to be chosenfrom a group of several representations. For example, a pro-jection which corresponds to a surface development is se-lected from a group of several projections.%st analysis (see [2]) shows there is no geometricallycomplete and valid instrument for measuring spatial compe-tencies, that is, no instrument which includes the geometric

tient selabore en accord avec les lois geometriques. Cepen-dant, le developpemen t des competences spatiales geome-triques d emeure peu connu a u dela de lenfance. Aussi, afinde mieux connaitre cette evolution, les performances degroupes de sujets dages differents ont ete mesurees. Suite aun bref rappel des principaux travaux portant sur les compe-tences spatiales, les resultats obtenus sont present& apreslexpose de la methodologie.Outre les travaux de Piaget, Tares sont les etudes qu icherchent a mesurer levolution des competences spa-tiales ou prennent en consideration leurs aspects geome-triques. En effet, plusieurs ch ercheurs (Cooper, Glaser,Kosslyn, Pellegrino, Shepard, Sternberg et dautres : voir[lo]) sefforcent de mieux comprendre les processus debase sous-jacents a la solution de problemes exigeant descompetence s spatiales mais aucun ne semble prendre enconsideration les qualites geometriques intrinseques desrepresentations etudiees. De meme le travail des psycho-metriciens, qui repose sur des analyses statistiques compa-rant les resultats obtenus par une masse de sujets a desitems donnes, ne prend aucunement en consideration cesqualites geometriques. 11s herchent plutot a identifier lescomposantes principales de ces resultats.

Ainsi, selon McGee [8], Lohman [5,6], Eliot et Smith [2 ],Pellegrino, Alderton et Shute [lo], qui ont effectue des ana-lyses a la fois exhaustives et detaillees de lensemble desetudes publiees en ce domaine, il ressort, en resume, quela notion de competence spatiale, telle que mesuree parles tests psychometriques generalement employ& lors desrecherches, est composee de deux facteurs principauxcommunem ent appeles (( elation spatiale )) et (( isualisa-tion spatiale N.Le generique de ((relation spatiale )) efere ala capacite detablir et de comprendre les relations quiunissent les divers elements dun ensemble (( patial Net decomparer des ensembles entre eux . Par exemple, le sujetdoit reconnaitre une figure ou une forme presentee dunpoint de vue different.

Le generique de ((visualisation)) refire, quant a lui, a lacapacite de manipuler mentalemen t des objets tridimension-nels. Cette competence est le plus souvent mesuree a partirde representations bidimensionn elles (projections). Le sujetdoit transformer mentalemen t une premiere representationde facon a choisir parrn i plusieurs autres representationscelle qui illustre le resultat dune operation. Par exemple, il



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Film 1Reprbentations dun polyhh(((prisme triangulaire 1,).Representations of a PolyhedronCtiangular prism).

levels (see next section) and which can be used to evaluatethe spatial geometric c ompetencies de velopment. ?bpologi-cal, projective and afKne aspects are almost always over-looked for the sake of metric aspects. Over 80% of the testsretained as suitable for geometric analysis (212/256) weremetric. Of the rest, 11.6% were affine, 2% were projectiveand only 4% were topological. There are even a few tests(2%) which contain blatant geometric errors [9].Since the aim of this study was to acquire a better un-derstanding of the spatial geometric competencies devel-opment, a new, geometrically valid research instrumenthad to be developed before groups of subjects at differentages could be tested.Geometric Repre sentations of SpaceThe geometric representations of space relie on axiomsgrouped into four geometric levels which overlap each other:the topological, projective, affine and metric levels. The hier-archical structure proceeds from the topological level to themetric level. Consequently, for a metric rep resentation, theproperties that characterize the topological, projective andaffine levels are necessarily retained, whereas a topologicalrepresentation does not display any projective, affine, or met-ric properties per se. In fact, a same family of polyhedrons(prisms, for example) can be depicted using four distinct andgeometrically exact representation types (see F&WE 1).

A metric representation also contains topological tiorma-tion since topological properties are previous in the hierar-chy. On the contrary, a topological repre sentation does notnecessarily contains the information relative to the projec-tive, affine and metric levels.

Principal topological characteristics include the number of

01

,/i ----------

/NH

/

AmTopolog que ProjectifTopological Projective AffineAffine MbriqueMetric

doit choisir parmi plusieurs projections celle qui corresponda une forme p resentee a laide dun developpemen t-plan.

Par ailleurs, lanalyse de la panoplie de tests utilises pourmesurer les competence s sp atiales (voir [2D permet de cons-tater que, dun point de vue geometrique, il nexiste aucuninstrument de mesure geometriqueme nt complet et valide.Aucun ne couvre lensemble des niveaux geometriques (voirla section suivante) et ne permet d evaluer le developpementdes competence s s patiales geometriques. Les aspects topolo-giques, projectifs et affines y sont presque toujours ignoresau profit des aspects metriques. Plus de 82 % (212/256) destests retenus, parce que pouvant etre analyses dun point vuegeometrique, sont de nature metrique, 11,6% sont affines,uniquement 2 % sont project& et seulement 4 % sont denature topologique. U n petit nombre (2 %) contient memedes erreurs geometriques flagrantes [9].

Lobjectif de cette etude &ant de mieux connaitre ledeveloppement des competence s spatiales geometriques,il a done ete necessaire delaborer un nouvel instrumentde recherche geometriqueme nt valide avant de tester lesperformances de groupes de sujets dages differents.Reprhsentations g6om hiques de IespaceLes representations geometriques de lespace sappuient surdes axiomes regroup& en quatre niveaux geometriques im-briques. I1 sagit des niveaux to pologique, projectif affine etmetrique. Limbrication se fait du topologique au metrique.Ainsi une representation metrique conserve necessairementtoutes les proprietes propres aux niveaux topologique, projec-tif et affine, tandis quune representation topologique ne rendcompte daucune propriete propremen t projective, affine oumetrique. En fait, une meme fam ille de polyedres, prisma-tiques par exemple, peut etre representee par quatre types derepresentations distinctes et geometriquement exactes (voirfigure 1).

Une representation metrique contient egalement les infor-mations topo logiques car les proprietes topologiques sonthierarchiquement prealables. Par contre, une representationtopologique ne contient pas necessairement les informationsrelatives aux niveaux projectif, affine et metrique.

Les principales caracteristiques topologiques sont lenombre de faces, de sommets et daretes et les proprietesdadjacence e t de connexite. La representation sous formede graphes conserve ces caracteristiques et proprietes et

17

Spatial geometriccompetenciesdevelopmentLe dheloppementdes comp&ences spatialesghom&riques


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sides, vertices a nd edges and the properties of adjacency andconnection. While the representation in graph form retainsthese characteristics, it does not display the characteristics ofhigher levels. When the graph is continuously distortedthrough stretch ing, shrinking, bending and twisting, its exact-ness remains unchanged. Thus, when the graph of a prism iscontinuously distorted, th e adjacency of sides is respected, asis the number of vertices, edges and sides.The projective level corresponds primarily to the pmper-ties of incidence of straight lines and planes. These proper-ties are retained in a representation created from a centralprojection.

The affine level corresponds primarily to the properties ofparallelism and convexity. These properties are retained in arepresentation created from a parallel projection.

The metric level corresponds principally tothe study ofproperties related to distance and angle measurement. Throprojections need to be used to recover the distances and an-gles measurements.W&king HypothesisIn his work on the development of space in the child, Piagetargued that spatial competencies develop accord ing to a setof successively overlapping geometric lev els. Developmentproceeds from the topological to the metric, via the projectivelevel; the notion of proximity precedes the other Euclideanaxioms, while intuition based on interior@ and exterior&yprecedes abstraction of a volume. However, little is knownwith any certainty about the order in which developmentoccurs after 11 to 12 years of age.The working hypothesis states that spatial geometric com-petencies development continues to progress according tothe overlapping of geometric levels until a dulthood. accord-ing to this hypothesis, performance at the topological levelshould, therefore, remain higher than at the other levels untiladulthood. Sim ilarly, performance at the metric leve l shouldimprove with age and remain lower than at other levels,while perfo rmance at the projective and affine levels shouldlie between the topological and metric levels.

The observations of Lunkenbein [6,7] deviate from ourhypothesis. He notes, using didactic activities, that childrenseem to find topological graphs easier than metric represen-tations (surface developments), whereas adults find metricrepresentations easier and tend to reject graphs as possible

ignore celles des niveaux superieurs. Des deformationscontinues te lles que letirement, le retrecissemen t, lepliage et la torsion naffectent pas la justesse du graphe.Ainsi lorsque le graphe dun prisme est deforme de faGoncontinue, ladjacence des faces est respectee, les nombresde sommets, daretes et de faces demeurent inchanges.

Le niveau projectif correspond principalement aux pro-prietes d incidence des droites e t des plans. Ces proprietessont conservees dans une representation c&e suite a uneprojection centrale.

Le niveau affine correspond principalement aux proprie-tes de parallelisme et de convexite. Ces proprietes sont con-servees dans une representa tion creee suite a une projectionparallele.

ti niveau metrique c orrespond principalement a letudedes proprietes reliees aux distances et aux mesures desangles. Lutilisation de deux projections permet de retrouverces distances et ces mesures angulaires.Hypothhe de travailPiaget dans ses travaux concernant la genese de lespacechez lenfant soutient q ue son developpement respectelimbrication des niveaux geometriques. Ce developpe-ment procedera it du topologique vers le metrique, en pas-sant par le niveau projectif: la notion de voisinage interve-nant avant les autres axiomes euclidiens, lintuition desdimensions fondle sur linteriorite et lexteriorite interve-nant avant labstraction dun volume. C ependant lordredes developpements ulterieurs, cest-adire de lage de 11-12 ans a Page adulte, demeure peu connu et incertain.Lhypothese de travail est que le developpemen t des com-p&ences spatiales geometriques continue a se calquer surlimbrication des niveaux geometriques jusqua Page adulte.Ainsi, selon cette hypothese, les performances de niveautopologique devraient toujours deme urer su perieures jusqualage adulte a celles observees pour les autres niveaux geome-triques. De meme, les performances de niveau metrique de-vraient saccroRre avec lage et demeurer inferieures auxautres, tandis que celles des niveaux projectif et affine de-vraient sintemaler en tre les niveaux topologique et metrique.

?butefois, les observations de Lunkenbein [6,7] secartentde notre hypothese . 11 ote, a partir dactivites didactiques,que les enfants s emblent avoir plus d e facilite avec les gra-phes topologiques quave c les representations metriques que



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representations of polyhedral objects. He also observes thatboth children and adults find it more natural to recognizeperspective drawings than to recognize other forms of repre-sentations such as graphs, surface developments and or-thogonal projections. These projections are probably themost accessible. Consequently, the entry point fo r recogniz-ing geometric properties is likely projective, not topological,for reasons apparently unrelated to geometric properties butrather for purely perceptual and figurative reasons.Lunkenbein, however, made use of relatively familiar axono-metric project ions (a ffine representations of cubes, pyramids,and so on), which cannot be used to state that projective no-tions are actually acquired.

Also, to verify that spatial geometric competencies devel-opment proceeds effectively from the topological to the met-ric level until adulthood, we wanted t o test groups of subjectsat different ages to see how they performed at the four geo-metric levels using a geometrically valid method.MethodologyAs no geometrically valid and complete t est was identifiedamong the many sp atial tests inventoried by Eliot and Smith[2], a new battery of tests was developed which minimallycovers the four geometric levels. It consists of four tests, eachcontaining ten items. Each of the forty items is made up oftwo-dimensional representations of three-dimensional ob-jects. The representations are in the form of either graphs orcentral, parallel or orthogonal projections. All the items inthe four tests are of the paper-pencil variety for reasons ofefficiency in subject examination, as several subjects can betested at the same time. This type of item is also in keepingwith the tradition of psychome tric testing for spatial compe-tencies.

?I validate item content, large numbers of question-prob-lems were sent to several people involved in teaching geom-etry for their comments and reactions. These comments andreactions were used to make final adjustments to the ques-tion-problems and accompany ing examples.The final selection of question-problems for final itempreparation was made by a select committee of five experts.For a question-problem to be selected it had to be geometri-cally correct, and simple. Each item involve s a different taskat the same geometric level. The item is accompanied by thesimplest pos sible example to illustrate the item in question.

sont les developpemen ts-plan, tandis que les adultes ont plusde facilite avec ces derniers et ont tendance a refuser lesgraphes en tant que representations possibles dobjets polye-driques. De meme, il observe que tous, enfants ou adultes,reconnaissent plus naturellement les dessins en perspectiveque les autres formes de representation : graphes, developpe-ments-plan, p rojections orthog onales. Ces projections se-raient les representations les plus access ibles. Ainsi, le pointdentree pour la reconnaissance des proprietes geometriquesne serait pas topologique mais projectif et ce, evidemmentpas pour des raisons liees aux proprietes geometriques maisplutot pour des raisons strictement perceptuelles et figura-tives. Lunkenbein utilisait toutefo is des projections cavalieres(representations affines) relativement familieres (cube, pyra-mide, etc.), lesquelles ne peuvent pas permettre daffirmerque les notions projectives sont veritablement acquises.

Aussi, afin de verifier si effectivement le developpemen tdes competences spatiales geometriques procede du topolo-gique au metrique jusqua Page adulte, nous avons voulutester dun point d e vue geometriquement valide les perfor-mances spatiales de sujets de diRerents ages aux quatre diffe-rents niveaux geometriques.M&hodologiePuisque aucun instrument de mesure geometriquementcomplet et valide na ete identifie parmi la masse de testsspatiaux recenses par Eliot et Smith, [2], une nouvelle batte-rie de tests, couvrant minimalement lensemble des niveauxgeometriques, a ete developpee. Elle est constituee de quatretests comportant chacun dix items. ?bus les quarante item ssont constitues de representations geometriques bidimen-sionnelles dobjets tridimensionnels. n sagit soit de graphes,soit de projections centrales, affines ou orthogonales. ? bus lesitems de chacun des quatre tests sont done de type ((crayon-papier )).Ce choix a ete fait pour des raisons defficacite auniveau de la procedure dexamination des sujets. 11s ermet-tent de (( ester ))plusieurs sujets simultanement. Par ailleurs,ce type ditem sinscrit dans la tradition des tests psychome-triques utilises pour mesurer les competences spatiales.

Afin dassurer la validite de contenu de lensemble desitems, une masse de ((questions-probleme s 1) ete soumisea plusieurs pe rsonnes concernees p ar lenseignement de lageometric. Suite a leurs comme ntaires et reactions, les((questions-problemes ))et les exemples les accompagnant

Spatial geometricdklopment

Le dheloppementdes comp&ences spatialesgbom&iques


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ExempZeLes figures ci-dessouscorrespondent & unemtime forme, Les liensentre les sommets A, B,C et D sont les m&me&

Example:The figures belowcorrespond to the sameshape. The connectionsbetween vertices A, B,C and D are the same.

Question :Lequel des graphesci-dessouscorrespond % aforme representeeci-contre.

Question:Which graph belowcorresponds to theshape represented

Indiquez votreIndicate youranswer.AB

ElCD

Fiiute 2Exempleet question accompagnantun item topologique.Exampleand que stion accompanyinga topological item.Figure 3Exempleet question accompagnantun item projectif.Exampleand question accompanyinga projective item.Figure 4Exempleet quest ion accompagnantun item affine.Example and question accompanyingan affine item.

Exemple : Example:Si on prolonge dans lespace les If the edges of the shapearktes de la forme reprhsentee represented below arecidessous, certaines dentre extended into space, someelles se croiseront tandis que of them cross while othersdautres ne se rencontrerontjamais. Par exemple lar6.a never meet. For example,the edge a does not meetne rencontre pas larett&D.Par contre elle croise CE enun point.edge CD. Howevsq at onepoint it crosses CE.

Question :En examinant la pyramiderepr&entCe ci-dessous, quelleest la paire dar&es qui secroiseront en un point 2lext&ieur de la forme, si onles prolonge dans lespace ?

Question:Looking at the pyramidrepresented below, whichpair of edges will cross eachother at a point outside theshape if the edges areextended into space?Choix de reponses /Choice of answers1) a et/and BC ;2) BC3) AB4) AD

Indiquez votrereponse.Indicate youranswer.12El4

FigW3rFour of these items-one item per level-are shown inFigures 2,3,4 and 5. Tests are marked o n a pass-fail basis.The sam ple is made up of six regular classes of students,ranging from prima ry level to university: a first group of45 children (11 and 1 2 years old), a second group of 23 ado-lescents (15 and 16 years old), three groups of young adults(104 individuals) between 18 and 24 years of age, and a finalgroup of 20 adults between the age s of 30 and 35 years. Thetest was given to 192 people in all. None of them had re-ceived any specific training in geometry.Of the total sample, the female component is slightlylarger than the male component: 65% female, 35% male.The breakdown for children is 62% female, 38% male. Foradolescents, the breakdown is 39% female, 61% male. The

Exemple :On peut grouper des formesselon le parallelisme de leursfaces.Example:Shapes can also be groupedby the parallelism of theirsides.

Aucune face parall&leNo parallel side Une paire de faces parallelesWo parallel sidesQuestion :Lequel des honcks suivantsgroupe correctement selon leparall6lisme de leurs faces escinq formes representkescidessous ?1) L.es formes ti A, B et C M nsemble ;2) Les formes u B et E M nsemble ;3) Les formes u A, C et E Hensemble ;4) Les formes u A, C et D M nsemble.

Question:Which statement correctlygroups the five shapesrepresented below by theparallelism of their sides?1) Shapes A, B and C together;2) Shapes B and E together;3) Shapes A, C and E together;4) Shapes A, C and D together.

A) Indiquez votrereponse.Indicate youranswer.

6iiiire4ront M mis au point.Finalement, un cornit restreint de cinq experts a effect&une demi&re selection des ((questions-problemes 1)mis aupoint pour la confection finale des items. Les &&es de se-lection f&n t la justesse gkom&rique de la tiche et sa simpli-citk. A chaque item correspond une tiche diffkrente et deniveau geom&rique difErent. Cet item est accompagne dunexemple le plus simple possible illustrant la tiche demandke.Ainsi, les figuxuzs 2, $4 et 5, presentent quatre de ces items,un par niveau geom&rique. La correction seffectue selon lemode suc&s ou echec.Uchantillon est composk dWves de six classes r&-uli&esreparties du niveau primaire jusquau niveau universitaire :un premier groupe de 45 enfants de 11 et 12 ans, un



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Exemple : JIxample:Les vues de face et de dessus de Using the side and top viewla forme representee cidessouspermettent de connaitre la of the shape representedbelow, the real size of thevraie grandeur des faces et des shapes sides and edges cana&es, On peut alors faire un be determined. A surfacedevelmppement plan qui, une development can then befois replie, per-met dobtenir la made which, once folded,forrne illustree. results in the illustratedVue de face shape.

dessus u?bp view 1 1 DCveloppement planSurface development

Questim : Question:Lequel des developpements- Which surface developmentplan ci-dessous permet below can be used to get thedobtenir la forrne representee shape represented by thepar les vues de face et de side and top views showndessus ci-contre ? opposite?

answer.ABEl3D

As expected, there is a very strong correlationbetween total performance and age: r = .67.The confide nce level is greater than .Ol. Theperformance of the groups of subjects does notimprove uniformly but, instead, grows sharplybetween childhood and adolescence and thenstagnates (see ltrble 1).

No significant d ifference in performance isobserved between the sex es (a = .05), nor is

any sign&ant difference betw een the sexes observed at anyone geometric level.Figure 5 AFiium 5Exemple et question accompagnantun item m&.rique.Example and question accompanyinga metric item.

breakdown for young adults is 68% female,32% male. The adult group consists entirely offemales. Subjects were tested in groups. Themaximum time allowed for completion of allitems was three hours. Subjects could use apencil, ruler, compass and eraser.

The forty items are significantly correlatedat a confidence level of a = .Ol with the total ofthe test. The Cronbach [l] reliability index forall forty items toge ther is .886. The items aregrouped in four blocks according to geometriclevel. Each block co ntains ten questions. Ailitems are significantly correlated to the totalsof their respec tive blocks. Similarly, the coeffi-cients are good, given that the items involve avariety of tasks and response modes: .n for thetopological level, .63 for the projective level, .72for the affine level, .60 for the metric leve l.Results

The correlation between age and the total on the test isreflected in the way performance, according to geometriclevel, progresse s from one age group to next. When all 192subjects are considered to gether, a significant correlationexists between age and performance according to geometriclevel: topological = 558, projective = .432, affine = .675,metric = .52.

In line with overall performance, results compiled accord-ing to age can be broken down into two quite distinct groups,with children on one side and the other age groups on theother (see Eible 2).

The averag es for the children are low at all geometric lev-els and tend to decrease from the topological to the affine

deuxieme groupe de 23 adolescents de 1 5 et 16 ans, puis troisgroupes de jeunes adultes (104 individus) entre 18 et 24 anset finalement un demier groupe de 20 ad&es entre 30 et 35ans. Un total de 192 personnes ont passe e test. Aucune nasuivi de formation particuliere en geometric.

Au total, lechantillon comporte legerement plus defemmes que dhommes : 65 % de femmes, 35 % dhommes.Chez les enfants, la repartition entre les sexes est de 62 %de filles p our 38 % de garGons. Chez les adolescents, elleest de 39 % de filles pour 61% de garqons. Chez les jeunesadultes, elle est de 68 % de femmes pour 32 % dhommes .Le groupe de sujets adultes examines est compose exclusi-vement de femmes. Les sujets ont ete examines en groupe.Le temps max imum alloue etait de trois heures pour len-semble des items. Les sujets avaient dro it a un crayon, uneregle, un compas et une gomme a effacer.

Chacun des quarante items est significativement correle aun niveau de confiance de a = 0,Ol avec le total du test. Lin-dice de fiabilite de Cronbach [l] pour les quarante items prisdans leur ensemble est de 0,886. Les items sont regroupes enquatre blocs selon les niveaux geometriques. Chaque regrou-pement comporte dix questions. ?bus les items sontsignificativement correles au total de leur regroupementrespectif. De meme, les coefficients de chacun d e ces regrou-pements sont bons, compte te nu qu e les items s ont de tacheset modes de reponse varies : 0,71 pour le nive au topologique ;0,63 pour le niveau projectif; 0,72 pour le niveau affine ; 0,60pour le niveau metrique.

R&ultatsConformement aux attentes, la performance totale aux testsest tres fortement correlee a Page r = 0,67. Le niveau de con-fiance es t de plus de 0,Ol. ?butefois, la performance desgroupes de sujets ne saccro3 pas uniformement. Elle le faitplutot brusquement entre lenfance et ladolescence puisstagne ensuite (voir tableau 1).

Aucune d ifference significative (a = 0,05) de performanceentre les sexes na ete observee. De plus, il nexiste aucunedifference de performance significative entre les sexes quelque soit le niveau geometrique.

Evolution des performances selon les niveaux geome-triques a travers les groupes dage reflete la correlationobservee entre Page et le total au test. Pour lensemble des192 sujets, lage et les performances observees selon les

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Spatial geometriccompetenciesdevelopmentLe dheloppementdes comp6tences spatialesg4omhiques


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Tableau 1 /Table 1Performance tota le selon Mge.Total performance according to age.

and increase slightly at the metric level. Their average issignificantly lower than those of other age groups for thetopological, projective and affine levels. At the metric leveltheir performance cannot be differentiated from the perform-ance of adolescents. The greater success that children showwith top ological items and the gradual deterioration in theirperformance at the projective and affine levels are in linewith our hypothesis an d fit the extended Piagetian model.While improved performance at the metric level seems tocontradict this model, it can be explained by the impact ofschool programs, which are concerned primarily with metricnotions: regular figures and shapes, the measurement of areaand volume, etc.

It is important, first of all, to note that the difference inperformance between adolescents, young adults and adults,increases as one approaches the metric level. Con sequently,topological ability does not appear to improve with age, whileslightly greater progress is observed at other geome tric levels.The results that most impro ve with age are those for affineitems; results on projective items show the least improve-ment, but an improvemen t which is nonetheless significant.

It is possible that, here too, the situation is only a reflec-tion of the almost exclusive attention given to the metricnotions, or isometric projections, employed at all stages ofacademic training.

A marked dec line in performance is observed at theprojective level. Contrary to our hypothesis, which supposed

lntervalle de confiance : 95 % l Interval of confidence: 95%25 - Enfants (1 l-l 2 ans)22.5 Adolescents (15-l 6 ans)Jeunes adultes (18-24 ans)20 Adultes (30-40 ans)17.5 Children (1 -l 2 years)

s Adolescents (15-l 6 years)zi 15 . Young Adults (18-24 years)Adults (30-40 years)12.5

II

EnfantsChildrenI I

Adolescents Jeunes adultesAdolescents Young AdultsGmupe d%ge l Age Group

IAdultesAdults

niveaux geometriques sont significativement correles :topologique = 0,558 ; projectif = 0,432 ; affine = 0,675 ;m&rique = 0,52.

En accord avec les performances g&&ales, les resultats,lorsquils sont compiles selon Page, se divisent en deuxgroupes bien distincts regro upant dune part les enfants etdautre part les autres groupes dage (voir tableau 2).

Quel que soit le niveau geometriqu e, les moyennes desenfants sont farbles et ont tenda nce a baisser encore du topo-logique 5 laffine puis a remonter legerement au niveau me-trique. 11s btiennent une moyenne significative ment infe-rieure aux autres groupes dage pour les niveaux topologique,projectif et affine. Au niveau metrique, leurs resultats nepeuvent etre distingues de ceux des adolescents. Cette reus-site superieure des enfants aux items topologiqu es et labaisse progressive de leurs performances aux niveaux projec-tif et affine, sont en accord avec notre hypothese et se situentdans le prolongement du modele piagetien. ? butefois, la re-montee d es performances au niveau metrique semble encontradiction avec ce modele. Elle peut cependant sexpli-quer par limpact des programme s scolaires, lesquels fontessentiellement reference a des notions m etriques : figures etformes regulieres, mesure de surface et de volume, etc.

Chez les adolescents, les jeunes adultes et les adultes, ilfaut dabord souligner que les resultats se distinguent de plusen plus entre les groupes dage au fur et a mesure que lonseleve vers le niveau me trique. Ainsi, les performances topo-logiques ne semblent aucun ement progresser avec Page,tandis que les performances semblent progresser un peuplus a chacun des autres niveaux ge ometriques. Les items deniveau affine sont ceux dont les resultats saccroissent le plusen fonction de Page et les items de niveau p rojectif sont ceuxdont les resultats saccroissent le moins mais ils saccroissenttout de meme significativement.

Encore une fois cette situation ne reflete peut& re que lepeu dattention portee aux notions au tres que metriques, ouaux dessins en perspective autres que cavaliere, tout au longdes apprentissages academiques.

Ce qui doit etre aussi note, cest la baisse marquee desperformances au niveau projectif. Contrairemen t a notrehypothese de depart qui supposait un affarblissement con-tinu des performances du topologique au metrique, nousassistons globalement a une chute significative des perfor-mances au niveau projectif et a une homogeneisation des



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Tableau 2 llhbk 2Perfom?ances selon Mge et lesniveaux g6!h3m&riques.Petfom?ance according to ageand geometric level.

that performance would continuously weaken from the topo-logical to the metric level, there was a significant all-rounddeterioration in performance at the projective level, whileperformance at the other three levels-topological, affine,and metric-tended to become more uniform. From this wecan posit that the projective level is the one least directlycalled upon during a persons many daily learning activitiesor that the projective level is the most specifically spatial, inthat it seems more difficult to solve projective items thanitems at other levels (topological, aff ine or metric) using ana-lytical strategies (observation, c ounting, m easuring, et c.). XIsolve projective items, th e subject has to use strategies thatare more specifically spatial in nature, requiring operationsand transformation s in space characteristic of what psychom-etrists call visualization; items for other levels, meanwhile,would be considered more analytical as defined by the re-lations fac tor, which is concerned with the ability to recog-nize shapes.

More spe cifically, the averages of adolescents increase inrelation to those of children at the topological and affine lev-els, but their perform ance remains poo r at the projective andmetric levels. Adolescents have as little success with projec-tive items as they do with metric items. The results for youngadults (18-24 yrs) are significantly lower (a = .Ol) for projec-tive items. Success among adults (30-35 yrs) is relatively uni-form ac ross the board. The difference betw een averagesbecomes less apparent. Adults continue to be less successfulonly with projective items.

lntervalle de confiance : 95 % l Interval of confidence: 95%

01 I I I I 1Topologique Projectif Affine MetriqueTopological Projective Affine Metric

I 1Enfants--------- ChildrenAdolescentsAdolescentsJeunes adultesYoung AdultsAdultes- Adults

Enfants (1 l-12 ans)Adolescents (15-l 6 ans)Jeunes adultes (18-24 ans)Adultes (30-40 ans)Children (1 l-l 2 years)Adolescents (15-l 6 years)Young Adults (18-24 years)Niveaux g6ombiques l Geometric Levels Adults (30-40 years)

performances entre les trois autres niveaux geo metriques:topologique, affine et metrique. Ce phenomene nous porte asupposer que le niveau projectif est peut-&re le moins direc-tement sollicite pa r les diverses activites dapprentissages dela vie quotidienne ou quil est peut-etre le plus specifique-ment spatial en ce sens quil semble plus difEcile de resoudreles items project& que les autres items (topologiques, affinesou metriques) a laide de strategies analytiques (observer,compter, mesurer, etc.). Les items project& semblent faireappel a des strategies plus proprement spatiales exigeant desoperations e t des transformations dans lespace propre aufacteur (cvisu alisation)) dentifie par les psychometricien s,tandis que les items de s autres niveaux seraient plus ((analy-tiques ))au sens du facteur (( elation )).11s eraient plus appel aune capacite de (( econnaissance Ndes formes.

Plus specifiquemen t, chez les adolescents, les moyennesaugmentent par rapport a ux enfants aux niveaux topologiqueet affine, mais leurs performances aux niveaux p rojectif etmetrique restent farbles. Les items de niveau projectif sontaussi peu reussis que ceux du niveau metrique. Les eunesadultes (18-24 ans) ont des result& significativement plusfarbles (a = 0,Ol) aux items de niveau projectif. Les adultes(30-35 ans) reussissent de facon relativement plus homo-genes a travers les niveaux ge ometriques. Les differencesentre les moyennes sattenuent. Seuls les items d e niveauprojectif demeurent moins bien reussis.

Dans les deux premiers cas, topologique et projectif, il nya pas de difference significative entre le

globally rigid symmetric tensegrities by connelly, terrell, 1995

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