N-ZONOTOPES AND THEIR IMAGES:FROM HYPERCUBE TO ART IN GEOMETRY

László VÖRÖSUniversity of Pécs, Hungary

ABSTRACT: The image of the four-dimensional cube – incorporated in the logo of this conferencetoo – and the common part of five cubes constructed into the platonic dodecahedron inspired thearchitect, teaching descriptive geometry and CAD for architect students, to seek after the possiblemodels of the hypercube. This process by involving the rule polygons and solids resulted in the3-dimensional models, whose hull, as it turned out, is the so called zonohedron.The edges of this zonohedron form easily recognizable bar-chains, whose binding points join on onehelix each. Increasing the number of the sections in the bar-chain infinitely, a continuous helix iscompleted whose polar distribution means the rotation around the lead. Therefore the hull of then-cube’s 3-model is generated as the surface of a solid of rotation – may be called n-zonotope – inwhich any k-dimensional cube’s 3-model can be constructed. According to these our lattice 3-modelof any k-cube can also be produced either as ray-groups of the edges or as sequences of bar-chainsoriginated from a separate helix.Combining 2<j<k edges, we can build 3-models of j-cubes, as parts of the k-cube. An investigated3-dimensional tiling holds always a 3-model of the k-cube and its necessary j-cubes but some k-cube3-model self can also create space-filling polyhedra. The space-filling arrangement of these modelscan further be dissected and reordered by inner lower-dimensional 3-models and Boolean operations.For any k each vertex can lies in planes parallel to the base plane of construction, therefore always anwell ordered plane tiling appears in the parallel intersections. These tiling can further be dissectedand reordered. We can gain frames for an animated 2-dimensional tiling from the space-fillingarrangements of the k- and j-cubes by intersecting planes moved continuously.These constructive geometrical results can be related to several fields of art from architecture andsculpture up to the research and education of theory and perception of color and space.The creation of the constructions and figures required for the paper was aided by the AutoCAD andAxisVM programs as well as Autolisp routines developed by the author.

Keywords: constructive geometry, modeled hypercube, zonotope, space-filling, plane-tiling

1. INTRODUCTIONThe descriptive geometry and introductoryCAD courses for architecture studentssupplement each other usefully in praxis. Withhelp of the CAD program we can quicklycreate variations for the exercises where depthperception is needed, which is a good devicefor practicing the 2- and 3-dimensionalconstruction with computer. The result can beseen in various perspectives, even rotating in

space. An obvious and frequent exercise is themodeling of platonic and Archimedean solids.In connection of this came the followingquestion: What solid is created as the commonpart of the five cubes written into a pentagondodecahedron. The answer is a rhombictriacontahedron which is in proportion of thegolden section to the body drawn byconnecting the vertices of the dodecahedronand icosahedron halving each others edges

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(Fig. 1-3). If in all of the vertices of this body,we draw parallels to every different-angled

edge, we will form the 3-dimensional model ofa 6-dimensional cube [3].

Figure 1 Figure 2 Figure 3 Figure 4

This discovery suggested the need for k-dimensional cubes delineated in 2 or 3dimensions. The cover figure of Journal forGeometry and Graphics (Figure 4), whichalso appears in the emblem of ICGG,suggested a basis for the solution of theproblem. According to the standardized results:if the edges starting from a common vertex ofthe 3-dimensional model (3-model) of thek-dimensional cube (k-cube), standing in acertain angle to the plain of a rule polygon andtheir projections are parallel on this plain tothe diagonals or edges of the k or 2k sidedpolygon, than we will get an image similar tothe normal cube’s isometric axonometry.These two discoveries have led to furtherstudying of the 3-model of the hypercube, andits possibilities within arts and education.The creation of the constructions and figuresrequired for the paper was aided by theAutoCAD and AxisVM programs as well asAutolisp routines developed by the author.

2. THE RULED 3-MODELS OF THEHYPERCUBE

2.1 Rule polygons as bases of constructionLifting the vertices of a k sided regularpolygon from its plane, perpendicularly by thesame height, and joining with the centre of thepolygon, we get the k edges of thek-dimensional cube (k-cube) modelled in thethree-dimensional space (3-model). Fromthese the 3-models or their polyhedral surface

(Fig. 5) can be generated as well in differentprocedures [3,4]. Each polyhedron from thesewill be a so called zonotope/zonohedron [2],i.e. a „translational sum” (Minkowski-sum) ofsome segments [3]. This structure keeps thenormal cube’s central symmetry and rotationalsymmetry too related to the diagonal joiningthe starting vertex referred to the groups ofany j<k dimensioned element. This diagonal isfurther on related to as main diagonal (Fig. 5).The number of the vertices ( j = 0) of thek-cube is 2k. In case of further values of j thenumber of the elements is 2k-jCk;j. (C means:combination without repetition.)

Figure 5

It is also possible to get to the endpoint of themain diagonal from the starting point alongeasily recognizable bar-chains, whose bindingpoints (the outer vertices of the model) join onone helix each. The common lead of thesehelices is the main diagonal. According tothese attributes, such a chain of bars can –

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distributed around the main diagonal in thenumber equal to the number of his elementsand the given distribution mirrored to thecentre point of the common lead – generatethe outer edges of the k-cube’s 3-model (F. 6).

Figure 6

Figure 7

Increasing the number of the sections in thebar-chain infinitely, a continuous helix iscreated, whose polar distribution means,according to the procedure so far, the rotationaround the lead, therefore the shell of then-cube’s 3-model is generated as the surfaceof a solid of rotation in which anyk-dimensional cube’s 3-model can beconstructed. The surface of revolution isgenerated as an infinite amount of central orplane-symmetrical double helices, similar tothe straights appearing on the surface of ahyperbole rotated around the imaginary axis.According to the construction rules so far, thefull 3-model of the n-cube generates a solid ofrotation, which is optionally expandable indirection of the rotation axis (Fig. 7). Thus theheight of the solid is also determinable, as itholds the 3-cube’s 3-model with the

proportions of the normal cube [4, 7].The vertices on our 3-model of the k-cube arereplaced on k+1 levels as vertices of k-1 rulepolygons each one with k edges and endpointsof the main diagonal. We can indicate themodel’s inner vertices by these polygons’vertices. This is a complex procedure, but itgives a descriptive picture of the polygons’proportions to the starting helix and oneanother (Fig. 8). Finding the inner vertices ismade easier, if we know how many verticesthere are in the k-folded polar distribution oneach level to be looked for (Fig. 9).

Figure 8

Figure 9

Because the vertices are following a k-foldedpolar distribution on level k+1, and there arealways k edges conjoining in one vertex, it canbe pointed out, that there are k edges startingfrom the vertex on the main diagonal’sendpoint, and there are also k edges startingfrom those endpoints as well, still, only k-1 of

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these are reaching the next level. Two edgesare ending in each vertex of the level, and k-2are leading from it, etc. This tendency getsopposed in middle-height by the model’ssymmetry. On behalf of the k-folded polarity,the points equal to the remainder of thedivision of the number of vertices on a givenlevel by k, coincide in the centre point of thelevel. The multiple coinciding of the innervertices’ polar groups is also possible.Following the arrangement of edges andvertices as described above, a new algorithm ispossible to determine to create the model. Noedge may start as continuing to the next levelfrom the edges arriving in a given vertex of alevel. It is enough to define these illegal edgesin one vertex of each polar group, becausethey are also arranged circularly symmetrical.The usable edges continuing from a givenvertex arrive in one vertex of each of the nextlevel’s polar groups (Fig. 9).

2.2 Separate helix as base of constructionInterpreting the construction of the k-cube’s3-model as a sequence of dispositions, theincreasing dimensioned cubes’ 3-models canbe easily separated, for example starting fromthe upper end of the main diagonal (Fig. 10).

Figure 10

In this model of the k-cube, the edges of the0,1,…k cube elements’ model sequence areparallel to the k elements of the bar-chainapproaching the starting helix, and the 1,2,…kdispositions’ vectors are following oneanother along this bar-chain.

Therefore the model’s 0,1,…k-1 segments canalso be interpreted as intersections of two fullmodels each (Fig. 11), and the equaldimensioned parts are positioned around themain diagonal of the model, symmetrical to itscentre point. Summarizing all our experiencesso far, a symmetric 3-model of the k-cube canbe described originated from a separate helixas it follows (Fig. 12):

- replacing the helix with a bar-chain of ksections of identical length, whose verticesare joining the helix

- the sections of the bar-chain replacing thehelix are designated as a1-ak, their disposedcopies as b1-b2

-generating the bar-chains A3(b3,b4, ... bk),A4(b4,b5, ... bk), ... Ak(bk)

-generating the groups of bar-chainsB1(A3,A4, … Ak), B2(A4,A5, … Ak), … Bk-3

(Ak-1,Ak) by moving the bar-chains into thesuitable common starting point, whosedisposed copies will later be called C1-Ck-3.

-creating C1 in the upper endpoint of a1

-creating C2 in the upper endpoint of a2

-creating C3 in the upper endpoints of a3 andall b3 segment of all so far created C

- ...-creating Ck-3 in the upper endpoints of ak-3

and all bk-3 segment of all so far created C-distributing all a and C k-folded around thestarting helix’s lead

-mirroring the copies of all created elementsto the plane halving the lead and beingperpendicular to it

- rotating the mirrored elements 180º aroundthe lead if k is odd

-eliminating the coinciding elements and givethe edges the colour of the startingbar-chain’s elements parallel to them.

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Figure 11 Figure 12

2.3 Solids as bases of constructionThe edges of the k-cube’s 3-model joining in avertex can be chosen with different lengthsand in random spatial arrangement. If we wantto gain a more ruled form, we can select theedges parallel to the edges of the platonic andArchimedean solids and of solids originatedfrom them as we could see by the case of the6-cube’s 3-model described in the introduction.These experiments can lead to well known andnew solids and connections. An exampleshows the figure 13.

Figure 13

The power of the applied computer can hinderthe construction of the k-cube’s 3-model fromthe gained edges. I have developed for thiscase an algorithm to construct only the hull ofthe model [3].More on the topic of this subsection can beseen in [7] too.

2.4 Some related fieldsMathematics, for the sake of largestandardization, uses the quantitativedescription of our world’s phenomena. Arts’effects lie in the selection of man’s senses oron the contrary, on the synthesis, theharmonizing of the intensity of these senses.Those branches of geometry, which representthe numerical relations with graphical basicelements, are visually, or considering size- andform-perception, a connecting point to finearts. Giving these elements other attributes,other senses could strengthen this connection,by creating more and more possibilities forassociation and a more complex artistic effect.This connection could also mean a practicaladvantage. Next to, or more or less instead ofmathematical signs and formulas, the

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description of constructions’ algorithmic stepsand the uncovering of their connections canhelp artists and architects with a more visualimagination, in form of color scales and mixesaccording to the logic of phase pictures andthe connection of geometric elements. (Thefigures of this paper are also created on behalfof this theory, but in this case vividness was astronger factor than artistic harmony.)Another connecting field in arts and educationis color theory.We could see, that the hull of the k-cube’s3-model can be constructed from bar-chainsjoining on helices. Each bar has an identicallength and angle to the base plane. The heightof the lattice structure can be chosen free.These seem to be good manufacture and staticproperties.

Figure 14

Figure 15

A preliminary investigation verifies that thisstructure with static stable vertices can becertainly applied. If the vertices can to someextent move freely, they have to be fastened toeach other with horizontal bars. A bigstructure can be built this way too. (Diameter:86,10m, hight: 36,00m, tubes on steel: 30x30or 20x20cm cross-sections, length of the

oblique bars: 5,10m) The above figures showthe torsions of the structure under verticalload (Fig. 14) and the drawing and pressingtensions produced in the bars characterized bycold and warm colors (Fig. 15).A distant nice association with the mirroredpairs of helices in our models is the type ofdouble spiral stairs built by the lodge of PeterParler in cathedrals of the medieval Hungary.

3. SPACE-FILLING, PLANE-TILING

3.1 Space-filling based on the 3-model ofthe k-cube and its rearrangementThe 2-dimensional orthogonal projection ofour 3-models indicates the idea how toconstruct space-filling with these models.However our 3-model of the 6-cube forexample does not fill the space. The projectedgrid of the 3-cube joins our grid above and thecube fills the space well known. The edges ofthe cube can be selected from the convenientlifted edges of the 6-cube’s 3-model. With theselected four edges of the grid we can buildthe 3-model of the 4-cube. The hull of this is arhombic dodecahedron which fills the spacebut this arrangement has not any rotationalsymmetry without additional assumptions.We can however replace a cube in the hole ofthe rotational-symmetrically arranged rhombicdodecahedra and continue the filling in a sixfold polar array with a rhombictriacontahedron, which contains our 3-modelof the 6-cube (Fig. 16).

Figure 16

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It can be seen, that we can fill the space withthese solids. The basic stones are to cut from ahoneycomb by symmetry-planes [5].We can follow the construction of thespace-filling mosaic based on the 3-model ofthe 8-cube by the next figures (Fig. 17-18).The construction method is similar to the caseof the 6-cube. In the last figure we can see theplaces of the next repeating base-models.

Figure 17

Figure 18

For odd k it is advisable to originate theconstruction from a k-1 sided polygon. In thiscase the kth edge is perpendicular to the baseplane but its upper endpoint must be in thecommon plane of the other edges’ endpoints.We obtain a degenerated axonometricprojection. Some sides fall into commonplanes but will not be identical. We gain onthis way a 3-model which has similarsymmetric-properties like this one in the casefor even k. The method and the space-filling

arrangement is adapted up to k =9 [6, 7].The construction based on the 3-model of the5-cube is very simple. The repetition of thegiven solid fills the space alone [7]. As weknow the 3-models of the 3-cubes originatingfrom higher-dimensional spaces and therhombic dodecahedrons (3-models of the4-cubes) are also space-filling solids. Thecommon part of the platonic hexahedron andoctahedron fills the space too if its edges havean identical length (Archimedean greatrhombicuboctahedron). This polyhedron canbe the hull of a 3-model of the 6-cube but thearrangement of the edges can be originatedfrom the common part of the platonichexahedron and octahedron (Archimedeancuboctahedron) if their edges halve each other[3, 7].By the solution for k =10, we obtain also adegenerated axonometric projection with thearrangement of the 10 edges joined in the basevertex. This one and the procedure of tilingare merged in figure 19 [6,7]. This methodmay be a possible way to generalize the fillingof the space with higher-dimensional cubemosaics based on 3-models of k >10 cubes.

Figure 19

Other possibility is to rearrange ourspace-filling, assembling the 3-models of thek- and j-cubes from lower-dimensional cube3-models. From the given edges we cancombine the 3-models of 2<j<k cubes. In casek=6: 4 of the 3-cubes, 3 of the 4-cubes and 1of the 5-cubes.

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Their additions (Fig. 20) can replace the3-models of the above k- and j-cubes in ourmosaic.

Figure 20

3.2 Plane-tiling based on the sections of thespace-filling arrangementsAs it follows from our modeling way, thevertices lie in planes parallel to the basic planeof the construction, therefore a rule tilingappears on these horizontal sections of ourspace-filling solid-mosaics based on the3-model of the k-cube. These tiling can furtherbe dissected by the projected edges of theintersected solids. A similar phenomenoncould be seen in the projection of the inneredges of the j-cubes’ 3-models.

Figure 21

In the figure 21 we have combined the grids ofthree and finally of all four horizontal sections

by k=6. This is further dissected by theprojected edges. These sections are describedin [5].The sections of other space-filling mosaicsbased on our 3-model of k- and j-cubes can begenerated on the horizontal planes of thevertices and on planes of symmetry similarly tothe above way [7] but we can choose naturallyany plane in the space to gain a plane-tiling.Their orderliness depends on the regularity ofthe applied solids and the alignment of thecutting planes with these ones. Any vertices liefor instance in planes parallel to any two edges.We can produce frames for an animated2-dimensional tiling from the space-fillingarrangements of the k- and j-cubes byintersecting planes moved continuously.

3.3 Some related fieldsIt is apparent by the task of space-filling andrearrangement that the thinking in space couldbe developed well by the k- and j-cube modelsas real building blocks or bars and nodalelements (for instance MERO-Novum inbuilding and Zome-tool as a toy and aneducational aid in one). On the 2-dimensionalscreen a researcher, a student, or a playermust have more imaginative power and isdemanded to learn and apply the basic CADcommands cleverly.Plain divisions, which can be produced withendless number of schemes and can beenriched with projections, overlapping andsliding, can provide help in numerous fields ofarchitecture and arts in case of productdevelopment or planning concepts(lattice-works, tiling, covering, textile andwallpaper schemes, windows’ pattern on glass,projected backgrounds, etc.) The easyproduction of plain compositions can help thestudy of color harmonies and their preference[1].The lighting of the schemes, their central andparallel projection on plains and differentshapes, is opening up further possibilities andleads us back to the field of geometry.

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The architectural use and installation of themodels’ space lattices is also an obvious idea.The combination of transparent bodies, colorand light, can give the basics for particularlyinteresting sculptural and interior designworks, as the moving of the spectator or thelight source continuously creates differenteffects. The moving of the models, therhythmical change of the animated elements’color and shape, or the synesthesia of colorsand sounds could urge us to enrich the artisticeffect with music.

4. CONCLUSIONSThe visual language of architecture as anapplied art is strongly related to geometry,which again is the most important connectionto the engineering and natural sciences, and tothe industrial environment and production, therealization of a building. The simplearchitectural educational question, describedin the introduction, led to answers that areconnected through descriptive geometry tonumerous fields of broader geometry andmathematics. The results can have an effect oneducation, architectural space-creating andwith the help of design on production. Withthe participation of basic artistic devices,numerous branches of arts can be given newimpulses, hopefully leading to works that areproviding a rich connection between variousfields, even mathematics.

ACKNOWLEDGMENTSI give my thanks to professors Dr. EmilMolnár, Dr. Helmuth Stachel and Dr. GunterWeiss for their help and encouraging of thelectures and publications on this topic and toZsolt Sándor for the preliminary staticverification of the structure mentioned in thesubsection 2.4.

REFERENCES[1] Nemcsics, A., Dynamics of Color, in

Hungarian: Színdinamika, AkadémiaiKiadó, Budapest, 1990.

[2] Towle, R.,http://home.inreach.com/rtowle/Zonohedra.html

[3] Vörös, L., Reguläre Körper undmehrdimensionale Würfel, KoG 9 (2005),21-27.

[4] Vörös, L., A SymmetricThree-dimensional Model of theHypercube, Symmetry: Culture andScience, Vol. 17, Numbers 1-2, 75-79(2006)Proceedings of the Symmetry-Festival2006.

[5] Vörös, L., Two- and Three-dimensionalTilings Based on a Model of theSix-dimensional Cube, KoG, 10 (2006),19-25.

[6] Vörös, L., Two- and Three-dimensionalTiling on the Base of Higher-dimensionalCube Mosaics, Proceedings of the 7th

International Conference on AppliedInformatics, Eger, Hungary, 2007, Vol. 1.,185-192

[7] http://icai.voros.pmmf.hu

ABOUT THE AUTHOR

László Vörös DLA, Dipl.-Architect teachesdescriptive geometry and CAD at the Instituteof Architecture at the University of Pécs,Hungary. His research interests areconstructive geometry, CAGD and CAAD.The topic of his dissertation is the computeraided geometric design of the auditoriums oftheaters in interest of the undisturbed visibility.He can be reached by e-mail:vorosl@witch.pmmf.hu or through postaladdress: PTE, PMMF Kar, 7624 Pécs,Boszorkány utca 2., Hungary.

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