308

integration through computationacadia 2011 _proceedings

Compu ta t i ona l l y, t he re ex i s t s i gn i f i can t po ten t i a l s t o i n t eg ra te pe r i od i c ( r epea t i ng )

and ape r i od i c ( non - repea t i ng ) t esse l l a t i ons i n a rch i t ec tu r a l app l i ca t i ons . Wh i l e

exp lo r a t i on o f two-d imens iona l and t h ree -d imens iona l t esse l l a t i ons appea r i n

h i s t o r i ca l l y s i gn i f i can t wo r ks , t oday, h i ghe r-d imens iona l t esse l l a t i ons a re capab le

o f be i ng gene ra ted compu ta t i ona l l y wh i ch may be use fu l i n va r i ous a rch i t ec tu r a l

app l i ca t i ons . Th i s pape r, a co l l abo ra t i on be tween an a rch i t ec t and ma thema t i c i an ,

e xp lo res t hese p rocesses and po ten t i a l s . I n s i gh t s w i l l be o f f e r ed i n t o t h i s ea r l y

s t age exp lo r a t i on r ega rd i ng t he c rea t i on and use o f h i ghe r-d imens iona l geome t r i es

f o r a r ch i t ec tu r a l app l i ca t i ons—such as pa t t e r n i ng , vo l ume t r i c desc r i p t i ons , and

modu l a r assemb lages .

Keywo rds

D ig i t a l A rch i t ec tu re , Ma thema t i cs i n A rch i t ec tu re , H ighe r-D imens iona l Ob jec t s i n

A rch i t ec tu re , Des ign Compu ta t i on and Ma thema t i cs

Potentials for Multi-dimensional Tessellations in Architectural Applications

David Celento

Pennsylvania State University

Edmund Harriss

University of Arkansas

ABSTRACT

309

Fig. 1

Fig. 2

Fig. 3

1 Introduct ion

Given t he h i s t o r i c uses o f t esse l l a t i ons i n a rch i t ec tu re , we exam ine t he po ten t i a l s

f o r t he use o f h i ghe r d imens iona l ma thema t i cs and compu ta t i on i n a rch i t ec tu r a l

app l i ca t i ons . W i t h i nc reas i ng compu ta t i ona l powe r, we wonde r i f h i ghe r-

d imens iona l t esse l l a t i ons o f f e r use fu l o r mean i ng fu l poss ib i l i t i e s i n a r ch i t ec tu re , f o r

such t h i ngs as su r f ace pa t t e r n i ng , spa t i a l o rgan i za t i ons , and modu l a r assemb l i es .

The r eade r i s assumed to be f am i l i a r w i t h a rch i t ec tu r a l p r i nc ip l es , bu t l e ss so w i t h

advanced ma thema t i ca l t heo r y ; t hus , we beg i n w i t h a b r i e f i n t r oduc t i on t o t he

h i s t o r i ca l con tex t f o r pe r i od i c , non -pe r i od i c , and ape r i od i c t esse l l a t i ons , o f f e r i ng

some use fu l de f i n i t i ons and i ns i gh t s . Fo l l ow ing t h i s , we i n t r oduce t he p rope r t i e s

o f h i ghe r-d imens iona l geome t r i es . We conc l ude by r e v i ew ing t he r esea rch t ha t

has been pe r f o rmed and o f f e r specu l a t i ons on poss ib l e a rch i t ec tu r a l app l i ca t i ons .

2 Context

I t i s we l l known t ha t t he use o f r epea t i ng ( pe r i od i c ) pa t t e r ns have se r ved du rab l e

r o l es i n many cu l t u r es—pa r t i cu l a r l y t hose o f ea r l y G reek , Roman , Japanese ,

Ch i nese , and I s l am ic wo rks ( F igu res 1 -3 ) . These pa t t e rns have been l a rge l y

composed o f two-d imens iona l and t h ree -d imens iona l geome t r i es t ha t t esse l l a t e

i n e i t he r r epea t i ng ( pe r i od i c ) o r non - repea t i ng ( non -pe r i od i c ) pa t t e r ns , w i t h

t he f o rme r be i ng t he dom inan t s t r a t egy due t o t he i r ease o f v i sua l i z a t i on and

cons t r uc t i on . I n bo th o f t hese s t r a t eg i es , t i l i ng i s ach i eved t h rough t he use o f a

l im i t ed numbe r o f p ro to t i l e s ( F igu res 1 , 2 ) .

Non -pe r i od i c t i l i ngs a re an o ld concep t , and no t pa r t i cu l a r l y i n t e res t i ng t oday f r om

a ma thema t i ca l po i n t o f v i ew. I n t he 1960 ’s a new concep t eme rged , ape r i od i c

t i l i ngs , t ha t we re h i gh l y o rde red bu t no t pe r i od i c . T i l e s used t o make t hese

ape r i od i c pa t t e r ns w i l l neve r f i t i n t o a pe r i od i c t i l i ng howeve r t hey a re a r r anged .

The f ac t t ha t t h i s i n t e res t i ng p rope r t y i s e ven poss ib l e was d i scove red by R .

Be rge r a s t uden t o f Hao Wang who was cons ide r i ng “Wang Dom inoes ” , o r squa res

w i t h co l o red edges ( F igu re 3 ) .

Be rge r ’s i n i t i a l so l u t i on , howeve r, had ove r 10 ,000 t i l e s and i s l i t t l e mo re t han a

t heo re t i ca l cu r i os i t y. The numbe r o f ape r i od i c t i l e s was r educed by Be rge r h imse l f

[ 104 t i l e s ] , Dona ld Knu th [ 92 t i l e s ] , and R . Rob inson [6 t i l e s ] . The mos t f amous

examp le , was f ound by Roge r Pen rose who b roke away f r om us i ng squa re t i l i ngs

t o cons ide r f i v e - f o l d s ymmet r y, and managed to f i nd an examp le w i t h j us t two t i l e s

( F igu re 4 , l e f t ) . I t i s s t i l l an open p rob l em whe the r one s i ng l e ape r i od i c t i l e e x i s t s ,

t he so -ca l l ed “E i ns te i n ” p rob l em (de r i ved f r om the Ge rman ph rase f o r ‘ one t i l e ’ ) ,

a l t hough an examp le was f ound r ecen t l y i f you a l l ow t i l e s w i t h mo re t han one pa r t

by Soco l a r and Tay l o r ( 2010 ) .

I f a co l l ec t i on o f shapes canno t t i l e pe r i od i ca l l y t he na tu r a l ques t i on a r i ses how

i t can be shown t ha t i t t i l e s a t a l l . The Pen rose t i l i ng , a l ong w i t h a l l t he ea r l y

e xamp les , used a p rocess ca l l ed subs t i t u t i on (Bonne r 2003 ; C romwe l l 2009 ) . Fo r

ou r pu rposes i n s t udy i ng h i ghe r-d imens iona l t i l i ngs , a d i f f e r en t me thod d i scove red

by t he Du tch ma thema t i c i an de B ru i j n i s pe rhaps o f i n t e res t ( 1981 ) . He showed t ha t

t he Pen rose t i l i ng can be cons t r uc ted as a p l ana r s l i ce o f a l a t t i ce o f h ype rcubes

i n 5 -d imens ions . Th i s me thod , ca l l ed t he Canon i ca l p ro j ec t i on me thod , i s qu i t e

gene ra l—an i n t e r sec t i ng f l a t p l ane i n any d imens ion o f space can be used to

gene ra te a 2D t i l i ng ( F igu re 4 , r i gh t ) , wh i l e an i n t e r sec t i ng cu r ved su r f ace wou ld

c rea te a 3D t i l i ng ( F igu re 5 ) .

3 Higher Dimensional Mathematics

Fo r t hose no t f am i l i a r w i t h f ou r-d imens iona l ( and h i ghe r ) ma thema t i cs , t hese

t heo re t i ca l d imens ions may be exp lo red abs t r ac t l y and compu ta t i ona l l y, e ven

i f v i sua l i z a t i on i s no t eas i l y ach i eved i n t h r ee -d imens ions . Wo r t h no t i ng i s t ha t

t he f ou r t h d imens ion o f a 4D ma thema t i ca l space i s o f t en t hough t t o be t ime .

Th i s i s ac tua l l y i nco r r ec t . I n t h r ee -d imens iona l Euc l i d i an space , a po i n t has t h ree

va r i ab l es : X , Y, and Z . I n t he f ou r t h d imens ion , wh i ch i s s t i l l ( i n t heo r y ) Euc l i d i an ,

form, geometry and complexity

Fig. 4

Fig. 5

Figure 1. Seikaiha pattern used for both historical

Japanese kimonos and ancient Chinese maps to

denote oceans (image from paperdemonjewelry.

wordpress.com)

Figure 2. Ceramic tile mosaic, Vakil mosque,

Shiraz, Iran, built in 1187 AH (photo by Richard

Henry, used with permission)

Figure 3. The current record holder for “Wang

Dominoes”, 13 tiles that can tile the plane

aperiodically (Kari 1996; Culik 1996)

Figure 4. On the left, 2D Penrose tiling with straight

edges (note that either of the tiles could also tile

periodically) On the right, a Canonical Projection

of a 5D hypercube lattice intersected with a flat

plane to create a similar 2D Penrose tiling. (image

by authors)

Figure 5. Two Penrose tiles and a patch made from

them using a technique often employed by M.C.

Escher - No matter how the shapes are assembled

a periodic patch will never appear - This provokes

the (as yet unanswered) question as to whether

higher dimensional lattices of such complexity are

similarly possible and/or useful (image by authors)

310

integration through computationacadia 2011 _proceedings

a point has four var iables. This same rule holds true for even higher dimensions—a f i f th-

dimensional point has f ive var iables, a sixth-dimensional point has six, and so forth.

Despite the impossibi l i ty of stat ical ly visual iz ing objects with dimensions greater than

three, we can computat ional ly generate these abstract structures and visual ize var ious

2D intersections with planes, or 3D t i l ings on various surfaces. Mathematicians began

this process in the 19th century and soon considered their subject to be the study of

abstract ion i tself, rather than simply abstract ions direct ly made from the world. A simple

example is seen using the concept of volume. In one dimension this is the same as length.

So we have a l ine of length L. In two dimensions we have a square with an area of L2.

Continuing to three dimensions we now have a cube with volume L3. In higher-dimensional

space, we can simply continue this sequence. A 4D-cube, often cal led a tesseract, would

thus have volume ( i f one could cal l i t a volume) of L4, and so forth for higher dimensions.

To make four dimensional structures visible in three dimensions we use tr icks simi lar to

when we use 2D drawings (on paper or a computer screen) to visual ize 3D models that we

intuit ively understand. This process can be compared to the shadow that a 3D wire frame

object casts on a f lat surface. Just as the f lat shadow of a 3D object is not the object

i tself, the 3D “shadow” of a 4D object is not the object i tself, but merely one stat ic view

of i t (Figures 6, 7).

Whi le we cannot stat ical ly represent higher dimensional geometr ies in our three-

dimensional world with any degree of intuit ive understanding, descript ions of 4D objects

are often suggested rather successful ly through animations that show what appears to be

a transforming shape. However, the shape is not transforming; rather, i t is simply rotat ing

in this higher-dimensional space. As such, the length of one side (L) of a 4D object always

remains unchanged, even i f i t appears to be growing or shrinking in these rotat ional

animations. The result ing forms and t i l ings, from the project ions of hypercube latt ices in

higher dimensions are cal led zonotopes (Ziegler 1995; Eppstein 1996; Towle 1996)

4 Architectural Potent ia ls of 4D Tessel lat ions

Given the potential for 4D objects to tessel late, along with the fact that wireframe

project ions of 4D objects onto a surface are possible, we ask the question whether three-

dimensional surfaces could, i f intersected with four-dimensional objects, provide useful or

meaningful results for architectural purposes.

We began our investigat ions with a simple system using Mathematica by Wolfram

Research. We consider a latt ice of cubes in some higher-dimensional Eucl idean space.

Figure 6. Projection of a surface of 4d cubes to a

2d plane (image by authors)

Figure 7 . Shadows of 3D cubes (on right) and

a “shadow” of a 4D cube (on left) - (image from

decoder.moy.su)

Fig. 7

Fig. 6

311

Within this space we consider a three dimensional subspace. Within this 3D space we

can place any object we want, from a simple plane to a highly complex NURBS surface.

To i l lustrate, in the famil iar world of three-dimensions we know we can approximate a 3D

sphere or 2D plane composed of 3D cubes (Figure 8).

Now consider things from the point of view of a computer. I t has no spatial understanding

beyond what we can describe. In order to make the above 2D pictures, the computer

looked at 3D points described by three integers, connects the points into cubes in a t i led

fashion, then cul ls the data that was behind the reference geometr ies (the sphere and the

plane) so we don’t see these cubes. Abstract ly, we can simply do the same thing with

points described by four (or more) integers. Accordingly, we can also consider a sphere

in four-dimensional space in much the same way we can think about a circle in three-

dimensional space.

We do have to be careful about some things, though. Think again about this circle. For the

purposes of visual izat ion, let ’s cal l this circle a “r ing”, simi lar to a piece of jewelry. I f this

r ing were drawn in a 2D plan view we acknowledge that the r ing has an “ inside” and an

“outside”, yet we can easi ly understand it as a 3D object. I f we l ived on a theoret ical two-

dimensional planet (one that lacked a third dimension), to go from a point inside the r ing to

a point outside the r ing, one must cross the boundary between inside and outside. I f this

r ing exists in 3D space (such as on our current planet), one can move from the inside to

the outside of the r ing—like a f inger that can be inserted into the r ing, or be moved to the

outside of the r ing—without having to physical ly penetrate the r ing i tself (or the boundary).

By extension, the same holds true of a sphere in 4D. Though we cannot picture 4D space,

a computer can understand 4D space as easi ly as i t does 3D space simply based on the

number of integers for a point. Thus, i f a sphere is described in 4D space, one could, in

theory, move from the inside to the outside without crossing the boundary in the same

fashion as our previous example with a r ing in 3D space.

Assuming this makes some sense, imagine a computer with a 4D model in i ts memory.

We now need to take that back to 3D so we can actual ly use it. To do this, as described

above, we can simply project a 4D object into three dimensions shown below (Figure 9).

You notice that the edges of the paral lelograms now go in four dif ferent direct ions. These

are the project ions of the edges of a 4D cube into 3D. Each pair of direct ions gives a

paral lelogram. In this case there is actual ly only one paral lelogram as the four direct ions

are symmetr ical, but i f the edge lengths of the object were not uniform, we might have

twelve dif ferent paral lelograms in the system.

In fact we could choose any four directions in space and then use this method to f ind an

approximation of a surface using those directions and the paral lelograms they generate.

We do not even need to stop at 4, we can choose any number of directions. The number of

dimensions that we need to use is the same as the number of directions (Figures 10, 11).

The process for creating useful data in 3D is worth describing for those interested in

experimentation. The projected 3D geometries created in Mathematica may be exported

as STL f i les, then imported into one’s desired modeling environment for manipulation.

Below is shown a 17D surface projected to a 3D plane in which we used both Maya and

Rhinoceros (Figure 12). Rhinoceros provided useful diagnostics, indicating a val id mesh

and that the geometry did not have any: degenerate faces, zero edge lengths, duplicate

faces, disjoint pieces, or unused vert ices. Some attr ibutes are worth noting, the most

signif icant of which are some non-uniform face normals, signif icant differences between

face and vertex normals, and a mix of both surfaces and sol ids. These issues obviously

may require edit ing, depending upon one’s goals. For these, Maya proved convenient for

repair.

5 Architectural Potent ia ls

The abstract form we are considering is inf luencing architecture in several ways. As an

example, Foster and Partners’ Swiss Re building (dubbed, “The Gherkin”) in London uses

an approximation of a zonohedron to give its shape. A more direct example is Olafur

El iasson’s facade for Harpa, the new concert hal l in Reykjavik, Iceland. Although this does

form, geometry and complexity

Figure 8. 3D sphere (left) and 2D plane (right)

approximated by 3D cubes (image by authors)

Figure 9. A sphere (left) and plane (right)

approximated in a 4 dimensional space and then

projected to 3D (image by authors)

Figure 10. Wavy surface approximated in 7D and

projected to a 3D space (image by authors)

Figure 11. Negatively curved surfaces

approximated in 7D (left) and 15D (right), projected

to 3D (image by authors)

Figure 12. A 17D surface projected to 3D plane

(image by authors)

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

312

integration through computationacadia 2011 _proceedings

not consider dimensions higher than 3, it uses a periodic t i l ing of three dimensions by a

single shape. Intersections of this t i l ing with a plane create the side walls (Figure 13).

6 Conclusions

I f theoret ical mathematical explorat ions in mult i-dimensional space have been taking place

for at least a century, the obvious question is why has this work not received greater

attent ion in architectural venues? We propose two possible explanations, both of which

are no longer the l imitat ions they once were due to the inherent potentials of current

computat ional tools.

First, is the chal lenge of visual izat ion. Early mathematical work in mult i-dimensional space

involved cryptic and approximated drawings of stat ic rotat ions of higher-dimensional

objects. These were accompanied by complex notat ions which could only be readi ly

understood by mathematical ly advanced part ies. I terat ive drawings of mult i-dimensional

objects were both laborious and relat ively ineffect ive at communicating objects in mult i-

dimensional spaces, thus, they were rarely done. Today, a century later, computat ional

programs which permit robust explorat ions of mult i-dimensional space are now avai lable

to broad audiences outside the f ield of higher mathematics. These programs—when

combined with computat ional ly animated rotat ions of mult i-dimensional objects—allow

users the abi l i ty to better understand these non-intuit ive mult i-dimensional objects.

The second l imitat ion was the chal lenge of real izat ion. With manual means of production

defining most of the work product of the past century, physical ly achieving the complex

and non-intuit ive potentials of higher dimensional objects was even more chal lenging

than visual iz ing them. Quite simply, their ordering systems were dif f icult to describe,

let alone bui ld in physical space. Today, digital fabricat ion techniques permit the direct

use of computat ional data in numerous fabricat ion processes through a wide variety of

CNC (Computer Numeric Control led) equipment. CNC processes much more easi ly and

accurately enables the physical creat ion of mult i-dimensional objects intersected with

three-dimensional objects.

While the most obvious potential for these types of inquir ies is for surface assemblies and

pattern-making, there also appears to exist potentials for novel ordering and structural

systems which chal lenge Eucl idian norms. These require greater explorat ion. Given the

early nature of this act ive research project—along with the need to communicate stat ic

visual information that is non-intuit ive—the examples i l lustrated are somewhat simpl ist ic.

Our current trajectory is dedicated to explor ing signif icantly greater complexity. Further

work may be seen at www.designandmath.com. Part ies interested in further detai ls to

assist in their own experimentat ion are invited to contact the authors.

Fig. 13

Figure 13. Harpa Concert Hall in Reykjavik, by

Olafur Eliasson (image by authors)

313

References

Bonner, J . 2003. Three t rad i t ions o f se l f -s im i la r i t y in four teenth and f i f teenth centu r y

Is lamic geomet r ic o rnament . Ed. R. Sarhang i and N. F r iedman. Proceed ings o f ISAMA :

B r idges: Mathemat ica l Connect ions in Ar t , Mus ic and Sc ience. 1-12. Granada.

de Bru i jn , N. G. 1981. A lgebra ic theor y o f Penrose ’s nonper iod ic t i l i ngs o f the p lane,

I , I I , I n Neder l . Akad. Wetensch. Indag. Math . 43:39-52, 53-66.

Cromwel l , P. R. 2009. The search fo r quas i -per iod ic i t y in I s lamic 5- fo ld o rnament .

The Mathemat ica l In te l l igencer 31 (1 ) : 36-56.

Cu l i k , K. I I , e t a l . 1996. An aper iod ic se t o f 13 wang t i l es . Discre te Mathemat ics 160

(1-3 ) : 245-251.

Eppste in , D. 1996. Zonohedra and zonotopes. Mathemat ica in Educat ion and

Research 5 : 15-21.

Kar i , J . 1996. A sma l l aper iod ic se t o f wang t i l es . D iscre te Mathemat ics 160 (1-3 ) :

259-264.

Soco la r, J . E . S. and J . M. Tay lo r. 2010. An aper iod ic hexagona l t i l e . arX iv P repr in t

arX iv :1003.4279.

Towle , R. , R. E . Maeder, D. B. Wagner, H. Mur re l l , I . Va rd i , L . G. Hector J r. , K . B.

L ipper t , and J . M. F r idy. 1996. Po la r zonohedra . Mathemat ica Journa l 6 :8-17.

Z ieg le r, G. M. 1995. Lectu res on po ly topes. In Graduate Tex ts in Mathemat ics .

7 :198-208 rev ised, i l l us t ra ted, repr in t ed. New York : Spr inger.

form, geometry and complexity