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hapterolyhedraolyhedral


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olyhedra andolyhedral SurfacesPolyhedra and polyhedral surfaces are shapes bounded by planar faces. They arefundamental to many modeling purposes and are often found in architecture Figure3.1).Actually, most architecture features polyhedral surfaces because planar partsare easier to build than curved ones. We star t our discussion with classical polyhedrasuch as pyramids and prisms that can be generated by extrusion of a polygon. To

Fig. 3.1polyhedra and polyhedral surfaces narchitecture.Left) The Seattle Public Library 1998-2004) by Rern Koolhaas and JoshuaRarnus.Right) Part of the glass roof of theDubai Festival Centre 2003-2007)by lerde and HOK image courtesy ofWaagner-Biro Stahlbau AG .

better understand why only a certain limited number ofpolyhedra can be realizedwith congruent regular faces, we study Platonic and Archimedean solids and some oftheir properties. Several of the Archimedean solids can be generated by appropriatelycutting off vertex pyramids of Platonic solids. One of the Platonic solids, theicosahedron, is also the polyhedron from which geodesic spheres are usually deduced.Parts ofgeodesic spheres are well known in architecture as geodesic domes. Geodesicspheres derived from an icosahedron consfst of triangular faces only. Polyhedralsurfaces are of great recent interes tin architecture for the realization of freeformshapes e.g., as steel-and-glass structures with planar glass panels.)


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Polyh edra an d Polyhe dra l Surfaces . Apolyhedron is a 3D sha pe chat consists ofplanarfices, straight edges an d vertices. Eac h edge is shared by exactly tw o faces an d aeach v e r t e w leas t three edges and three faces meet Figure 3.2 le f t) . The bou ndedvolume enclosed by the polyh edron is sometim es considered par t of the polyhedron.,4polyhedraf s;rrJI?ce s a unio n o f finitely many pl ana r polyg ons again cal1edf;lcesj thacneed no t enclose a volume Figure 3.2 right). A polyh edral surface can have boundaryvertices o n bo undary edges. Th e latter o nly be long to on e single face.Before studying polyh edra in m ore detail, we n ote that the ter m polyhedron iscom mon ly used for solid and surface models of objects bo und ed by planar faces. Acareful distinction tends to result in comp licated formulatio ns whic h we want toavoid. The same holds fo r the termpolygon w hich we use for polygons an d polylines.

a)polyhedron edges

Fig. 3.2The geome tr ic en t i t ies of a c) The Sp i t telau Apartm ent Housesa) polyhedron and a 2004 -2005 ) in Vienna by Zaha Hadid.b) polyhedral surface are its faces, d) The Booster Pump Stat ion Eastedges, and vertices. 2003 -2005 ) in Amsterdam East byBekkering and Adams.

polyhedr l surf ce


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yramids and rismsPyramids. One prominent type ofpolyhedron used in architecture is thepyramid.The base of an Egyptian pyramid is usually a square, and its other four faces aretriangles. general pyramid consists of a base polygon in a plane hat is connectedby triangular faces to the apex v not lying in Figure 3.3 . Thus, the mantle M of apyramid is a polyhedral surface with triangular faces. We obtain apyramidalfistumby cutting the pyramid with a planeEparalle1 toP.The pillar of an obelisk is oneexample of a pyramidal frustum Figure 3.3 .

Fig. 3 . 3pyramid consists of a base polygon B we get a pyramidal frustum. If wethat is connected by triangular faces to put a pyramid on top of a pyramidalthe apex v If we cut the pyramid with frustum we have an obelisk.

plane E parallel to the base plane P

pyr mid pyr mid l frustum obelisk


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Prisms. prisnl is a polyhedron whose b ott om and top face arc translated copies ofeach ot her in parallel planes Figure 3.4). The vertices of the bo tto m an d the top faceare connected by parallel straight line segments. Thus th e side faces of a p i s m areparallelograms an d the ma ntle form ed by the m is a polyh edral surface. If the top face isobtained by a trans lat ion or th ogo ~lalo the plane co ntaining the bo ttom face we havethe special case of a rightprism all side faces are rectangles). A special right prism isthe cuboid previously encountered in C hap ter 1

Example:M o d e l i n g p y r a m id s a n d p r i s m s u s i n gextrusion . Special pyramids an d prismsare often included as basic shapes inC A D systems. To mod el a pyramid orprism w ith a general basepolyg on B weuse the extrusion comm and Figure 3.4).We draw the polygon B in a p lane e g ,the xy-plane) and extrude it. A parallel

p r llel extrusion

prism right prism

extrusion takes the profile polygon B andextrudes it along parallel straight lines.If the extrusion direction is orthogonalto the reference plane we generate aright prism. Otherwise if the extrusiondirection is not parallel to the referenceplane) we generate a prism. A centralextrusion takes the profile polygon B in

and extrudes it along the connectinglines with a chosen extrusion centerFigure 3.4). hus a central extrusion

generates a pyramid or pyramidalf rus tum whose shape depends on B andon the posi tion o f the apex u Examplesof pyramids a nd prisms in architectureare shown in Figures 3.5 an d 3 6

Fig. 3 4Prisms and pyram ids can be generatedby parallel and central extrusion o aplanar polygon.centr l extrusion\pyr mids


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nter

Fig. 3.5pyramids in architecture:a) The ancient Egyptian pyramids ofGiza around 2500 B.C.).b) The main pyramid in front of theLouvre 1989) in Paris by I.M. Pei: Itsbase square has a side length of 35meters and it reaches a height of 20.6meters.c) The Transamerica pyramid 1969-72 in San Franciso by Wil liam Pereirais four-sided slim pyramid witha height of 260 meters. t has twowings on opposing sides containing anelevator shaft and a staircase.d) The Taipei 101 1999-2004) inTaipei by C. Y. Lee. The 508 metershigh building features upside down

pyramidal frustums.


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Fig. 3.6Prisms in architecture:a) Castel delM onte i n Bari by the Holy

Roman Emperor Frederick I around1240).b) The Jewish Museum Berlin 1998-

2001) by Daniel Libeskind.


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latonic Solids

convex sets A set is convex if it contains with any pair ofiu points also the straight line segment connecting themFigure3.7 .A polygon o r a polyhed ron is convex if it is the

boundary of a convex set.

Convex and non-convex domains in D

A cube is a special case o f a cub oid w hose faces are congruen t squares. Geometrically acube is a polyhedron with 6 square faces, 12edges and 8 vertices. In each vertex of thecube thr ee squares meet. All dihedr al angles i.e., the angles between faces that meet in acom mon edge) are equal to 9 degrees. A cube has several symmetries tha t can be usedto reflect nd rotate it see Chap ter 6 o that it is always transformed on to itself. A cubealso has the pro pert y to be a convex polyhedron.

A regularpolygon has its vertices equally spaced on a circle such that the n on-overlapping edges are of equal len gth. Examples of regular polygons are the equilateraltriangle, the square, and the regular pe ntagon . All regular polygons are convex.Pyram ids and prisms w hose base is a regular polygo n are convex polyhedra. If the basepolygon is non-convex the pyramid or prism is a non-convex polyhedron.

con vex non convex convex non convex


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Now we devote ourselves to the following question: Are there any convex polyhedraother than the cube with congruent regular polygonal faces such that the same numberof faces meet at each vertex? The answer is yes and it can be shown thac there are fivesuch polyhedral solids. They are called the latonic solids (Figure 3.8 ,named afcerthe ancient Greek philosopher Plato (circa 428-348 BC). The fiv Platonic solids areknown as the tetrahedron, the hexahedron (cube), the octahedron , the dodecahedron,and the icosahedron. The prefix in the Greek name of a Platonic solid tells us thenumber of faces thac make up the polyhedron: cettares means four, hex six, okto eight,dodeka twelve, and eikosi twenty. Because other polyhedra with the same number offaces commonly carry the same name, the Platonic solids are ofcen distinguished byadding the word regular to the name. n his section a ,,tetrahedrona always means aregular tetrahedron and the same applies to the other Platonic solids. Let us begin withthe construction of paper models of the Platonic solids.

Paper models of the Platonic solids. We can construct paper models of the Platonicsolids by first arranging their faces in a plane as shown in Figure 3.8. These ,,unfoldedpolyhedra are cut out of paper or cardboard and then folded and glued together alongthe edges. For a tetrahedron, we need four equilateral triangles, for a cube, six squares,for an octahedron, eight equilateral triangles, for a dodecahedron, twelve regularpentagons, and for an icosahedron, twenty equilateral triangles. O f course, there aredifferent possibilities in arranging the faces of each l'latonic solid on a cut-out sheet.

tetr hedron

cube

icos hedron

Fig 3 8The five Platonic solids are thetetrahedron cube octahedrondodecahed ron and icosahedron. Wecan bui ld them out of paper using theshown cut-ou t sheets.

oct hedron

dodec hedron


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We

Fig 3.9The vertex pyramid of the polyhedronvertex v is formed by all edges andfaces joining in v We also show theunfolded vertex pyramids.

Ho w to find the five Platonic so lids? A con-vex polyhedron is a Platonic solid if thefollowing criteria hold.

All faces are congruent regular polygons.At each vertex, the same number of faces meet.

Three is the minimum number of polygons and edges) that have to meet in a vertex vof a polyhedron. All faces and edges that meet in v form the vertexpyramid Note thatby vertex pyramid we mean the mantle of a pyramid and this pyramid may have a io n-planar base polygon Figure 3.9). A polyhedron vertex is convex if the sum of anglesbetween consecutive edges is less than 360 degrees. This is clearly the case for the cubeFigure 3.10) because we get 3-90 270 degrees for the three squares joining in each

vertex. Ifwe put together four squares the angle is 4.90 360 degrees and all foursquares are contained in a plane. Therefore, we do not get a spatial polyhedron. Five ormore squares will not work either. Thus, the cube is the only Platonic solid made up ofsquares. Let us now use equilateral triangles instead of squares.

Three equilateral triangles meeting in a vertex are allowed 3.60 180degrees). We get the tetrahedron by adding a fourth equilateral triangle asa pyramid base Figure 3.10). The resulting polyhedron has four faces, fourvertices, and six edges. It fulfils the previously cited criteria and is thus aPlatonic solid.Four equilateral triangles meeting in a corner are also allowed 4.60 240degrees). We obtain the octahedron simply by gluing together two suchvertex pyramids along their base squares Fig 10). Again both criteria citedpreviously are hlfilled and we have derived another Platonic solid. It has 8faces, vertices, and 12 edges.

vertex pyr midscon vex

vertex pyr mids unfolded

fl t s ddle sh ~e d


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ixequilateral triangles meeting in a vertex form a planar object (6.60 360 degrees),and more than six cannot form a convex vertex anymore. Thus, we have derived allPlaton ic solids whose faces are mad e up o f equilateral triangles or squares.

icos hedron

Five equilateral triangles me etin g in a cor ner are the m ost we can achieve(5.60 300 degrees). But now we are no t allowed to simply glue two suchvertewpyramids along their pentago nal bases. The criterion wo uld n ot befulfilled for the base vertices. It turns o ut that we have t o conn ect tw o suchpyramids with a band of 10 additional equiiateral triangles (Figure 3.10j t oobtain a n icosahedron. No w bo th criteria hold again an d we have found afourth Platonic solid. It has 20 faces, 12 vertices, and 30 edges. Note thatactually all vertex pyramids of th e icosahedron are form ed by five equilateraltriangles and are congruent.

Fig 3 10How to derive the five Platonic solidsusing their vertex pyramids

dodec hedron


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The next regular planar polygon is the regular pentagon with five edges and an innerangle of 108 degrees. Because 3-108 324 < 360 degrees we can form a convcx cornerout of three congruent pentagons. Then the dodecahedron is obtained by connectingfour such caps consisting of th ~e ec on ~i ue nr :entagons Figure 3.10). Both criteria arefi~lfil led nd we have derived the fifch Platonic solid. It has 2 faces, 20 vertices, and 30edges. Ifwe pu t four pentagons together, we have 4.108 = 432 > 360 degrees which nolonger yields a convex corner.Are there Platonic solids with regular polygonal faces other than triangles, squares, orpentagons? The answer is no. The following explains why this is the case. For a regularhexagon the face angle is 120 degrees and thus three hexagons meeting in a corner arealready planar 3.120 = 360). Since the face angles of the regular 7-gon, 8-gon, and so onare becoming larger and larger we can no longer form convex vertexpyramids anymore.

Solids Platonic solids in high r dimensions The mathematicianSchlafli proved in 1852 that there are six polyhedra thatI fulfill the properties of the Platonic solids in 4D space. InI spaces of dimension n = 5 and higher there are always onlythree such polyhedra. The three Platonic solids that exist inany dimension are the hypocube n-dimensional cube), thesimplex n-dimensional tetrahedron), a nd the crosspolytope

Fig. 3 11Photo of a 3D image of the 600-cell atTU Vienna.

n-dimensionaloctahedron).In dimension 3 we additionallyhave the dodecahedron a nd the icosahedron. In dimension4 there are rhe 24-cell he 120-cell and the 600-cell.We are

ll used t o seeing2D images of 3D objects. Similarly we cancreate 3D images of 4 D objects. Such a 3D image of a 600-cell is shown in Figure 3.1 1.


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Convex polyhedra whose faces are equi latera l triangles. Note that there are actuallyeight different convex polyhedra consisting of equilateral triangles. Three of them arethe Platonic solids tetrahedron, octahedron, and icosahedron. The other re nolonger as regular and have 6, 10, 12, 14 and 16 equilateral triangular faces Figure3.12) . They are obtained as follows. Gluing together two tetrahedra gives the first newobject. Gluing together two pentagonal pyramids generates the second one.Splitting a tetrahedron into two wedges and stitching them together with a band ofeight equilateral triangles yields the third one. Attaching three square pyramids to theside faces of a triangular prism generates the fourth. Finally, by attaching two squarepyramids to a band of eight equilateral triangles we obtain the fifth one. Note that allfaces that are glued together are then removed from the generated polyhedron.

Fig 3.12tetrahedron The eight different convex polyhedrathat are made up of equilateraltriangles Only three of them arePlatonic solids

_ y

octahedron

icosahedron


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ropertiesoflatonic SolidsThe faces of a Platonic solid are congruent equilateral triangles (tetrahedron,octahedron, icosahedron), or congruent squares (cube), or congruent regularpentagons (dodecahedron). Table 3.1 summarizes the number of faces fl,henumber ofvertices v ), and the number of edges e that form each Platonic solid.Table 3.1 Number of Faces, Vertices, and Edges Associated with Platonic Solids

Platonic sol idTetrahedron 1 4 1 4 1 6Cube 1 6 1 8 1 1 2

.The Euler formula It is easy to verify that for the five Platonic solids the number of

OctahedronDodecahedronIcosahedron

vertices minus the number of edges plus the number of faces s always equal to 2:

This polyhedral formula, derived by the mathematician Leonhard Euler (1707-1783),actually holds for all polyhedra wi thout holes. We verify it for the pyramid with a

81220

square base: v 8 = In Chapter 14 we will learn more about theFig. 3 13 Euler formula and other so-calledtopologic lpropertiesof geometric shapes.Platonic solids and their duals. Thetetrahedron is self-dual. The cube andthe octahedron are duals of each other.The same holds for the icosahedronand the dodecahedron.

62012

123030


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Platonic solids a nd thei r duals. The face midpoints of each Platonic solid are thevertices of another Platonic solid, called the solid s utrrzl (Figure 3.13). Let us firstderive the dual o f a tetrahedron. Each of the four vertex pyramids consists of threeequilateral triangles. Because oft he symmetry of th e tetrahed ron the three facemidpoint s of each vertex pyramid form another equilateral triangle. Thus, we againobtain a convex polyhedron consisting of four equilateral triangles. Thus the dual of atetrahed ron is again a (smaller) tetrahedron contained in th e original one.Let us now derive the dual of a cube. For each vertex pyramid of the cube (consistingof three congru ent squares) we connect the face midpoints by an equilateral triangle.Thus each vertex of the cube gives rise to an equilateral triangle (a face of t he dualpolyhedro n), and each face of the cube yields a vertex of the dua l (which is the facemidpoint of the square). Hence the dual of a cube consists of eight equilateral trianglesthat form an octahedron. Obviously the number of vertices of a Platonic solidcorresponds to the number of faces of the dual and vice versa. The number of edges isthe same for a Platonic solid and its dual (see Table 3.1). Not e that the dodecah edronand the icosahedron are duals of each other.

Spheres associated wi th P lato nic solids. There are three spheres with t he same centernaturally associated with each Platonic solid (Figure 3.14). O ne sp here contains allvertices (th e circumsphere), the second touches all faces in their face-midpoints (theinsphere), and the thir d touches t he edges at their edge-midpoints.

Symm etry prop erti es. The vertex pyramids pf a single Platonic solid are congruentwith one another. By construction, ll faces are congruent regular polygons an d thusll edges have the same length. This means that if we want to co nstruct a Platonic solid

we have the following advantages.We only need o ne type o f face.ll edges have the same length.

The dihedral angles between neighbouring faces are equal.ll vertex pyramids are congruent . Fig. 3 14Each Platonic solid has three associatedspheres with the same center. Herewe show the circumspheres and theinspheres.


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The olden SectionThe g old en s ectio n. Thegolden section (also known asgolden ratid or divineproportion) is the nu mber

p = ( 1 + 6 ) / 2 - 1.618033989We o btain the exact golden section if we divide a line segment in to two parts (a largerone of length c, and a smaller one of le ngth d o that the following holds (Figure3.15): the ratio of c and d is the same as the ratio of c d an d c. Stated anot her way,larger to smaller as total to larger. Formally, we write

c : d = ( c + d ) : c .E w e use a new variable p = c/d we have p = 1 1/ p which leads to the quadraticequation p2- p 1 = 0. The positive solution (1+ 6 ) / 2 i s the golden sec t ion .Interestingly, the gold en section can be appro ximat ed by the ratio o f two successivenumbers in the so-cal led Fibonacciser ies of numbers 1, 1, 2, 3, 5 8 , 13 ,21,3 4,Although 3:2 = 1.5 is a rough approximation of p, 5 3= 1.666 is already a little bitbetter . Co ntin uing in the same fashion we obtain better approxim ations. For example,t h e v al ue 1 3 4 = 1.625 already approximates the golden rat io within 1% accuracy.

' Ihe gol den rectangle . T he dimensions o f a golden rectangle are always in the goldenratio p : 1.To co nstru ct a golden rectangle, we start with a square o f side length c. Asshown in Figure 3.15 we obtain a largergolden rectangle with dimensions (c d ) :an d a smaller one with dimensions c :d.

Ihe Fibo nac ci spiral. L et us contin ue the above constru ction as follows. We dividethe smaller rectangle in to a square of side length d and an other go lden reccangle ofdimensions d c d . Ifw e con tinue in the same fashion the result is a whirling squarediagram. By connecting op posite corners of the squares with qu arter circles we obtainthe Fibonacci spira l (Figure 3.1 5).


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History:The golden section in art and architecture. In art thegolden section has been found in numerous ancient Greeksculptures-including those by Phidias (fifch century BC )whose name motivated the choice o fp as the symbol for thegolden section. The golden section also appears in paintingsranging from Leonardo da Vinci s Mona Lisa (1503) toMondrian s Composition with Red M o w and Blue (192 1 .

golden section golden rectangle

.> , ;::: ;.? . ?< 2 .*? . ......;l:. : : -< -:..*... --*2'.> :' I..iec .;,.>-.:. ; . ,

= - . , ,-.. . ^c/Z c/Z ::+:. d i ;.

In architecture it is argued that the golden section w sused in the Cheops pyramid (around 2590-2470 B.C.), nthe Parthenon temple (447-432 B.C.) in Athens, in thePantheon (118-125) in Rome, in various triumphal arches,or in the facade of the Notre Dame cathedral (1 163-1345)in Paris. In the twentieth century, Le Corbusier developedthe golden-section-based modular system for architecturalproportions and applied it in his famous building Unit;dJHabitation (1952) in Marseille (Figure 3.16).

Fig. 3.15The golden section, the goldenrectangle, and the golden spiral.

Fibonacci spiral

Fig. 3.16Unite dfHabitation (1952) byLe Corbusier.


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wa,in

345)

niti

a ) Modeling a tetrahedron by cuttingit out of a cu be.b) Modeling an octahedron bychoosing its six vertices as the unitpoints on the axes of a C artesian

system: 1,0,0), 0,1,0), -1,0,0),O,-l,O), O,O,l), O,O,-l).

Mod eling th e Pla tonic sol ids . C A D s o h a r e ohen provides Pla tonic solids asfunda men tal shapes. Ift hi s is not the case all five Platonic solids can also be modeled inthe following way. The m ost diff icultpar t here is the construct n of an icosahedron andits dual, the dodecahedron-which exhibits a beautiful relation to the golden ratio.

To mo del a cubc, we can use parallel extrusion of a square with edge le ngth sin a direction or thogo nal to its supporting plane to a heights.To m odel a tetrah edron , we cut it out of a cube as shown in F igure 3.17. Tneplanar cuts g enerate a polyh edron whose six edges are diagonals of the cubesfaces which are congrue nt squares. Thus, the n ew edges are of equal length.The four faces of the new p olyhed ron are congruen t equilateral triangles.Hen ce the polyhe dron is a tetrahedron.To m odel a n octah edron (Figure 3.17), we select in six vertices as unit pointso n [he axes of a Cartes ian system: (1,0,0) , (0,1,0), (-1,0,0), (0,-1,0), (0,0,1),(0,0,-1). Alternatively we could also mode l it as the du al of a cube.

pl nes 0

cube tetr hedron


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TOmodel an icosahedron we use three congruent golden rectangles which.we position mutually orthogonal with coinciding centers in the origin ass h o w ~ nigure 3.18. The twelve vertices of the three golden recrangles haveCartesian coordinates kp,?l,O), 0, kp ,? l), +1,0, +p and are the vertices ofan icosahedron ofedge lengths 2.We model a dodecahedron as the dual of an icosahedron Figure 3.18 ).The twenty face midpoints of the icosahedron are the vertices of thedodecahedron.

The cube is the most widely used Platonic solid in architecture. On e unorthodox use isillustrated in Figure 3.19a. The tetrahedron also finds its way into ar ch ite c~u reFigure3.19b). In material following we will learn how geodesic spheres can be derived froman icosahedron.

z zicosahedron dodecahedron j

icosahedron with side length s=

Fig. 3.18Left) Modeling an icosahedron of edge

length s 2: The vertices of threecongruent golden rectangles of width2 and length 2qdefine the twelvevertices of an icosahedron.Right) Modeling a dodecahedron as

the dual of an icosahedron.

Fig. 3.19a) The Cube Houses 1984) in

Rotterdam y Piet Blorn feature cubeswith one vertical diagonal resting onprisms with a hexagonal base polygon.b) The r t Tower 1990) in Mito yArata Isozaki can be modeled as astack of tetrahedra.


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a

rchimedean SolidsArchimedean solids are convex polyhedra that are consisting of two or more types ofregular polygons so that all vertex pyramids are congruent. These special polyhedrawere known to the ancient Greek genius Archimedes more than 2000 years ago. As inthe case of Platonic solids, all edge lenghts are equal in an Archimedean solid. Thus,the real difference is that more than one type ofplanar face appears. Each face still hasto be a regular polygon but no t all of them have to be congruent.

Corner cutting of Platonic solids By cutt ing off the vertices of a Platonic solid wecan generate some of the Archimedean solids. For a better understanding ofpossiblecucs we first discuss corner cu tting along straight lines) for regular polygons. For eachregular polygon we can perform two different corner cuts so that we again obtain aregular polygon Figure 3.20 :.

Type 1:Cuts tha t generate a regular polygon with the same number of edges.Type 2: Cuts that generate a regular polygon with twice as many edges.

Fig. 3.20 corner cuts typeThe two types of corner cuts of regularpolygons that generate another regularpolygon. We illus trate th e cuts for t heequ~lateral r~an gle, he square, andthe regular penta gon.

corner cuts type 2


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The first type of cut passes throug h the edge m idpoin ts, and the second type of cu t hast o b e ~ e r f o r r n e dn such a way that the generated polygon with twice as many edges isagain regular. \Ve can perform these two types o f corner cuts for a Platonicso lid in ananalogous way and thus generate Archim edean solids. In we do so we cut off parts ofthe vertex pyramids.

C o r n e r c u t s o f t y p e 1 Let us start with cuts throug h the edge mid points Figure 3.21 .For a te t rahedron we chop o f our smaller regular te t rahedra and whatremains is an octahe dron, Thus, we again have a Platonic solid and n ot a newtype of polyhedron.For a cube, we obtain t he so-called cuboctahedron consisting of six congruen tsquares which remain from t he six faces of the cube) and eight congru enttriangles which remain from the eight corners of the cube).Ifwe cut off the corners of an octahedron, we obta in a polyhedronwhose faces are eight con gruen t equilateral triangles one for each ofthe octagon faces) and six congrue nt squares one replacing each cornerof the octahed ron). This is again a cuboctahe dron. Indeed , by cornercutting throug h the edge midp oints dua l Platonic solids generate thesame polyhedron. This also holds for the dod ecahed ron and its dual, theicosahedron.W ith planar cuts through the edge midpoints of a dodecahedron we obta ina polyhedron wh ose faces are 12 congr uent regular pentagons one foreach of the twelve faces of the dode cahed ron) a nd 2 0 congruen t equilateraltriangles one for each of the 20 vertices of the dodecah edron) . The derived~ o l ~ h e d r o ns known as the icosidodecahedron ts name reveals the fact thatit can also be generated from an icosahedron via corner cuttin g thro ugh theedge midp oints Figure 3,21).

tetr hedron

cornercutting

oct hedron

Fig 3 21New polyhedra generated b y cornercut t ing of Platonic solids wi th cutsthrough the edge midpoints

cube oct hedron dodec hedron icos hedron

corner cornercutting

cuboct hedron

cutting

icosidodec hedron


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Fig 3 22From the 13 Arch imedean so l ids theseven named in th is f igure can begenerated from Platonic sol ids viacorner cutt ing The classic soccer bal lis one of them

orner cuts of type 2 Wi th corner cuts of the second type we can generate onefurther Archimedean solid for each of the five Placonic solids (Figurr 3.22). Figure3.20 indicates by means o f a single face how we have t o perform the necessary cllcsthat chop off parts of the vertex pyramids of each Platonic solid. Archimedean solidsgenerated by rruncacion are called truncated tetrai~e dron runcated cube truncatedoctahedron truncated dodecuhedron and truncated icosahedron.The trunca ted icosahedron is very likely the most recognized Archimedean solidbecause ic is represented in the classic shape of a soccer b all (Figure 3.22). The samepolyhedron was also named a buckyballby chemists, because it resembles the shape ofthe geodesic spheres ofBuckminister Fuller which we study in the next section. It canbe generated from the icosahedron by corner cutting so that we chop off 1 /3 of eachedge on both edge ends. We generate for each of the 12 vertices a regular pentagon,and fo r each of the 20 triangles of the icosahedron a regular hexagon. C orner cuttingis a fundamental idea in generating new shapes from existing ones. We encounter thisidea again in future chapters including Chapter (on freeform curves) and Chapter1 1 (on freeform surfaces).In total, there are 13 different Archimedean solids (Figure 3.22) other than certainprisms and anti-prisms (discussed in material following). Ihree of the Archimedeansolids even consist of three different types of regular polygonal faces.

trunc tedtrunc ted dodec hedronoct hedron


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Remark. Note-that the classical definiton of an Archimedean solid is also fulfilled bya 14th polyhedron. It was found 2000 years afier Archimedes by J C. P iMiller andVG Ashkinuze. This polyhedron is obtained by cutting a rhombicuboctahedron thepolyhedron marked by an n Figure 3.22) in ha[ rotating one part by 45 degrees, andthen gluing the two parts together.Some Archimedean solids inherently closely resemble the shape of a sphere, with thefurther advantage that they can be made of struts of equal length. From Platonic solidswe derive polyhedra that resemble the shape ofa sphere even better. These are thegeodesic spheres and spherical caps of them are called geodesic domes.

Example:Prisms and anti-prisms tha t are congruent regular polygons as bottomArchimede an solids. prism whose and top faces. The top face is a rotatedtop and bottom faces are congruent and translated copy of the bottom faceregular polygons and whose side faces and both are connected by a strip ofare squares is an Archimedean solid triangles. If we use equilateral trianglesFigure 3.23, top). Recall that for an we obtain polyhedra that fulfill the

Archimedean solid all edge lengths properties of an Archimedean solidhave to be equal. An anti prism has Figure 3.23, bot tom).

octahedron

cube prisms

anti prisms

Fig. 3 23Examples of prisms and anti prismsthat are Archimedean solids.


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eodesic SpheresAgeodesicsphere is a polyhedron with an alm ost spherical s tructure. The name isderived from the fact that

all vertices lie on a com mon sphere S an dcertain sequences ofvertices are arranged on great circles ofS.

The great circles of a sphere are the shortest paths that c onnec t two distinct points ona sphere. These shorte st paths o n a surface are calledgeodesics, and thus the p olyhedrawe stud y in chis section are called geodesic spheres. All faces of a geodesic sphere aretriangles. Howev er, not all of them are congruent. Geodesic dom es are those partsof geodesic spheres Figure 3.24) that are actually used in architec ture Figure 3.25).Built do mes range from sizes covering almost a full geodesic sphere to only half ageodesic sphere. The latter are called hemispherical dom es.

dome at the Milan Triennale-an interna tional exhibi tiondedicated to present innovative developments inarchitecture, design, crafts, and city planning. The 42-footpaperboard geodesic dome of Fuller gained worldwide

geodesic sphere

attention and Fuller won che Gran Premio. BuckminsterFuller built several domes including a large on e in M ontrealFigure 3.25) to house the U.S. Pavilion at Expo 1967 Fuller

also proposed enclosing midtow n Ma nhattan with a 2-mile-wide dome.

hemispheric l geodesic dome


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Fig. 3.25a) The Geodesic ome 1967) inMontreal by Buckminster Fullercomprises three-quarters of a geodesicsphere. The outer-hul l is made ou t oftriangles and is Jinked to the inner hullconsisting of hexagons.b) The Spaceship Earth geodesicdome 1982) in Orlando is almost fullgeodesic sphere.c) The esert ome 1999-2002) inOmaha is hemispherical.


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Fig. 3.26Subdividing a triangle into smallertriangles: By splitting each edge into2, 3, 4 equal segments we get 4 9 16smaller triangles. I n general, spli ttingeach edge into n qqual segments yieldsn2smaller triangles.

Fig. 3.27Projecting the edge midpoints of anicosahedron onto its circumsphereresults in a geodesic sphere.

icosahedron

Starting w ith Platonic solids we derive different form s of geodesic spheres by applyingthe following iterative process: we subdivid e each face in to .I iegular patte rn oftriangles and p roject t e 13r..v vertices on to the c ircumsphere a&the lnic solid.From our considerat ions 'latonic solids i t is evident that geodesic ~lo m es annot bebui l t wi th all tr iangular f~ c c s eing congruent. Zoweve r, we try to obtain o nly a smallnum ber of different faces.

Geo d es i c d o m es d e r iv ed f ro m a n i co sah ed ro n : Al te rn a t iv e 1 Because theicosah edron closely approximates its circumsphere it is often used as a sta rting pointfor deriving geodesic spheres. We sta rt by subdividing the triangular faces intosmaller triangles (Figure 3.26b). An equilateral triangle can be split in to four smallerequilateral triangles by adding the edge midpoin ts as new vertices. ?he n three newedges parallel to th e origina l edges are inserted.We d o th is subdiv is ion for each of the 2 0 congruent faces o f the icosahedron toobta in a total of 80 20.4 triangles. Now we project the newly inserted 30 vertices(the mid points o f the 30 edges of the icosahedron) radially from the center of theicosahedron o nt o i ts circumsphere (Figure 3.27). From the 8 0 triangles of thisgeodesic sphere, 20 are still equilateral. The rem aining 6 0 are only isosceles. Thus thisgeodesic sphere can be made up of two types of triangles (colored differently in Figure3.28, level 1 .

vide each

geodesic sphere

:

project new vertonto circumsphere


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Ifw e want to con struct a larger geodesic sphere, it is advantageous to have mor e triangles.Let us again derive such a geodesic sphere from an icosahedron. We subdivide eachedge of the icosahedraa i nto three equal parts and ob tain for each triangle nine smallertriangles (instead offour, as previously; Figure 3. 26 ~ ). hen we project the 80 newvertices (2 on each of the 30 edges of the icosahedron, and on e for the midpoint ~ f e a c hof the 20 faces) again radially on to the circumsphere of the icosahedron.This geodesic sphere has a total of 9 2 vertices, 20.9 = 180 triangles, and 27 0 edges(Figu re 3.28, level 2). It follows from the co nstr ucti on t ha t all triangles are isosceles.However, there are two different types of triangles: 60 cong ruent ones tha t form thevertex pyramids aroun d the 12 original vertices of the icosahedron a nd 120 congruen tones that form the 20 vertex pyramids arou nd the displaced m idpoin ts of theicosahe droni triangles.

I No te th at there are only three different edge lengths involved. If we subdivide eachtriangle of the icosahedron i nto 16 triangles (Figure 3.26 d), and th en project all newvertices on to the circumsphere, we ob tain a geodesic sphere at level 3with 20.16 = 320triangles (Figure 3 .28, level 3 . fw e keep subdividing the triangles of the icosahedron inthis fashion, we generate a geodesic sphere at level k wit h 2O .(k+l) triangles.

Geodesic domes derived from an icosahedron: Alternative 2 A n alternativeapproach in generating geodesic spheres also starts with an icosahedron. However, itrecursively splits each triangle of the geodesic sphe re at th e previous level into f oursmaller triangles and then projects th e new vertices ont o the circumsph ere (Figure3.29). The first subdivision level gives exactly th e same geodesic sphere w ith 2 0.4 =80 triangles as before. But the se cond a nd third level already produce geodesic sphereswith 20.16 = 320 , and 20 . 64 = 128 0 triangles respectively. Thus, by th e secon dsubdivision approach a geodesic sphere of level k has 2 0 .8 criangles. At level 1, bothalternative constructions return the same geodesic sphere. However, at higher levelswe obtain geometrically different results because the order i n which we perform thesubdivision a nd pr ojection steps is different.

.

.

Fig. 3.28Geodes ic spheres genera ted by b) At leve l 2 we have 180 isosce lessubd iv id ing th e t r iang les of a n t r ia n g les o f t wo t y p e s m a r k e d w i t hicosahedron and p ro jec t ing the new dif ferent colors.ver t ices: c ) At leve l 3 we have 320 t r iang les o fa ) At leve l 1 we h a v e 8 0 t r i a n g le s o f f i ve d i f fe ren t t ypes . On ly 20 o f themtwo types 20 equ ila te ra l, 60 isosceles are e qui lateral , a l l othe rs are isosceles.in d i f ferent colors) .

icosahedron geodesic sphere level geodesic sphere level 2 geodesic sphere level 3

.-; . ., . , . . ' , .;, .

, . . : :: . , ,? ,. .. . .,> . - .... ,-.


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Figure 3 30 shows a circular section and the different resulrs obta ined by changing theorder in which subdivision and projecrion are star ting with a single edge.Thus, although the number of triangles o fa geodesic sphere at level 2, 3, 4 , ...with thesecond approach is the same as the number of triangles o fa geodesic sphere at level3, 7, 15, generated wit h the first approach, the geometiy of the resulting objec:s isslightly different. By varying the generation process (subdivision and projection) wecan obta in even othe r variants of geodesic spheres.

Remark Note that a geodesic sphere still contains the 12 vertices of rhe icosahedronfrom which it is constructed. These 1 2 vertex pyramids consist ofon ly 5 triangles.All other vertex pyramids o f a geodesic sphere consist of 6 triangles which yields anatural relation t o hexagons that can be formed around those vertices. Recent projects(such as Eden by Grimshaw and partners) are revisiting large spherical roofs using ahexagonal pattern (see Chapt er 11 for details).

Fig. 3.29Geodesic spheres ge nerate d wit h thesecond subdivision alternat ive I n eachstep the t r iangles of the prev ious stepare spl i t into four sm aller tr iangles a ndthe new vertices are projec ted againradially onto the c ircums phere o f theicosahedron.

icos hedron geodesic sphere level geodesic sphere level2 geodesic sphere level 3

Fig 3.30The order c?f subd ivision and pro jec tionsteps matters. This is i l lustrated athand of a circular section. ltern tive ltern tive 2


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rhombic dodec hedr honeycombs Fig 3 31Examples of space filling polyhedra:rhombic dodecahedra andhoneycombs


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Space illing PolyhedraObviously the cube is a space-filling polyhedron. This means that we can stackcongruent cubes and completely fill 3D space with them. Actually, the cube is the onlyPlatonic solid that has this property. Nevertheless, there are other polyhedra that havethe space-fillingproperty. O ne example is the following: Ifwe add square pyramidsof height s/2 to each face of a cube with edge lengths , we obtain a so-called rhombicdodec hedron (Figure 3.31a). ll faces of a rhombic dodecahedron are congruentrhombi which are non-regular polygons of equal edge length in the form of a shearedsquare.Let s examine why the rhombic dodecahedron is also a space filling polyhedron. Wetake a cube and attach six congruent cubes to its faces. Replacing these six cubes byrhombic dodecahedrae, the initial cube is completely filled by six congruent pyramidalparts of the adjacent rhombic dodecahedrae. Thus, ifwe replace in a space-fillingassembly of cubes every second cube (in a 3D checkerboard manner) with a rhombicdodecahedron, we find a space-fillingby dodecahedra. Space-fillingpolyhedra appearin nature (e.g. as regular six-sided prisms that form the building blocks of honeycombs,Figure 3.3 lb ).It is of course possible to fill space with non-congruent polyhedra, although it becomesmuch more complicated to do this in a meaningful way. For practical applications wesometimes want to fill a certain volume with polyhedra. For simulation purposes one

Fig. 3.32Volume filling tetrahedra image


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ohe n uses tetrahedr a Figure 3.32 .On e recent example in archit ecture that employsspace-filling polyhedra of varying shape is the National Swimming Center in BeijingFigure 3.33 .An architectural design that employs polyhedra derived from so-called

Voronoi cells see Chapter 17 ir shown in Figure 5.34.

Fig. 3 3 3Volume filling polyhedra inarchitecture: National SwimmingCenter in Beijing.

Fig. 3 34An architectural design based onirregular space filling polyhedra imagecourtesy o 6 Schneider).


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Fig. 3.35Approximation of cyl inder s an d conesby polyhedral surfaces.a) Simple strip models.b) The str ips are furth er subdividedinto planar quadr i laterals.c) Since 99 the Melbourne ShotTower is covered by a 84m high conicalglass roof designed by Kisho Kurokawa.

yhedra urfacesIfa smo oth surface is approximated by a polyhedral surface we also speak of a discretesu ce. n architec ture disc rete surfaces are ofspecial interest in the realization of a design.

Approx imation of cylind ers an d cones by polyh edral surfaces. The first idea is totake a smoo th cylinder surface and replace ir with a strip model Figure 3.35a). Wecan also divide each strip int op lan ar quadrilaterals. The same holds for a cone surface.Figure 3.35b shows a strip mod el an d a model w ith planar quadrilaterals. Note thatth e quadrilaterals are plana r because two oppo sing edges are lying in the same plane byconstruction.

non congruent quadrangles


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Example:model o f the courtyard roof of the in the middle and two congruent

Abbey in Neumunster. The previous conical sections at either side. Theidea was used to cover a rectangular circular arc c is the base curve of twoarea y a curved roof with planar glass adjacent surfaces. Then, a parallelpanels in an interesting way Figure extrusion generates the cylindrical3.36a). Th e shape of the roof consists part-and a central extrusion withof three parts: a cylindrical section vertex generates the conical one. Th e


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cyl inder surface is approx imated by i f we use a different decom posi t ioncongruent planar quadri la tera ls us ing int o quadri la tera ls , they are no longerthe genera tors of the cyl inder . p lanar. Thu s , to real ize chis des ign withIf we app roxim ate the conica l parts by planar glass panels on e has to subdivideplanar quadri la tera ls us ing the cone each nonplanar quadri la tera l in to twogene ra to rs , t he ou tcome migh t be t ri ang les F igure 3 . 36 ~ ) .undes irable Figur e 3 .36b). However,

Fig. 3 36a ) The courtyard roof of the Abbey in

Neumunster 2003) by Ewart-HaagenLorang Arch itects,built by RFR.b) A design only using the gen eratorsof the cylinder and the cones.c) An alternative design thatdiscretizes the co nical pa rts in totriangles.


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In general, the approximation of a freeform shape by triangles is much simpler than byplanar polygons. This fact is intensively used in com puter graphics, in which objectsare decomposed in to triangles e.g. for rendering purposes, recall Cha pte r 2 . Inarchitecture, freeform-shapes are enjoying increased popularity . For the design of roofsconst ructed as steel-and-glass structures with planar panels, the simplest polygons aretriangles Figure 3.37).

Fig. 3.37The Zlote arasy (pol ish for GoldenTerraces ) in Warsaw b y JerdePar tnersh ip In terna t iona l opened in2007. The f reeform shape roof isgeometr ical ly a polyhedral surface witht r iangular faces ( image s courtesy ofWaagner -B i ro Stahlbau AG).


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Whereas three vertices of a triangle in space are always contained in a single planechis need n ot be the case for four or more points. The design of polyhedral freeformsurfaces with planar faces other than triangles is a difficult task that is a topic ofcurrentresearch. As illustrated in Figure 3.38 computer graphics has already developedalgorithms for the approximation of arbitrary shapes by polyhedral surfaces. Howeverthese methods generate a variety o f differently shaped planar polygons a nd thus mayno t be the best solu tion for architectural design. Therefore in Chapte r 9we study adifferent approach t hat becter meets the needs ofarchitecture.

Fiq. 3.38Shape approximat ion by polyhedralsurfaces applied to Michelangelo sDavid ( imag e cour tes y of P ierre Al l iez) .

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