Archimedean Polyhedra and Geodesic Forms · Eleven of the Archimedean polyhedra can be derived from the ﬁve Platonic polyhedra by truncations of vertices and/or edges. The construction

Information Sciences and Computer Engineering, Vol. 2, No. 1, (2011) 35–45

International Journal ofInformation Sciences and Computer Engineering

j o u r n a l h o m e p a g e : http://www.ijisce.org

Archimedean Polyhedra and Geodesic Forms

Dimitra Tzourmakliotou∗

Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece, 67100

Abstract– Geodesic forms allow effective use of material and space and maybe employed to create architecturally interesting and economic buildingstructures. They are presently used in a number of specialized areas of con-struction such as domes for arenas, cultural centres, exhibition halls andOlympic facilities. Their widespread use has been obstructed by the diffi-culty in defining their geometry. This problem has presented a challengefor engineers and architects for decades. Many attempts have been madethroughout the world to evolve techniques that deal with the data generationof geodesic forms. However, the approach presented in this paper providesa methodology that allows data generation for geodesic forms of all kinds tobe handled with ease and elegance.

Keyword: Archimedean Polyhedra, Geodesic forms, formex algebra, Formian.

1. Introduction

The fascinating world of polyhedral has a long and diverse his-tory. These archetypical geometrical configurations have influ-enced numerous aspects of art and science. As can be seen, thetheory has not only led to the solution of a number of interest-ing structural morphological problems but also produced a toolfor the design of efficient structures with the possibility of greatvisual and architectural qualities. Polyhedra have been the sub-ject of fascination and interest since ancient times. They havebeen studied throughout the ages by mathematicians, philoso-phers, and artists and they play an important role in a numberof branches of science and technology [1, 2, 3]. The interest inpolyhedra in this paper stems from the fact that they provide abasis for the generation of an important class of structural formsknown as geodesic forms.

Most of the existing geodesic domes have been obtained fromthe radial projection of the triangular faces of the regular poly-hedron called icosahedron on a sphere. However, in this paper ageodesic dome may be obtained by projecting a polyhedric con-figuration based on the Archimedean polyhedra on a surface. Forthe projection the tractation retronorm will be applied. Differenttypes of surfaces such as spheres, ellipsoids, circular or elliptical

∗Corresponding author:Email address: [email protected], Ph: +30 2541079714

paraboloids and different types of projections such as central, ax-ial, or radial will be used to generate intersecting geodesic domeconfiguration.

An important aspect of this paper is the establishment of theconcepts and constructs through which polyhedric and geodesicconfigurations may be created with the polyhedron function. Thepolyhedron function provides a basis for the configuration pro-cessing of geodesic configurations in a compact and readily un-derstood manner and it allows one to work with the same setof tools in all data generation problems eliminating the need forthe employment of an assortment of programs which are dealingwith specific problems. Hence, the emphasis is on the manner inwhich the concepts of formex algebra are employed for the gen-eration of polyhedric and geodesic domes rather than the detailsof the formulations and no prior knowledge of formex algebrais necessary for following the material. The paper does containsome formex formulations but these are included to give a feelfor the appearance of formex formulations rather than their de-tails being essential for understanding of the material.

Formex configuration processing provides a reliable mediumfor the processing of configurations of all kinds. The conceptsare general and can be used in many fields. In particular, theideas may be employed for generation of information about var-ious aspects of structural systems such as element connectivity,nodal coordinates, loading details, joint numbers and support ar-rangements. The information generated may be used for variouspurposes, such as graphic visualization or input data for structuralanalysis. Formex configuration processing uses the concepts offormex algebra through the programming language Formian togenerate and process configurations. The preliminary conceptsand ideas of formex configuration processing were evolved dur-ing the last four decades [4, 5].

2. The Archimedean Polyhedra

The Archimedean or “semiregular polyhedra” are what iscalled “facially” regular polyhedra. This means that every faceis a regular polygon though the faces are not all of the samekind. However, every vertex is to be congruent to every other ver-tex that is, the faces must be arranged in the same order aroundeach vertex. The Archimedean polyhedra were discovered in an-cient Greece and were described by Archimedes thus they are

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called the Archimedean polyhedra. His writings on the semireg-ular polyhedra were lost together with the knowledge of the fig-ures. During the Renaissance they were gradually rediscoveredand were described by such people as Pierro della Francesca andAlbrecht Durer, though the description of the Archimedean poly-hedra did not appear until Johannes Kepler’s Harmonices Mundiwas published in 1619. The Archimedean solids are given in Fig-ures 1, 2 and 3 [6, 7].

These polyhedra consist of various combinations of triangles,squares, pentagons, hexagons, octagons and decagons, Figure 5.The letter which is given in each vertex of a polyhedral face isa constituent of the face code and will be explained below. Tenof the Archimedean polyhedra utilize only two kinds of polygonsand the remaining three utilize three kinds of polygons, as shownin Figure 4. In these Figures each polyhedron is shown togetherwith the global Cartesian x − y − z coordinate system. The originof the coordinate system is at the centre of the polyhedron and isindicated with a large dot. The point where the positive side ofthe x−axis intersects the polyhedron is indicated by a little circlewith an enclosed x. This point is referred to as the “x−point”.The positions of the positive directions of y− and z− axes areindicated by arrows with z being always vertical. Each face ofthe polyhedron is identified with a “face code” which is given atone corner of the face. A face code consists of a number followedby a letter and possibly followed by an asterisk. The number in aface code is the identification number of the face. The letter in aface code determines the points A, B,C, · · · , etc of the

configuration that is to be placed on the indicated corner of theface. If a face code has an asterisk, it implies that the config-uration which is to be placed on the face is the reflection, withrespect the x − y plane, of the given configuration [8].

Eleven of the Archimedean polyhedra can be derived from thefive Platonic polyhedra by truncations of vertices and/or edges.The construction of the last two semiregular polyhedra, the snubcube and the snub dodecahedron is still based on the truncationof the edges of a regular polyhedron. Both of them have faceswhich are located on the faces of the original regular polyhedron,comprising regular polygons with the same number of sides butsmaller and rotated in a particular direction. This process is re-ferred to as “snubbing”. The rotation maybe right handed or lefthanded. Therefore, the snub cube and snub dodecahedron occurin two versions, each version is the mirror image of the others. Itwill always be a right handed (dextro) or a left handed (laevo) ver-sion of the polyhedron. Such polyhedra are called “enantiomor-phic”.

The right handed versions of the snub cube and the snub do-decahedron have been used in this paper. The sheer length ofmany of the names of the Archimedean polyhedra may detersome people but they are descriptive and can be easy to remem-ber.

The cuboctahedron can be seen to have obvious relationshipsto both the cube and the octahedron. The icosidodecahedron hasrelationships to both the icosahedron and the dodecahedron. Theword “truncated” in such names as truncated tetrahedron clearlydescribes a polyhedron, in this case a tetrahedron which has hadits extremities removed or truncated. The great rhombicubocta-hedron and great rhombicosidodecahedron are sometimes calledthe truncated cuboctahedron and the truncated icosidodecahe-dron. But their former names are used in this paper. The prefixes

Fig. 4. The Archimedean Polyhedra and the polyhedron code P for each one

Fig. 5. Types of polygons in Archimedean Polyhedra

Tzourmakliotou/Information Sciences and Computer Engineering, Vol. 2, No. 1, 2011 37







Fig. 6. A graphical representation of the cuboctahedron

“great” and “small” in the names of the rhombicuboctahedra andrhombicosidodecahedra differentiate between figures in terms oftheir sizes. The word “snub” is an old Norwegian word, a literaltranslation being “snub-nosed” or “flat-nosed”.

3. Polyhedron Function

Consider a polyhedron and let some given configurations beplaced on its faces. The result is referred to as a “polyhedricconfiguration” or “polyhedric form”. This term may also be usedto refer to a portion of a polyhedric configuration.

A polyhedron which is used as the basis for the creation of apolyhedric configuration or form is referred to as the “base poly-hedron” of the polyhedric form. This is a formex function called“polyhedron function” that can be used to crate formices repre-senting polyhedric configurations [9].

In the case of Archimedean polyhedra the faces are not all ofthe same shape. In spite of this, one may create a polyhedric formby mapping a single configuration on all or some of the faces ofan Archimedean polyhedron.

However, in creating polyhedric forms that are based onArchimedean polyhedra, normally one would use different con-figurations for different face shapes. For instance, the polyhdericconfiguration of Figure 6 is based on a cuboctahedron and is ob-tained by mapping the configuration of Figures 7a and 7b onto thetriangular and square faces, respectively. To elaborated considera single layer triangular and a square configuration. The triangu-lar and square configurations together with the normat U1-U2-U3for the formex formulation are shown in Figures 7a and 7b, re-spectively. These configurations may be represented in terms ofthe formex variables P5 and P6 which are given as,

P5 = GENID(6, 6, 2, S QRT |3, 1,−1)|...ROS AD(1, S QRT |3/3, 3, 120)|[0, 0; 2, 0]

andP6 = LUX([6, 6])|ROS AD(6, 6)|BB(1, 1/S QRT |3)|P5Let it be required to map this configuration onto the faces of

a cuboctahedron. A Formian statement describing this operationmay be given asC = PEX |(POL(7,10,[0,0; 12,0], [1; 8] |P5 # . . .

POL(7,10, [0,0; 12,0], [9; 14]) |P6)The construct

POL(7, 10, [0, 0; 12, 0], [1; 8]is a formex function representing a rule for transformation of

a given formex P into a formex C. The parameters 7, 10, [0,0;12,0], [1;8] are parts of the rule defining the particulars of thetransformation and are referred to as canonic parameters.

The above function is referred to as a “polyhedron function”.The polyhedron function can be used to create single layer ormulti layer polyhedric configurations. The general form of thepolyhedron function for single layer polyhedric configurationsmay be written as

POL(P,R, [A1, A2; B1, B2] <<, F1, F2, , Fn >>)where the first canonic parameter P is referred to as the “poly-

hedron code”. The polyhedron code specifies the type of polyhe-dron which is to be used as the basis for the operation. Figures1, 2, and 3 list the code numbers for the Archimedean polyhedra.The integer 7 given as the polyhedron code in the above polyhe-dron function specifies a cuboctahedron. The “radius specifier”determines the size of the polyhedron by specifying the radius ofthe circumsphere, that is, the sphere that contains all the verticesof the polyhedron. This parameter is given as 10 units of length.The “locator” specifies the manner in which a given configura-tion is to be mapped onto a face of the polyhedron. To elaborate,consider the configuration shown in Figures 7a and 7b.

Two corners of the triangular and square configuration are de-noted by the letters A and B. The configurations are intended tobe placed on the faces of the cuboctahedron in such a way that ABfits an edge of the cuboctahedron. This convention in conveyedby including the U1-U2 coordinates of A and B in the locator.

The last canonic parameter is referred to as the “face list”. Therole of the face list is to specify those faces of the polyhedrononto which the configuration is to be mapped. The face list is en-closed in special brackets which are referred to as option brack-ets. Absence of the face list in the polyhedron function impliesthat the configuration is to be mapped onto all the faces of thatpolyhedron.

However, one has the option of generating only a part of thepolyhedric configuration by specifying the required face numbersthrough the face list. The face numbers for the Archimedeanpolyhedra are given in Figures 1,2, and 3, respectively.

4. Geodesic Forms

4.1. An ExampleConsider the configuration shown in Figure 8(a). This is a view

of a geodesic form obtained by projecting the icosidodecahedralconfiguration of Figure 8(b) onto an ellipsoid using central pro-jection. A Formian statement describing this operation may begiven as

D = TRAC(1,0,0,-10,2,0,0,-5,11,15,20,13) — Fwhere the formex variable F represents the polyhedric form of

Figure 8(b) and may be given asF = POL(12,10, [0,0; 8,0],[1; 10]) |E1 # ...

POL(12,10,[0,0; 8,0], [21; 26]) |E2Formex variable E1 represents the compret of the pattern on

the triangular faces of the icosidodecahedral configuration Figure9(a). Formex variable E2 represents the compret of the patternon the pentagonal faces of the icosidodecahedral configurationFigure 9(b). The configuration of Figure 8(a) is an example of a“geodesic form” or “geodesic configuration”.


Fig. 7. a) A triangular configuration and b) a square configuration

Fig. 9. Graphical representations of formex variables a) E1 and b) E2

Fig. 8. a)View of a geodesic form based on an icosidodecahdron and b) view ofthe icosidodecahedron

The constituent parts of formex variable D are as follows:TRAC is an abbreviation for tractation and is followed by a

sequence of parameters enclosed in parentheses. The constructthat consists of TRAC together with the ensuing parameter list isreferred to as a “tractation retronorm”.The “projection specifier”specifies the type of projection to be used. The projection spec-ifier may have the value 1,2,3 or 4 indicating central, parallel,axial or radial projection, respectively. The projection specifierin Figure 10 is given as 1, implying central projection. The “sur-face specifier” specifies the type of surface on which projection isto be made. The surface specifier for formex variable D is givenas 2 implying an ellispoid and this is followed by the coordinatesof the centre of the ellipsoid (0,0,-5) and the numbers 11, 15, 20that specify the three semi axes a, b and c of the ellipsoid in thex, y and z directions. The items should follow the surface spec-ifier in different cases are as given in Table 1. The “selector”specifies the course of action to be taken when the projection of

Fig. 10. formex variable D that implies central projection) E2

a point cannot be determined uniquely. In this case the value of13 indicates the solution with the greatest z component.

A geodesic form may have more than one layer. For example,consider the configuration shown in Figure 11. This is a view of adouble layer geodesic form obtained by projecting the polyhedricconfiguration of Figure 12 on two concentric spherical surfacesusing radial projection. A Formian statement describing this op-eration may be given as

D3 = TRAC([4,1,4,1,0,0,0,10,13],[4,2,4,1,0,0,0,12,13]) — P2where the formex variable P2 represents the polyhedric form ofFigure 8 and may be given asP2 = PEX |POL(5, [10; 12], {[1, SQRT|3/3; 11,SQRT|3/3],...[0,0; 12,0]}, [1; 5]) |(T#B)

Formex variable T and B represent the compret of the doublelayer configuration of Figure 13 and may be given asT = GENID(6,6,2,SQRT|3,1,-1) |ROSAD(1,SQRT|3/3,3,120)|{[0,0,2; 2,0,2], [0,0,2; 1,SQRT|3/3,1]}B = GENID(5,5,2,SQRT|3,1,-1) |ROSAD(2, 2*SQRT|3/3...,3,120) |[1,SQRT|3/3,1; 3,SQRT|3/3,1]}


Table 1. The surface specifier and its coefficientSurfaceSpec-ifier(S)

Surface Description

1 Sphere The xc, yc and zc coordinates of thecentre of the sphere followed by theradius R of the sphere

2 Ellipsoid The xc, yc and zc coordinates of thecentre of the ellipsoid followed bythree semiaxes a, b and c of the ellip-soid in the x, y and z directions

3 Elliptic Paraboloid The xc, yc and zc coordinates of thecentre of the elliptic paraboloid fol-lowed by three semiaxes a, b and c ofthe elliptic paraboloid in the x, y andz directions

4 Hyperbolic Paraboloid The three semiaxes a, b and c of thehyperbolic paraboloid in the x, y andz directions

5 Circular Cylinder The xc and yc coordinates of the cen-tre of the directrix circle of the cylin-der followed by the radius R of the di-rectrix circle

6 Elliptic Cylinder The xc, yc coordinates of the centre ofthe directrix ellipse of the cylinder fol-lowed by two semiaxes a and b of thedirectrix ellipse in the x and y direc-tions

7 Parabolic Cylinder The xc and yc coordinates of the cen-tre of the directrix parabola of thecylinder

8 Plane specified by twopoints that define a vectornormal to the plane

The x, y and z coordinates of the twopoints N(n1,n2,n3) and T(t1,t2,t3) thatdefine a vector normal to the plane

9 Plane specified by threepoints

The x, y and z coordinates of threepoints, (x1,y1,z1), (x2,y2,z2) and(x3,y3,z3)


Fig. 11. A double layer geodesic form

Fig. 12. A double layer polyhedric configuration

In Figures 11, and 12 the top layer elements are drawn in thicklines and the bottom layer elements as well as the web elementsare drawn in thin lines.

4.2. Further Interconnection Patterns

The concepts and constructs presented in previous sections arecombined in this section to represent the compretic and normicproperties of a number of polyhedric forms and geodesic domeconfigurations. Configuration based on all Archimedean polyhe-dra will be described here in an attempt to explore the immenserange of possible shapes and forms for polyhedric and geodesicforms [10, 11, 12].

To begin with, consider the configuration of Figure 14. This isa plan view of a truncated icosahedral configuration obtained byplacing the configuration of Figures 15(a), 15(b) and 15(c) on thetop part of a truncated icosahedron. The configuration of Figure15(a) is placed on the pentagonal faces whereas the configuration

Fig. 13. A triangular double layer configuration

Fig. 14. A truncated icosahedral configuration

Fig. 15. The configurations of the pentagonal and hexagonal faces

of Figures 15(b) and 15(c) are placed on the hexagonal faces. Thegeodesic forms of Figures 16 and 17 are obtained by mappingthe polyhedric form of Figure 14 on a sphere and an ellipsoid,respectively.

Furthermore, consider the geodesic form of Figure 18. This isobtained by first mapping the patterns of Figures 19(a) and 19(b)on the top part of a truncated dodecahedron and then project-ing this on an ellipsoid. As it has been demonstrated so far thetype of polyhedron and the configuration that has been mappedonto the faces of a polyhedron may be changed. For instance,by placing the configurations of Figures 20(a) and 20(b) on thetriangular and the square faces of the top part of the snub cubethe polyhedric configuration of Figure 21 is obtained.

In order to obtain the geodesic forms of Figure 22 the poly-hedric configuration of Figure 21 is projected on a sphere. Fi-nally, consider the polyhedric forms of Figures 23 and 24. Theseare based on a great rhombicosi-dodecahedron. The polyhedric

Fig. 16. A truncated icosahedron projected on a sphere


Fig. 17. A truncated icosahedron projected on an ellipsoid

Fig. 18. A truncated dodahedron projected on an ellipsoid

Fig. 19. The configurations of the triangular and decagonal faces

Fig. 20. The configurations of the triangular and square faces

Fig. 21. A graphical representation of the top part of a snub cube

Fig. 22. A snub cube projected on a sphere

Fig. 23. A graphical representation of a great rhombicosidodecahedron

form of Figure 23 is obtained by placing the patterns of Figures25(a) and 25(d) on the square and the decagonal faces, respec-tively. In the case of the polyhedric form of Figure 24 the pat-terns of Figures 25(a), 25(b), 25(c) and 25(d) are mapped ontothe square, hexagonal and decagonal faces, respectively. Thegeodesic form of Figure 26 is obtained by projecting the poly-hedric form of Figure 24 on a sphere using central projection.

5. Conclusion

Polyhedra, spatial bodies of perfect geometric shape, have fas-cinated human beings throughout history. However, the process-ing of polyhedric configurations in pre-computer days was an ex-tremely difficult task. In spite of this, a number of gifted design-ers managed to deal with the problem and create many beauti-ful geodesic structures based on polyhedric configurations. Theconstraint of the processing difficulties however, did not allowthe designer to take full advantage of the whole spectrum of pos-sibilities and their scope remained rather limited. Even today,

Fig. 24. A graphical representation of a great rhombicosidodecahedron


Fig. 25. The configurations that is to be mapped onto the faces

Fig. 26. A great rhombicosidodecahedron projected on a sphere

the processing of polyhedric configurations is mainly carried outusing computer programs that lack generality and have many lim-itations and shortcomings.

In contrast, the conceptual methodology that has been pre-sented in this paper, combined with suitable computer softwaresuch as Formian, provides a means for dealing with the process-ing of any kind of polyhedric and geodesic forms in a readily un-derstood manner. One key factor in dealing with the processingof polyhedric configurations is the ability to generate face-objectsin a convenient manner. The creation of these objects in Formiancan be carried out using the concepts of formex algebra.

References

[1] R. Bury, Plato IX Timaeus, The Loeb classical library. HarvardUniversity press, 1975.

[2] B. Fuller and R. Marks, The Dymaxion World of BuckminsterFuller. Anchor Books, 1973.

[3] J. Baldwin and B. Works, Buckminster Fuller’s ideas for Today.New York: Wiley, 1996.

[4] H. Nooshin and P. L. Disney, “Formex configuration processing,”in Structural Morphology and Configuration Processing of SpaceStructures (R. Motro, ed.), pp. 391–443, London: Multi Sciencepublishers, 2009.

[5] H. Nooshin and P. Disney, “Formex configuration processing III,”International Journal of Space Structures, vol. 17, no. 1, pp. 1–50,2002.

[6] H. Kenner, Geodesic Math and How to Use it. University of Cali-fornia Press, 1976.

[7] A. Pugh, “Polyhedra: A visual approach,” Berkeley: University ofCalifornia press, 1976.

[8] D. Tzourmakliotou, Computer aided design of braced domes. PhDthesis, University of Surrey, United Kingdom, 1993.

[9] D. Tzourmakliotou, “The polyhedric configurations in spatialstructures,” in Proceedings of the Sixth Conference on Computa-tion of Shell and Spatial Structures, pp. 10–14, 2008.

[10] A. Greorghiu and V. Dragomir, Geometry of structural forms. Lon-don: Elsevier applied science publishers, 1978.

[11] R. Motro, “Review of the development of geodesic domes,”in Analysis, Design and Construction of Braced Domes (Z. S.Makowski, ed.), pp. 387–411, London: Granada publishing Ltd,1984.

[12] P. Huybers, “Prism based structural forms,” Engineering Struc-tures, vol. 23, no. 1, pp. 12–21, 2001.

Dimitra Tzourmakliotou received theDiploma degree in Civil Engineering fromDemocritus University of Thrace, Xanthi,Greece in 1990 and the Ph.D. degree inCivil engineering from University of Surrey,Guildford, U.K. in 1993. I have work inthe private sector for companies such asImpregilo Spa and YTONG Gmbh. I am

currently an assistant professor at Democritus University ofThrace, Laboratory of steel structures. My research interestsare, cad of spatial structures, sustainability of steel and spatialstructures.

Documents

Archimedean Polyhedra and Geodesic Forms · Eleven of the Archimedean polyhedra can be derived from the ﬁve Platonic polyhedra by truncations of vertices and/or edges. The construction