https://doi.org/10.3311/PPar.12375 Creative Commons Attribution b |99 Periodica Polytechnica Architecture, 49(2), pp. 99–108, 2018 The Truncated Icosahedron as an Inflatable Ball Tibor Tarnai1, András Lengyel1* 1 Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary * Corresponding author, e-mail: [email protected] Received: 09 April 2018, Accepted: 20 June 2018, Published online: 29 October 2018 Abstract In the late 1930s, an inflatable truncated icosahedral beach-ball was made such that its hexagonal faces were coloured with five different colours. This ball was an unnoticed invention. It appeared more than twenty years earlier than the first truncated icosahedral soccer ball. In connection with the colouring of this beach-ball, the present paper investigates the following problem: How many colourings of the dodecahedron with five colours exist such that all vertices of each face are coloured differently? The paper shows that four ways of colouring exist and refers to other colouring problems, pointing out a defect in the colouring of the original beach-ball. Keywords polyhedron, truncated icosahedron, compound of five tetrahedra, colouring of polyhedra, permutation, inflatable ball 1 Introduction Spherical forms play an important role in different fields – not even among the relics of the Romans who inherited of science and technology, and in different areas of every- many ball games from the Greeks. day life. For example, spherical domes are quite com- The Romans mainly used balls composed of equal mon in architecture, and spherical balls are used in most digonal panels, forming a regular hosohedron (Coxeter, 1973, ball games. Spherical inflatables such as air-supported p. 12; Wikipedia (a)). These balls were colourful, as seen in radomes (radar or radio antenna domes) and soccer balls the 4th century AD mosaic in the Villa Romana del Casale, have structures analogous to each other. They are made Piazza Armerina, Sicily (Wikipedia (b)); they could be com- from planar elements that are stitched (glued) together, posed of as many as twelve digonal pieces as demonstrated and after inflation, they achieve a shape approximating by toy balls, found in Egypt from the Roman period, kept in a sphere. The present paper is related to inflatable balls. the British Museum (2017). These multi-panel balls, which originally were stuffed During the Han (202BC-220AD) and Tang (618-907AD) with hair or feathers, have a long history, although their dynasties in China, tetragonal and hexagonal hosohe- beginnings are not known to us. We know that Plato (1892a) dral balls were used to play cuju (ancient Chinese foot- wrote in Phaedo (360BC): "... the earth, when looked at ball) (Cui, 1991). During the Tang dynasty, the hexagonal from above, is in appearance streaked like one of those hosohedral balls were modified. The two vertices of the balls, which have leather coverings in twelve pieces, and hosohedron were truncated. The ball obtained in this way is decked with various colours ...". Referring to Timaeus was built from two hexagonal and six quadrilateral panels. (Plato, 1892b), Robin (1935) remarked in a footnote that The Chinese claim that, during the Song dynasty (960- the number of faces of a regular dodecahedron is twelve. 1279AD), a 12-piece ball was developed where each panel Rassat and Thuillier (1996) went further, and explicitly was a regular pentagon (hwjyw.com). We do not know the stated that a dodecahedral leather ball is mentioned in the details of this, but it is well known that there are paintings quotation above. We prefer to think that this ball could and drawings from the Song to the Ming dynasties that be dodecahedral, but not necessarily. On the one hand, show people playing cuju with a ball on which (at least) the emphasised number twelve could refer to many other one pentagon is seen. things, and on the other, we do not know of any archaeo- We know of only one example of the existence of poly- logical evidence for the existence of dodecahedral balls hedral balls in the ancient times. This is a ball that can be Cite this article as: Tarnai, T., Lengyel, A. (2018) "The Truncated Icosahedron as an Inflatable Ball", Periodica Polytechnica Architecture, 49(2), pp. 99–108. https://doi.org/10.3311/PPar.12375 Tarnai and Lengyel 100|Period. Polytech. Arch., 49(2), pp. 99–108, 2018 seen in the 2nd century AD mosaic of the Baths of Porta Marina, Ostia (Rassat and Thuillier, 1996). It could be a hexadecahedron of D2 symmetry with 4 hexagonal and 12 pentagonal faces. Since manufacturing such a compli- cated ball must be difficult, Rassat and Thuillier (1996) thought that the mosaic is probably just a misinterpreta- tion of a dodecahedron. During the Renaissance, artists and scientists stud- ied the Archimedean polyhedra. Pacioli (1509) published them with the drawings by Leonardo da Vinci. One of these polyhedra is the truncated icosahedron of Ih sym- metry with 12 pentagonal and 20 hexagonal faces (Fig. 1). It looks quite round, and could serve as a basis for balls, but as far as we know it was not used for this purpose, but, Fig. 1 A lath skeleton model of a truncated icosahedron drawn by for instance, much later for decorating a church memo- Leonardo da Vinci (Pacioli, 1509). (Source: https://archive.org/details/ rial (Fig. 2) (Tarnai and Krähling, 2008). However, ball divinaproportion00paci) games were also played in the Renaissance. One piece of evidence for this is a leather ball in the Stirling Smith Art Gallery and Museum, Scotland, which was made sometime in the 1540s, and is considered as the world's oldest extant football (The Stirling Smith Art Gallery and Museum). It was constructed from only three panels: two equal cir- cles and an elongated rectangle. Prior to inflation it was shaped like a circular cylinder. Interestingly, this kind of ball is still in use in England in the traditional Shrovetide football game (Wikipedia (c)). With the growing pop- ularity of football, the number of different ball designs increased, and around the turn of the 20th century, designs showed great variety, and in addition to the hosohedral Fig. 2 A truncated icosahedron at the feet of the effigies on the ball (Fig. 3(a)), many variants occurred (Tarnai, 2005). monument to Sir Anthony Ashley in Wimborne St. Giles, England, (The ancient hosohedron form still exists in present-day erected in 1627. (Photo: Tarnai). The respected architecture historian Nikolaus Pevsner misdescribed this object as a "sphere of hexagons" beach-balls (Fig. 3(c)).) (Newman and Pevsner, 1975). The truncated icosahedral ball appeared only in the 1960s. Most people think that this 32-panel ball (Fig. 3(b)) is an Adidas invention, since Adidas made the first official and "we should make sure that every step to improve its FIFA World Cup match ball in 1970, which was a trun- qualities is free for all of us". As far as we know, the first cated icosahedral ball. However, the 32-panel truncated patent was given in 1964 for the black and white colouring icosahedral ball was, in fact, invented by Eigil Nielsen, of the ball (Doss, 1964). a former goal keeper of the Danish national football team The date of the invention of the truncated icosahedral and was introduced by his firm Select in 1962 (Select ball has been known to us for years. Consequently, it was a Sport (a), (b)). (Adidas started making footballs only in great surprise when, while watching a recent TV documen- 1963 (Wikipedia (d)).) We do not know the circumstances tary (Kloft, 2017), one of us unexpectedly saw a colourful of the invention or how Nielsen got the idea, but it is clear truncated icosahedral beach-ball (Fig. 3(d)) on the screen. that it required a certain level of geometrical knowledge. This documentary showed excerpts of several archive As Michael Karlsen of Select (personal communication) home movies in colour. The ball appeared in the home informed us, Eigil Nielsen knew a mathematical family. movie entitled Wörthsee, made sometime at the end of the His invention has never been patented. His philosophy 1930s. This means that a truncated icosahedral ball - here- was that the ball is the most entertaining toy in the world, after we will call it simply the "Wörthsee ball" - already Tarnai and Lengyel Period. Polytech. Arch., 49(2), pp. 99–108, 2018|101 (a) (b) (a) (b) (c) (d) (c) (d) Fig. 3 Hosohedral balls composed of digonal panels (a), (c), and Fig. 4 Different views of the Wörthsee ball. Colours in truncated icosahedral balls (b), (d). (a) Soccer ball in the second counterclockwise order around a pentagon: (a) R, K, Y, B, W, half of the 19th century, from the collection of the National Football (b) R, B, Y, K, W and R, K, Y, R, W, (c) R, Y, B, W, B, Museum, Preston, UK. (Photo: Tarnai). (b) Modern 32-panel soccer (d) B, Y, K, Y, K. (Kloft, 2017) ball. (Photo: Tarnai). (c) Beach-ball. (Available at: https://www. evapresent.eu/). (d) "Wörthsee ball", a beach-ball from the end of the 1930s. (Kloft, 2017). assume that this ball was made as a one-off, and not by mass production. It seems that beach-balls, similar to those existed more than twenty years before Eigil Nielsen's of the present day, were mostly hosohedra at that time. 32-panel ball, although this was not common knowledge. The skin of the Wörthsee ball was made of thin, The aim of the present paper is to describe the properties coloured textile pentagons and hexagons. The seams were of the Wörthsee ball, to point out a defect in its colouring and quite robust.