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A Quarta Dimens˜Ao: Da Sua Natureza Espacial Na Geometria Euclidiana

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A Quarta Dimens˜Ao: Da Sua Natureza Espacial Na Geometria Euclidiana Revista Brasileira de Ensino de F´ısica, vol. 43, e20210034 (2021) Artigos Gerais www.scielo.br/rbef c b DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0034 Licenc¸aCreative Commons A quarta dimens˜ao:da sua natureza espacial na geometria euclidiana `acomponente tipo tempo de variedades n˜aoeuclidianas The fourth dimension: from its spatial nature in Euclidean geometry to a time-like component of non-Euclidean manifolds Jos´eMaria Filardo Bassalo1,2, Francisco Caruso*3,4 , Vitor Oguri4 1Academia Paraense de Ciˆencias, Bel´em,PA, Brasil. 2Funda¸c˜aoMinerva, Bel´em,PA, Brasil. 3Centro Brasileiro de Pesquisas F´ısicas,Coordena¸c˜aode F´ısicade Altas Energias, 22290-180, Rio de Janeiro, RJ, Brasil. 4Universidade do Estado do Rio de Janeiro, Instituto de F´ısica Armando Dias Tavares, 20550-900, Rio de Janeiro, RJ, Brasil. Recebido em 27 de janeiro de 2021. Revisado em 12 de fevereiro de 2021. Aceito em 13 de fevereiro de 2021. Neste artigo, apresenta-se a evolu¸c˜aodas ideias sobre a quarta dimens˜aoespacial, partindo daquelas que surgem da geometria euclidiana cl´assicae abordando, em seguida, as que resultam do ˆambito das geometrias n˜ao euclidianas, como as de Riemann e Minkowski. Particular aten¸c˜ao´edada ao momento no qual o tempo real passa efetivamente a ser considerado como uma quarta dimens˜ao,conforme introduzido por Einstein. Palavras-chave: Quarta Dimens˜ao,Espa¸co,Geometrias n˜aoEuclidianas, Hist´oriada Geometria, Hist´oriada F´ısica. In this article, the evolution of the ideas about the fourth spatial dimension is presented, starting from those which come out within classical Euclidean geometry and going through those arose in the framework of non- Euclidean geometries, like those of Riemann and Minkowski. Particular attention is given to the moment when real time is effectively considered as a fourth dimension, as introduced by Einstein. Keywords: Fourth Dimension, Space, non-Euclidian Geometry, History of Geometry, History of Physics. 1. The Beginning of the History: Not “For, as the Pythagoreans say, the universe More Than Three and all that is in it is determined by the number three, since beginning and middle The first step in the history of the fourth dimension was and end give the number of the universe, actually an attempt to deny its existence. Indeed, the im- and the number they give is a triad. And possibility of a fourth dimension was actually sustained so, having taken these three from nature as by Aristotle of Stagira (384–322 b.C.). In effect, in his (so to speak) laws of it, we make further use De Caelo, which consists of Four Books, he treated this of the number three in the worship of the impossibility just right in the first paragraph of Book 1, Gods”. saying, in summary, that: Such kind of identification between the tri- “A magnitude if divisible one way is a line, dimensionality of space and God’s will be recurrent in if two ways a surface, and if three a body. the history of science. Johannes Kepler (1571–1630), for Beyond these there is no other magnitude, example, asseverated that three is exactly the number because the three dimensions are all that of dimensions due to the Holly Trinity [2, 3, p. 307]. there are, and that which is divisible in three The second necessary (but not sufficient) step toward directions is divisible in all” [1, p. 447]. the conception of the fourth dimension has to do with the systematization of the geometric knowledge in Ancient In this same paragraph, the Stagirite continues giving Greece. a cosmological justification of this number three by appe- The Greek mathematician Euclid of Alexandria aling to its divinization, sustained by the Pythagoreans. (c. 323–285 b.C.) wrote his famous treatise on Geometry, Quoting him, The Elements, which has influenced the development of Western Mathematics for more than 2000 years. This * Endere¸code correspondˆencia: [email protected] classical work contains Thirteen Books. From Book 1 to Copyright by Sociedade Brasileira de F´ısica.Printed in Brazil. e20210034-2 A quarta dimens˜ao:da sua natureza espacial na geometria euclidiana 10 just Plane Geometry is considered. In Book 1, Euclid Italian Art in the end of the Trecento and during the presents 23 definitions including those of point, line and following centuries. Actually, the first to introduce the surface, as following: (1) “A point is that which has no tridimensionality in Medieval Art [11] was the Italian part”; (2) “A line is breadthless length”; (...); (5) “A painter Giotto di Bondone (1266–1337). Giotto painted surface is that which has length and breadth only” [4, almost all the walls of St. Francis’ Lower Church, in p. 153]. Assisi. “The Legend of St. Francis”, whose authorship is The third dimension is treated only in the last three traditionally attributed to him, is the theme of a cycle Books, when a solid is defined in the following way: of 28 frescoes that are found in the Upper Basilica in “A solid is that which has length, breadth and depth” Assisi, probably painted between 1297 and 1300. The [5, p. 260]. frescoes painted by him in the Arena Chapel at Padua, Throughout the entire compendium, Euclid limits about the year 1305, mark an entirely new stage in himself to treat Linear, Plane and Spatial Geometry, the development of empirical perspective [12, p. 57]. He and he did not even consider the possibility of a fourth was also the first artist in that period to paint the Sky dimension. in blue, diverting up from the traditional golden Sky Some centuries later, the Greek astronomer Claudius characteristic of the Byzantine Art [13, 14]. Clearly, he Ptolemy (85–165), in his (lost) book On Distance, was inspired by St. Francis’ world view, who pointed out published in 150 a.C., gave a “proof” about the impos- emphatically that there was a second book (not only the sibility of the fourth dimension, based on the very fact Sacred Scripture) able to bring someone to God: it is the that it is impossible to draw a fourth line perpendicular Book of Nature. One should look at Nature as it is. This to three mutually perpendicular lines. This is indeed not attitude suggests and anticipates a transition to the new a proof, but rather reinforce that we are not able to relation between Man and Nature, which is a prelude to a visualize the fourth dimension from which one cannot new man that is still to be forged in the Renascence [15]. conclude about its non existence. The formal discovery of perspective is attributed to To the best of our knowledge, speculations and new the Italian architect Filippo Brunelleschi (1377–1446), ideas about the existence of a fourth dimension had who suggested a system that explained how objects to wait for the middle of the 16th century on to shrink in size according to their position and distance be strengthened, when a more propitious intellectual from the eye. In 1435, in a book named On Painting, atmosphere is to be found, as we will see all over this Leon Battista Alberti (1404–1472) provided the first paper. theory of what we now call linear perspective. The tri- For the moment, it is important to stress that the two dimensional representations of painting conquered then biggest syntheses of the Classical Greek Philosophy – a scientific aspect when painters and architects of the that of Aristotle and that of Euclid – considered impos- Quattrocento started to study the relationship between sible the existence of more than three spatial dimensions. Geometric Optics [16] and Perspective in the Euclidean This very fact is not meaningless, so far the enormous space, as did, for instance, Piero della Francesca [17]. influence of these two thinkers is considered. The long period in which there was no discussion why space could not have a different dimensionality may be, in part, 2. The First Ideas of a New Dimension attributed to Aristotle’s authority and, on the other side, in Space to the fact that the study of Euclid’s Elements in the Middle Ages, including different educational levels, was Back to the fourth dimension, the idea of a new spa- languished or quite neglected [6, p. 365–369]. tial dimension was revived by the studies of several The overcoming of the domination of the Aristotelian mathematicians in the 16th and 17th centuries. Indeed, influence and the abandonment of its Cosmos must the Italian physicist, philosopher, mathematician and still await the Renaissance [7, 8]. As Koyr´eemphasizes, physician Ge(i)rolamo Cardano (1501–1576) and the this implies and imposes the reformulation of the basic French mathematician Fran¸coisVi`ete(1540–1603) consi- principles of philosophical and scientific reason [9]. In dered such “additional” dimension in their researches on contrast, the revival of interest in Euclid’s Elements quadratic and cubic equations. The same did the French should wait the invention of printing press by Johannes mathematician and physicist Blaise Pascal (1623–1662) Gutenberg (c. 1400–1468) [10]. in his study named Trait´edes trilignes rectangles et le It is not out of place to remember that, since the leurs onglets [18, p. 260–298], when, generalizing his first prehistoric cave painting until the medieval period, “trilignes” from the plane to the space and beyond, he the World was pictured without any perspective in two wrote: “La quatri`emedimension n’est point contre la dimensional spaces, probably translating the difficulty pure g´eom´etrie.” [18, p. 269]. to represent the third dimension on a bi-dimensional Meanwhile, the French philosopher and mathemati- canvas, wall or any other surface.
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