{ "type": "Article", "authors": [ { "type": "Person", "familyNames": [ "Freitas" ], "givenNames": [ "Pedro" ] } ], "description": "In the 20th century, some artists took to using mathematical\nconcepts (such as the golden ratio) in their works, in the belief that\nthese would encapsulate a certain form of universal beauty. Portuguese\nartist Almada Negreiros was among them. However, he did more than absorb\nmathematical elements, he actually proved some mathematical results\nabout them. This paper addresses some of these discoveries, setting them\nin the context of the author’s views on mathematics and art.", "identifiers": [], "references": [ { "type": "Article", "id": "bib-bib1", "authors": [], "title": "\nS. P. Costa and P. J. Freitas,\nLivro de Problemas de Almada Negreiros, Leituras em Matemática, 14.\nPortuguese Mathematical Society (2015)\n" }, { "type": "Article", "id": "bib-bib2", "authors": [], "title": "\nLe Corbusier, Le Modulor. Birkhäuser Architecture, 2000.\n" }, { "type": "Article", "id": "bib-bib3", "authors": [], "title": "\nP. J. Freitas, Almada Negreiros and the regular nonagon. Recreat. Math.\nMag. 39–51 (2015)\n" }, { "type": "Article", "id": "bib-bib4", "authors": [], "title": "\nP. J. Freitas and S. P. Costa, Almada Negreiros and the geometric canon.\nJ. Math. Arts9, 27–36 (2015)\n" } ], "title": "The geometric theorems of Almada Negreiros", "meta": {}, "content": [ { "type": "Figure", "id": "S0-F1", "caption": [ { "type": "Paragraph", "content": [ "Almada Negreiros in the late 1940s" ] } ], "licenses": [ { "type": "CreativeWork", "content": [ { "type": "Paragraph", "content": [ "All rights reserved." ] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "freitas-Fig1.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p2", "content": [ "José de Almada Negreiros (São Tomé and Príncipe, 1893 – Lisbon, 1970) was a\nkey figure of 20th century Portuguese culture, in both visual arts and\nliterature. His visual work went through several stages: starting with mostly\nfigurative work, he became increasingly closer to geometric abstractionism,\nwhich he came to adopt completely in 1957, in four works displaying simple\ngeometric figures in black and white." ] }, { "type": "Paragraph", "id": "p3", "content": [ "This progressive change in style was not just the result of an aesthetic\nchoice, but also the consequence of a way of thinking about the relationship\nbetween art and geometry. In fact, Almada – this was his own choice of name –\nbelieved in a universal geometric system, underlying all visual art, throughout\ntime. He called his system “The Canon”.\n" ] }, { "type": "Paragraph", "id": "p4", "content": [ "This belief in the universality of mathematics as a foundation for art may\nremind us of Le Corbusier’s belief that his Modulor system, based on three\nsimple concepts – unit, the double and the golden section – would tap into an\nabstract and universal form of beauty. Other authors of his time, such as\nMatila Ghyka, also developed similar lines of thought." ] }, { "type": "Paragraph", "id": "p5", "content": [ "Almada’s system involved several geometrical elements, such as rectangles with\nknown proportions – such as ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "2,φ", "meta": { "altText": "\\sqrt{2},\\varphi" } }, " (the golden rectangle),\n", { "type": "MathFragment", "mathLanguage": "mathml", "text": "3,2,5,φ", "meta": { "altText": "\\sqrt{3},2,\\sqrt{5},\\sqrt{\\varphi}" } }, " – divisions of the circle into equal\nparts, and the golden angle. These were used by Almada to describe and\nunderstand artistic artifacts, and were seen as a sign that there was a\ncollection of such constructions that was used, consciously or unconsciously,\nby artists of all styles and origins." ] }, { "type": "Paragraph", "id": "p6", "content": [ "However, also like Le Corbusier, Almada also actually proved mathematical\nstatements. In his book ", { "type": "Emphasis", "content": [ "Le Modulor" ] }, " [", { "type": "Cite", "target": "bib-bib2", "content": [ "2" ] }, ", p. 37] Le Corbusier\nsuggests a construction for a right angle, placed within a rectangle, according\nto certain rules. The construction (which is not presented in full clarity and\ndetail) actually does lead to an approximate right angle." ] }, { "type": "Paragraph", "id": "p7", "content": [ "Almada goes much farther in his speculative geometry. In two collections of\ndrawings as well as some artist’s notebooks, comprising more than a hundred\ncompleted works as well as many additional sketches, he presents constructions\nof a geometric nature which can rightly be regarded as artworks, but which are\nat the same time geometric results related to the elements of the Canon,\nshowing their intrinsic proximity. Figures ", { "type": "Cite", "target": "S0-F2", "content": [ "2" ] }, " and\n", { "type": "Cite", "target": "S0-F3", "content": [ "3" ] }, " present two examples from the collection “Language of the\nsquare”, a set of about forty-five finished drawings on paper,\n", { "type": "MathFragment", "mathLanguage": "mathml", "text": "50×70", "meta": { "altText": "50\\times 70" } }, " cm." ] }, { "type": "Figure", "id": "S0-F2", "caption": [ { "type": "Paragraph", "content": [ "Figure 2" ] } ], "licenses": [ { "type": "CreativeWork", "content": [ { "type": "Paragraph", "content": [ "All rights reserved." ] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "freitas-Fig2.png", "mediaType": "image/png", "meta": { "inline": false } } ] }, { "type": "Figure", "id": "S0-F3", "caption": [ { "type": "Paragraph", "content": [ "Figure 3" ] } ], "licenses": [ { "type": "CreativeWork", "content": [ { "type": "Paragraph", "content": [ "All rights reserved." ] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "freitas-Fig3.png", "mediaType": "image/png", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p8", "content": [ "Both drawings can be regarded as protocols for geometric constructions, both\nstarting with a square divided in two equal parts by a horizontal line, with an\ninscribed quarter circle. The construction in Figure ", { "type": "Cite", "target": "S0-F2", "content": [ "2" ] }, " leads\nto two red lines, marked ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "5", "meta": { "altText": "\\sqrt{5}" } }, " and ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "φ", "meta": { "altText": "\\varphi" } }, ", indicating that these lines\nare diagonals of rectangles with those proportions, and sides parallel to the\nsides of the square." ] }, { "type": "Paragraph", "id": "p9", "content": [ "It is not difficult to verify the correctness of the proportions. If we take\nhalf of the side of the square as our unit measure, then, by the Pythagorean\nTheorem, the green diagonal measures ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "5", "meta": { "altText": "\\sqrt{5}" } }, ", and as this measure is\ntransported, by compass, to the top side of the square, we do obtain a\nrectangle with this proportion." ] }, { "type": "Paragraph", "content": [ "The small green horizontal line would then measure ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "5-2", "meta": { "altText": "\\sqrt{5}-2" } }, ", and as we add\nit to half the square, we get a length of ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "1+5-2=5-1", "meta": { "altText": "1+\\sqrt{5}-2=\\sqrt{5}-1" } }, ". The\nproportion of the rectangle having as diagonal the red line marked ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "φ", "meta": { "altText": "\\varphi" } }, "\nwould then be" ] }, { "type": "MathBlock", "id": "S0.Ex1", "mathLanguage": "mathml", "text": "25-1=2⁢(5+1)4=1+52=φ.", "meta": { "altText": "\\frac{2}{\\sqrt{5}-1}=\\frac{2\\bigl(\\sqrt{5}+1\\bigr)}{4}=\\frac{1+\\sqrt{5}}{2}=\\varphi." } }, { "type": "Paragraph", "content": [ "The numbers 9 and 10 on the right refer to the divisions of the circle into 9\nand 10 parts, which are achieved by the intersection of the red lines with the\ncircle arc. As it happens, the proportions of the rectangles are precise, but\nthe divisions of the circle are not: they represent very good approximations,\neach having an error of about 0.7 %. In the book [", { "type": "Cite", "target": "bib-bib4", "content": [ "4" ] }, "], the authors\npresent a few more analyses of this type concerning about 30 of Almada’s\ndrawings." ] }, { "type": "Paragraph", "content": [ "In Figure 3, a similar but more intricate construction leads to four\npoints marked ", { "type": "Emphasis", "content": [ "a" ] }, ", ", { "type": "Emphasis", "content": [ "b" ] }, ", ", { "type": "Emphasis", "content": [ "c" ] }, " and ", { "type": "Emphasis", "content": [ "o" ] }, ", and three\nlengths, which Almada interprets as follows:" ] }, { "type": "MathBlock", "id": "S0.Ex2", "mathLanguage": "mathml", "text": "a⁢b=⊙¯14,a⁢o=⊙¯10,a⁢c=⊙¯9.", "meta": { "altText": "ab=\\frac{\\overline{\\odot}}{14},\\qquad ao=\\frac{\\overline{\\odot}}{10},\\qquad ac=\\frac{\\overline{\\odot}}{9}." } }, { "type": "Paragraph", "content": [ "This is again a reference to the division of the circle into equal\nparts. In this case, the 10th part is exact, the 14th part has an error\nof 1 % and the 9th part has an astonishingly tiny error of\n0.001 %.", { "type": "Note", "id": "idm229", "noteType": "Footnote", "content": [ { "type": "Paragraph", "id": "footnote1", "content": [ "The article [", { "type": "Cite", "target": "bib-bib3", "content": [ "3" ] }, "] compares the approximation for\nthe 9th part of the circle achieved by this construction with that of\nother constructions and concludes this is one is the best of all those\nanalysed." ] } ] } ] }, { "type": "Paragraph", "id": "p12", "content": [ "In the history of mathematics, the problem of dividing the circle into ", { "type": "Emphasis", "content": [ "n" ] }, "\nequal parts with straightedge and compass has a respectable place. Thinking\nabout prime values of ", { "type": "Emphasis", "content": [ "n" ] }, ", it has known since ancient Greece that it is\npossible to divide the circle into 3 and 5 parts, but no method was found for\n7, 11 or 13 parts. It was Gauss, in 1796, who proved that it was possible to\ndivide the circle into 17 parts, a result that came to be included in Section\nVII of ", { "type": "Emphasis", "content": [ "Disquisitiones Arithmeticae" ] }, ".", { "type": "Note", "id": "idm242", "noteType": "Footnote", "content": [ { "type": "Paragraph", "id": "footnote2", "content": [ "There is an anecdote that\nGauss was so pleased by this result that he requested that a regular\nheptadecagon be inscribed on his tombstone. The stonemason declined, stating\nthat the difficult construction would essentially look like a circle." ] } ] }, "\nEventually, he found a sufficient condition for the division into ", { "type": "Emphasis", "content": [ "n" ] }, "\nparts, and stated that this condition should also be necessary. In 1837,\nPierre Wantzel proved the necessity, leading to the result that became known\nas the Gauss–Wantzel theorem." ] }, { "type": "Claim", "id": "Thmgwtheoremx1", "claimType": "Theorem", "label": "Gauss–Wantzel’s theorem.", "title": [ { "type": "Strong", "content": [ "Gauss–Wantzel’s theorem" ] }, { "type": "Strong", "content": [ "." ] } ], "content": [ { "type": "Paragraph", "content": [ { "type": "Emphasis", "content": [ "A circle can be divided into\n", { "type": "MathFragment", "mathLanguage": "mathml", "text": "n", "meta": { "altText": "n" } }, " parts with straightedge and compass if and only if" ] } ] }, { "type": "MathBlock", "id": "S0.Ex3", "mathLanguage": "mathml", "text": "n=2k⁢p1⁢⋯⁢pt", "meta": { "altText": "n=2^{k}p_{1}\\cdots p_{t}" } }, { "type": "Paragraph", "content": [ { "type": "Emphasis", "content": [ "where ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p1,…,pt", "meta": { "altText": "p_{1},\\ldots,p_{t}" } }, " are distinct ", { "type": "Emphasis", "content": [ "Fermat primes" ] }, ", that\nis, primes of the form ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "22r+1", "meta": { "altText": "2^{2^{r}}+1" } }, ". The only known Fermat primes\nare ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "3,5,17,257", "meta": { "altText": "3,5,17,257" } }, " and ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "62237", "meta": { "altText": "62237" } }, "." ] } ] } ] }, { "type": "Paragraph", "id": "p13", "content": [ "Going back to Almada’s constructions, we note that 7 is not a\nFermat prime, and ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "9=3×3", "meta": { "altText": "9=3\\times 3" } }, " (3 is a Fermat prime that occurs twice in\nthe factorization). Therefore, a circle cannot be divided precisely into\n9 and 14 parts using only straightedge and compass." ] }, { "type": "Paragraph", "id": "p14", "content": [ "The purpose of these constructions is mainly to show that the various\nelements of Almada’s Canon have a natural and harmonious\nrelationship amongst themselves, which is revealed by the elegance and\nsimplicity of the geometric constructions he presents. Thus, the aim is\nprimarily symbolic and philosophical rather than mathematical.\nNevertheless, these drawings also present original constructions for the\ndivisions of the circle and for producing rectangles with a given\nproportion, so in fact, they represent original mathematical results,\neven though some of them are approximate." ] }, { "type": "Paragraph", "content": [ "Many more drawings exist, some of them leading to general statements\nwhich one might regard as theorems, if it weren’t for the fact that\nthey represent approximations. One of these statements is" ] }, { "type": "MathBlock", "id": "S0.Ex4", "mathLanguage": "mathml", "text": "2⁢⊙¯9+⊙¯10=2⁢r.", "meta": { "altText": "2\\frac{\\overline{\\odot}}{9}+\\frac{\\overline{\\odot}}{10}=2r." } }, { "type": "Paragraph", "content": [ "The meaning of this equality is that 2 chords of the 9th part of the\ncircle plus a chord of the 10th part will equal the diameter of the\ncircle (two times the radius). The first side of the equation is\nactually equal to ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "1.986⁢r", "meta": { "altText": "1.986r" } }, ", an error of 0.7 %." ] }, { "type": "Figure", "id": "S0-F4", "caption": [ { "type": "Paragraph", "content": [ "A construction of the pentagram from the 9th part of the\ncircle" ] } ], "content": [ { "type": "ImageObject", "contentUrl": "freitas-Fig4.svg", "mediaType": "image/svg+xml", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p16", "content": [ "The presence of the numbers 9 and 10 in this equality (and in many geometric\nconstructions) is not fortuitous. Almada believed there was a special\nconnection between these two numbers, which he called the “Relation 9/10” and\nstrove to seek for them and to connect them with other elements of his Canon.\nThis is also the reason for the presence of these numbers in\nFigures ", { "type": "Cite", "target": "S0-F2", "content": [ "2" ] }, " and ", { "type": "Cite", "target": "S0-F3", "content": [ "3" ] }, "." ] }, { "type": "Paragraph", "id": "p17", "content": [ "The statement above is used in an approximate construction for the pentagram\n(another of Almada’s favourite figures, because of its relation to the\nPythagoreans). The construction starts with a circle with two elements marked:\na diameter and a 9th part, measured from one of the extremes of the diameter.\nIn Figure ", { "type": "Cite", "target": "S0-F4", "content": [ "4" ] }, ", the 9th part is the arc AP", { "type": "Subscript", "content": [ "9" ] }, ", on\nthe left, and the diameter is marked AB." ] }, { "type": "Paragraph", "id": "p18", "content": [ "The construction now goes as follows. The arc of a circle centred at A is drawn\nfrom the point P", { "type": "Subscript", "content": [ "9" ] }, " to the diameter, marking point D, through\nwhich one of the lines of the pentagram is drawn: CE, perpendicular to AB. This\npoint D is now the centre of the half circle AF. Point F now determines a new\narc of a circle with centre B, yielding points G and H, through which the\nremaining lines of the pentagram are defined." ] }, { "type": "Paragraph", "id": "p19", "content": [ "According to this construction, lines AD and DF are chords of the 9th part of\nthe circle, and line FB is the chord of the 10th part (which is the chord of\nboth arcs BG and GH). These three lines add up to a diameter, which illustrates\nthe previous equation, connecting it to the pentagram." ] }, { "type": "Paragraph", "id": "p20", "content": [ "We emphasize that this is not an accurate construction – it is actually\nimpossible for it to be accurate, according to Gauss–Wantzel’s theorem, since\notherwise the 9th part of the circle would be constructible, if we could start\nwith a pentagon and extract point P", { "type": "Subscript", "content": [ "9" ] }, " from it." ] }, { "type": "Paragraph", "id": "p21", "content": [ "The fact that some of these results are approximations, and that these\nappear among exact results with no distinction, is actually quite\nrevealing of Almada’s methods. We have reason to believe that\nAlmada didn’t actually compute the exact measures of the elements\nhe claimed to produce, but probably only checked them visually, using\ninstruments of measure or other elements of comparison. So, his way of\nestablishing geometric results is not the same as the one used by\nmathematicians. And even though Almada was aware that some of his\nresults were approximate, he stuck to them and never distinguished\nbetween exact and approximate ones. This was probably because he was\nmore interested in the visual aspects of such results, and for this\neffect, some approximations are acceptable." ] }, { "type": "Paragraph", "id": "p22", "content": [ "One of his last artworks, which can be considered his geometric legacy\nand a summary of many of his statements on this subject, is the mural\n", { "type": "Emphasis", "content": [ "Começar" ] }, " (", { "type": "Emphasis", "content": [ "To begin" ] }, ", 1968), which is located in the main\nhall of the building of the Gulbenkian Foundation in Lisbon. It is a\nvery intricate collection of lines and circles, inscribed in stone, with\ndimensions ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "12.87×2.31", "meta": { "altText": "12.87\\times 2.31" } }, " m (see Figure ", { "type": "Cite", "target": "S0-F5", "content": [ "5" ] }, ")." ] }, { "type": "Figure", "id": "S0-F5", "caption": [ { "type": "Paragraph", "content": [ "Mural ", { "type": "Emphasis", "content": [ "Começar" ] }, " by Almada Negreiros (1968)" ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by-sa/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [ "by ", { "type": "Link", "target": "https://commons.wikimedia.org/w/index.php?curid=35070536", "content": [ "Manuel V. Botelho" ] } ] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "freitas-Fig5.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p23", "content": [ "The mural is usually divided into five parts. The first one displays the\npentagram with the construction presented in Figure ", { "type": "Cite", "target": "S0-F4", "content": [ "4" ] }, " as the\nmain motif. Then, we find a 16-point star, which is an allusion to a drawing\nby Leonardo da Vinci appearing (apparently by mistake) in Geneva’s codex of\nLuca Pacioli’s ", { "type": "Emphasis", "content": [ "Divina Proportione" ] }, ". The central element is again a\npentagram, which Almada associates to a coin, minted by Portugal’s first king,\nwhich is set in the midst of many other constructions that Almada used to study\nsome 15th century Portuguese paintings. The right part of the mural presents\nreferences to the Minoan civilization and to a medieval poem associated with a\nguild of cathedral stonemasons. There is a virtual guided tour at\n", { "type": "Link", "target": "https://gulbenkian.pt/almada-comecar/en/", "content": [ "gulbenkian.pt/almada-comecar/en/" ] }, "\nwhere the reader can find more detailed information about this mural. However,\nwhat we wish to point out with these brief remarks is that Almada’s geometry is\ntruly an effort to unite all art and all cultures." ] }, { "type": "Paragraph", "id": "p24", "content": [ "A thorough study of the geometric works in Almada’s estate has been undertaken,\nin a collaboration between Simão Palmeirim and the author; some of its results\ncan be found in [", { "type": "Cite", "target": "bib-bib4", "content": [ "4" ] }, "]. We hope that this study can bring to light not only\nthe remarkable visual aspects of Almada’s geometric work, but also the\nmathematics behind it, which not only yield new geometric results, but also\nrepresent a powerful statement about the author’s thought regarding geometry\nand art." ] }, { "type": "Paragraph", "id": "p25", "content": [ { "type": "Emphasis", "content": [ "Acknowledgements" ] } ] }, { "type": "Paragraph", "id": "p26", "content": [ "We wish to thank the family of Almada Negreiros for allowing us to study his\nestate and for giving permission to use the reproductions in\nFigures ", { "type": "Cite", "target": "S0-F1", "content": [ "1" ] }, ", ", { "type": "Cite", "target": "S0-F2", "content": [ "2" ] }, " and ", { "type": "Cite", "target": "S0-F3", "content": [ "3" ] }, ". This\nstudy was done within the scope of project ", { "type": "Emphasis", "content": [ "Modernismo Online" ] }, ",\nUniversidade Nova de Lisboa\n(", { "type": "Link", "target": "http://www.modernismo.pt", "content": [ "www.modernismo.pt" ] }, "), which aims to\ncollect in digital form the heritage of Portuguese Modernism." ] }, { "type": "Paragraph", "id": "p27", "content": [ "Figure ", { "type": "Cite", "target": "S0-F5", "content": [ "5" ] }, " is by Manuel V. Botelho\n(", { "type": "Link", "target": "https://commons.wikimedia.org/w/index.php?curid=35070536", "content": [ "commons.wikimedia.org/w/index.php?curid=35070536" ] }, "),\nCC BY-SA 4.0\n(", { "type": "Link", "target": "https://creativecommons.org/licenses/by-sa/4.0/", "content": [ "creativecommons.org/licenses/by-sa/4.0/" ] }, ")." ] }, { "type": "Paragraph", "id": "p28", "content": [ "The author is supported by FCT, I.P., through Project UID/HIS/00286/2019." ] }, { "type": "Paragraph", "id": "authorinfo", "content": [ "Pedro J. Freitas has a PhD in mathematics and works in the Department of History and Philosophy of Sciences in the University of Lisbon. He is a member of the Interuniversity Center for the History and Philosophy of Sciences.\nHis teaching and research are mainly related to the history of mathematics, recreational mathematics and relations between mathematics and art. He is also involved in mathematical outreach.\n\n", { "type": "Link", "target": "mailto:pjfreitas@fc.ul.pt", "content": [ "pjfreitas@fc.ul.pt" ] } ] } ] }