Writers' Workshop for ARCH 531 / McGill University / School of Architecture / Fall 2012

Week Six: Luca Pacioli








Jacopo de’ Barbari’s symbolically charged painting Portrait of Fra Luca Pacioli (1495) remains a powerful lens through which to understand the significance of Pacioli’s thinking and writing.  Taking only one symbolic element within the painting, please elaborate on what the component means in relation to the thinking of Luca Pacioli.  Kindly do not choose an element that has been discussed by a previous blogger.


11 responses

  1. Trebuh

    Luca Pacioli was an Italian mathematician and a pioneer in the field of accounting. He also wrote the book, “De Divina Proportione” published in Venice in 1504 and illustrated by Leonardo da Vinci. This book was divided in three parts; the first being the study of proportions through the golden ratio and the study of polygons. The painting Portrait of Fra Luca Pacioli (1495) is one element that helps to understand Pacioli’s thinking. “There have been many attempts to understand the portrait of the mathematician Luca Pacioli which now hangs in the Museo di Capodimonte, Naples […]”. [1]

    In this painting mathematical topics are prominent and illustrated through various objects. The meaning of these objects is a topic for interpretation: “For centuries, historians of science have tried to identify the puzzling mathematical references and objects that are prominently displayed in the panel and that make it a favourite illustration for surveys early modern science.” [2]. This image exposes different mathematical elements, mostly geometrical tools, but also a dodecahedron model and a rhombicuboctahedron. The rhombicuboctahedron is a geometric shape first conceptualized by Pacioli and illustrated by Da Vinci in Pacioli’s famous book on mathematics. The hanging rhombicuboctahedron is composed of 26 individual faces, eight triangular and eighteen square. In this painting the rhombicuboctahedron appears fragile, built from glass materials. As such, it was likely used primarily for contemplation opposed to physical manipulations, as well as to “visualize geometrically complex constructions”. [3] Additionally, the polyhedron was partially filled with water, which concerns the visual and spatial perspective of those depicted in the painting. “In examining the suspended crystal, the beholder discovers both the picture’s vanishing point – where the orthogonals of its perspective system meet – and its preferred viewing point. In fact, the level of the water at the crystal’s sides reveals that the vanishing point in located in Pacioli’s head, at the height of the gentleman’s eyes.” [4]

    The complexity of the rhombicuboctahedron and its role as the subject of the painting is critical to our understanding of the image. It is clear that Pacioli is studying intensively the shape’s mathematical and geometrical properties, apparent from his eyes looking through the shape directly. However, from the image it is not as obvious what Pacioli is evaluating; the reflection, shadows, or lines? The content of the blackboard sketch in the foreground, as well as additional objects in the image, could give suggestions as to the thoughts of Pacioli.

    [1] MacKinnon, Nick, The Portrait of Fra Luca Pacioli, The Mathematical Gazette, Vol 77, No 479 (jul 1993), p. 130

    [2] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 83

    [3] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 95

    [4] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 95

    October 13, 2012 at 11:07 am

  2. Amalin

    As Trebuh has appropriately suggested, Jacopo de’ Barbari offers a very dynamic portrayal of Fra Luca Pacioli: “it is not a static representation of a mathematician surrounded by arbitrary icons that merely symbolise his craft, but an actual lesson in the geometry of the dodecahedron.” [1] Nonetheless, what is Pacioli illustrating on the blackboard? For some time, the consensus was that he revealed proposition 12 of Euclid’s Elements XIII, which states: “if an equilateral triangle [is] inscribed in a circle, the square on the side of the triangle is triple of the square on the radius of the circle.” However, when we look closely, the perspective view of the diagram in reference does not coincide flawlessly with the drawing.

    The most likely possibility is that “Pacioli is expositing a proposition from the so-called Book XIV. Books XIV and XV both extend Book XIII in various ways which were later subsumed by the extensions of Pappus, and both books are used extensively in Piero della Francesca’s De prospettiva pingendi and Trattato d’abaco, and in Pacioli’s De divina proportione. In particular we find Elements XIV, Proposition 2: The same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere.” [2] It certainly seems logical that Pacioli would try and combine the knowledge obtained from Euclid’s Elements XIII during the apparent study of the rhombicuboctahedron in this painting. “In summary the portrait shows Pacioli in the act of expositing Elements XIV.8. The 1482 Euclid is open at Book XIII.12, which is required in the proof of XIV.8, and the only disturbing issue is that the book is shown open half way, when Book XIII is nearly at the end. Practicalities explain this: the portrait is painted with the book blank and open near the middle where it will lie flat. The artist then adds the text very exactly.” [3]

    1 Mackinnon, Nick. “The Portrait of Fra Luca Pacioli.” The Mathematical Gazette. 77.479 (1993): 131.
    2 Ibid., 138.
    3 Ibid., 139.

    October 13, 2012 at 2:57 pm

  3. reckon

    The Portrait of Fra Luca Pacioli (1495) establishes Pacioli as key mathematician of his time. The painting shows both the “first ancient and the first modern mathematical books to be printed, respectively, Euclid’s Elements and Pacioli’s Summa” [1] I will focus on the open book, Euclid’s Elements, to which Pacioli is pointing. It is possible to establish that it is open specifically to Book XIII, “the book devoted to the construction of the regular solids,” [2] specifically at proposition 12, which is required in the proof of XIV.8, the geometry that Pacioli is drawing with his right hand, as discussed by Amalin.

    There are conflicting opinions as to which edition of Elements has been painted, one conclusion stating that the inconsistent details allow us to assume that the artist decided not to reproduce any particular edition, [3] rather “visually asserting Pacioli’s intimate knowledge of the Elements,” [4] perhaps even predicting his own publication of a revised edition of the Elements in 1509.

    One of the main differences between the standard printings of Elements and the artist’s reproduction is that he has left “slightly too much white space toward the bottom of the left hand page.” [5] The open and annotated (in red ink) margins visible on both pages “indicate the intended use of the pen offered to the viewer, […] encouraging beholders to go through their copy of Elements to annotate its propositions and figures,” [6] but most importantly that Pacioli has read the book “line by line and understood completely.”[7]

    [1] Baldasso, Renzo. Portrait of Luca Pacioli and Disciple: A New Mathematical Look, Art Bulletin, March-June 2010, p. 89.
    [2] MacKinnon, Nick. The Portrait of Fra Luca Pacioli, The Mathematical Gazette, Vol 77, No 479, July 1993. p. 131.
    [3] Baldasso, Renzo. Portrait of Luca Pacioli and Disciple: A New Mathematical Look, Art Bulletin, March-June 2010, p. 93.
    [4] Ibid., p. 93.
    [5] MacKinnon, Nick. The Portrait of Fra Luca Pacioli, The Mathematical Gazette, Vol 77, No 479, July 1993. p. 138.
    [6] Baldasso, Renzo. Portrait of Luca Pacioli and Disciple: A New Mathematical Look, Art Bulletin, March-June 2010, p. 93.
    [7] Ibid., p. 93.

    October 13, 2012 at 3:05 pm

  4. Esiolé

    At first, what is extremely thought provoking in this painting is the presence of the unidentified character with Fra Luca Pacioli. “The main difficulty in interpreting this painting, however, is to understand it as a whole, identifying its subject and meaning as explicated by its pictorial, iconographic, and cultural elements and their interrelations.” [1] Over time, even though no name was ever mentioned, some experts have narrowed down the possible identities of this persona. The most likely hypothesises link this figure to Guidobaldo da Montefeltro, duke of Urbino, and Albrecht Dürer.

    However, these two possibilities foster two very different understandings of the painting and of its function. “If the second figure were this or another artist […] the painting could have served as a record certifying a young painter’s intellectual relationship with an eminent mathematician If the second figure were the duke, this painting should then be considered a patronage item, possibly commissioned to make official a new relationship that established Guidobaldo, the dedicatee of the Summa, as Pacioli’s patron, rather than the pupil he had been in his youth when he was tutored by the friar.” [2] By observing the panel, we see that the stature and thinness of the second character pictures him as taller than the friar, thus enlightening us on the social status of the persona as a probable patron. Therefore, the theory of the character being the Duke of Urbino is the most likely one. Another important elements of the painting is probably the way the instruments are placed on the table: they are positioned for the viewer, out of reach of the two characters, as though the third persona, us, was also an important part of the panel hierarchy. The viewer is the one who sees the painting, thus making him the one to learn from it. We are the pupils and the two characters are the master and patron, opposed to a vision of a master with is pupil. That is what can implicitly be deduced of the peculiar mathematic tools organization and characters disposition in the painting. “The orientation of the compass and the square together with the pen’s position clearly indicate that these instruments are offered to the beholder: they are ours to write and draw with.” [3]

    [1] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 83.
    [2] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 86.
    [3] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 88.

    October 13, 2012 at 7:26 pm

  5. petersburg

    Several objects shown in the Portrait of Luca Pacioli are very intriguing with regard to this mathematician’s intention behind this scene. In fact, Pacioli haven’t asked Jacopo de’ Barbari to paint this setting just for the sake of leave a record of his creations. Spatially speaking, perhaps the most apparent is the glass polyhedron filled with water depicting volume, transparency, refraction, and reflection in a successful way. However, the spatial accuracy of the other polyhedron, simpler in term of shape and materiality (the wooden dodecahedron illustrated in the bottom right above the closed book), is incontestable, yet it requires a closer examination.

    From the viewer’s perception, one might doubt the perspective of the dodecahedron after careful observation; from the general composition and according to this one point perspective view, it seems to be wrong if compared to the water surface in the glass polyhedron. Since this transparent rhombicuboctahedron is closer to the eye level of the painter, one might think that the top face of the wooden solid has to be less flat and be vertically larger. This statement is interesting but again, the volume is correctly illustrated in the space. “As in the Platonic doctrine, here, too, appearances and shadows deceive […]. But to realize this implies that one has stereometric knowledge of Platonic solids, as well as the capacity to picture a dodecahedron in space […], to see that its top and bottom faces are not actually parallel.” [1] One could consult Summa, the open book in front of Pacioli, with illustrations of the polyhedron figures, “the first such [ones] to be disseminated by means of the printing press”. [2]

    This is just one example of “actual lesson in the geometry of the dodecahedron”. [3] Also, one symbol among many others proving “the artist [is] encouraging us to see the scene as a dynamic one”. In fact, the elements and the characters are real and “embedded in time and space but […] also in a social structure, in the logical structure of the elements of Euclid, and in the mathematics of the Renaissance”. [4]


    [1,2] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 95

    [3,4] MacKinnon, Nick, The Portrait of Fra Luca Pacioli, The Mathematical Gazette, Vol 77, No 479 (jul 1993), p. 131

    October 14, 2012 at 5:13 am

  6. labyrinth

    The ‘Portrait of Fra Luca Pacioli’ painted by the young Jacopo di Babari is full of symbolism that conveys Pacioli’s theories and methods. One intriguing aspect of the painting is the presence of the two books, and their relation to one another.

    The two books in the painting are Euclid’s ‘Elements’, which lies open as a reference at Pacioli’s fingertips, and Pacioli’s own ‘Summa de arithmetica geometria proportioni et proportionalita’ (1494), which lies closed and untouched on the table. It is interesting to note that Pacioli is not depicted as referencing his own writings, as was common in portraits at the time, but is instead using Euclid’s writings as a mathematical reference; this immediately gives prominence to the open book, rather than the closed ‘Summa’ [1].

    This gesture implies several things. First, it elevates Pacioli’s public image as a leading mathematician, by associating him with the writings of the ancients, as well as by highlighting the revised diagrams in the open book, which do not correspond with the printed versions at the time, and which Pacioli obviously corrected himself [2]. Secondly, the placement of the closed book among the mathematical tools scattered across the table, and it’s proximity to the edge of the painting and thus to the viewer, alludes to Pacioli’s ‘Summa’ as a tool for students and scholars to be used as an “introduction” to understanding the theoretical mathematics of Euclid’s ‘Elements’ [3]. Lastly, the importance given to Euclid’s ‘Elements’ demonstrates that it is the ultimate source of mathematical knowledge, which was seen as being synonymous with the cosmos in the Renaissance period. It is interesting to note that this “divine knowledge” is not coming from the Bible – it is coming from the writings of an ancient mathematician, clearly illustrating the shift to humanism in this time period.

    [1] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 92.

    [2] Ibid., p. 93-94.

    [3] Ibid., p. 94.

    October 14, 2012 at 11:58 am

  7. VDSY

    In this painting, heavily charged with symbols, many narratives can be put forward. Very good analysis of mathematical books in this painting were made here. These two books, Euclid’s Elements and Fra Luca Pacioli’s Summa are of great importance to the understanding of this painting and of the time that it has been painted. The scene focuses on a composition centered on Fra Luca Pacioli, a central triangle connects the head of Pacioli to Euclid’s book under his finger and then to geometrical demonstration he is drawing on the chalkboard. The rest of the scene seems to gravitate in periphery.

    Looking to the books presented on the table, specific details can give us ways of understanding. First, the book written by Pacioli, placed on the table, closed and partially off the canvas, possesses in its binding details that are not usual for the publication of treaties.This implies that this copy is probably unique. The representation of this book puts forward some extent on the use of the Treaty: “The” solidity “of the pages of the Summa would have caught the eye of original viewers, making them realize that this book was brand new: only freshly cut paper can be so precisely compact. “[1] No spaces or movement in the pages are apparent, noting that its utilization is not usual. The disposition of the pages at the corner of the book also allows to guess the binder made​: “[…] depending on the binding process, usually the resulting edge is either concave or convex; its straightness confirms the extra care in the binding and the fact that this book was probably never opened.»[2] The representation of this book, ornamented, allows us to deduce that it most likely doesn’t belongs to Fra Luca Pacioli, as the Euclid’s book under his hand. This as a binding «with a multicolor silk headband woven with no fewer than four threads of different colors» [3] demonstrates the quality of the book. All these elements highlighting the quality of the books presented in this painting raise the interrogation about their owner.

    It is possible to deduce that these books belong to the person illustrated behind Pacioli and that this one is receiving the teaching of the master in his quarter. With the help of the Pacioli’s treaty, the instruments pictured in the foreground, the observers could understand Euclid’s Elements, and discover the geometry of a dodecahedron (represented in timber) and reach the comprehension of the abstract idea of a rhombicuboctahedron, represented in translucent solid.

    [1] Baldasso, Renzo. Portrait of Luca Pacioli and Disciple: A New Mathematical Look, Art Bulletin, March-June 2010, p. 94.
    [2] Ibid., p. 94.
    [3] Ibid., p. 94.

    October 15, 2012 at 5:22 pm

    • Would you say the student is not worthy of Pacioli’s teachings, as he never opened his expensive book? Could this be interpreted as being a critique of wealth not being enough to possess a rich mind?

      October 15, 2012 at 6:18 pm

  8. The presence of one of Plato’s solids on the red book—the dodecahedron—informs the viewer that Fra Luca Pacioli is not only teaching Euclid’s Elements. The relevance of Plato’s Theory of Forms to the mathematical realm is undeniable and celebrated in subtle ways in the drawing. In the Allegory of the Cave and the Analogy of the Divided Line he describes the precedence of the Forms and the mathematical thinking over the physical world.[1] When the philosopher comes out of the Cave, their is a clear sense of ascent over the mere world of shadows. Physical ascension has for the longest time been impossible for humans, while the heavens are most often described as being a higher world. The only realm in which humans were able to rise was in the intellectual one, in which one would seek deeper comprehension of mathematics and philosophy, being able to see the Forms. By placing the vanishing point of the linear perspective just above Pacioli’s head[2], the painter underlines Euclid’s indebtedness to the founder of the Academy. He also places the viewer—who seems to be the real student, being offered the tools—slightly above Pacioli himself, suggesting that one rises through knowledge.

    [1] Plato, The Republic, 360 B.C.E, Book VI (translated by Benjamin Jowett)
    [2] MacKinnon, Nick, The Portrait of Fra Luca Pacioli, The Mathematical Gazette, Vol 77, No 479 (jul 1993), p. 132

    October 15, 2012 at 7:09 pm

  9. Trebuh introduces the rhombicuboctahedron and quotes various scholars. You could look farther into the object though. Who discovered the rhombicuboctahedron? Who do most analysts think actually painted it? Why has it been called “symbolic of the eternal truth and clarity of mathematics”? What are the reflections representing? This needs more fleshing out. Amalin brings up the tensions between the theorem that Pacioli is pointing to and the one that is exposited, the two being contingent upon each other. Reckon brings up this subject again. Esiolé insists that we are the pupil and the second figure is the patron. Be careful in your analysis Petersburg – the Summa is the closed book not the open one. Is it important that the dodecahedron consists of twelve pentagons? Labyrinth points to the non-biblical nature of the literature present and stresses the difference between open and closed books. VDSY brings up the books again to discuss binding and authorship. Madeck illustrates the significance of the vanishing point above Pacioli’s head.

    Overall as a group I think we need some more thorough scrutiny when looking at this painting.

    October 16, 2012 at 3:30 pm

  10. Trebuh

    The rhombicuboctahedron is certainly a complex element in Jacopo de’ Barbari’s painting, as evidenced by the fact that the focal point of the painting is Pacioli directly observing the polyhedron. My previous post focused mainly on the relation between Pacioli and the polyhedron depicted in the painting. Various theories have been generated as to the creator of the rhombicuboctahedron. Leonardo Da Vinci originally conceived the shape in the Da Divina Propotione, however opinion varies regarding the object’s origins. Some experts propose Da Vinci as the inventor of the shape, but other analysts believe that Da Vinci was simply reproducing the shape from wooden models that Pacioli originally conceived and designed. It has also been suggested that Da Vinci himself may have painted the rhombicuboctahedron in Luca Pacioli’s image. Ultimately, whoever may have painted the shape was certainly preoccupied by giving as much information as possible through the rhombicuboctahedron. Not only were they fully aware of the mathematical concepts behind the shape, they also went on to paint a reflection in the polyhedron. The reflections visible through the polyhedron suggest the physical location of the painting: “The translucent solid, half filled with what may be water, carries the same image of a princely mansion, probably intended to be Urbino’s palace (The Palazzo Ducale), sketchily reflected three times; it also contains the reflection of the green tablecloth together with Pacioli’s lighter and the nobleman’s darker presence.” [1]

    In addition to indicating the location of the image, I believe that there are further aspects of this shape that are open to interpretation as it forms both a central and critical subject in Jacopo de Barbari’s image.

    [1] Baldasso, Renzo, Portrait of Luca Pacioli and Disciple : A New Mathematical Look, Art bulletin, 2010, p. 87

    October 17, 2012 at 12:18 am