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Image of a copper plate engraving depicting an angle sitting with her head in her hands next to a strange shape. A banner above reads: MELENCOLIA I
Albrecht Dürer, Melencolia I. Engraving. 1514. Bradley Collection. Library of Congress, Prints and Photographs Division.

Art and Math: The Unusual Shape in Melencolia I

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“Angels can fly,” G. K. Chesterton once wrote, “because they can take themselves lightly.” If this twentieth-century author and lay-theologian is correct, then it is no wonder that the winged figure in Albrecht Dürer’s famous 1514 copper engraving Melencolia I appears grounded. Dürer’s subject has undertaken a daunting task. The celestial figure sits holding a pair of geometrical dividers in one hand while she rests her chin dejectedly in the other. Abandoned instruments for wood carving are strewn haphazardly in the foreground of the engraving, and the subject gazes toward the horizon with a stony expression. Clearly, this downcast figure is an artist, and what artist takes themselves lightly? Likely not the printmaker, painter, and art theorist, Albrecht Dürer (1471-1528) for whom this melancholy, celestial being is said to have been a spiritual self-portrait.

Image of a copper plate engraving depicting an angle sitting with her head in her hands next to a strange shape. A banner above reads: MELENCOLIA I
Albrecht Dürer, Melencolia I. Engraving. 1514. Bradley Collection. Library of Congress, Prints and Photographs Division.

The son of a goldsmith, Dürer had received little in the way of a formal education in his hometown of Nuremberg, Germany. Nevertheless, by 1514 Dürer had traveled twice to Italy; he was close friends with the leading humanist in Nuremberg, Willibald Pirckheimer; he was in contact with leading Italian artists like Giovanni Bellini; he had even received commissions from the Holy Roman Emperor Maximilian I. No mere journeyman, Albrecht Dürer was actively grappling with the ideas that he encountered within the larger cultural movement of Renaissance humanism.

Drawing showing a face and one large hand in scratchy brown lines with a crumbled pillow beneath. The face is a self-portrait.
Albrecht Dürer. Self-portrait, Study of a Hand and a Pillow (recto). Pen and brown ink on paper. Robert Lehman Collection. Metropolitan Museum of Art. Public Domain.

One of these ideas was the concept of perspective: the correct rendering of three-dimensional objects on a two-dimensional surface. Albrecht Dürer became fascinated with how the study of geometry and measurement enabled craftsmen like himself to theorize about the concept of beauty as a universal and to render beauty in the particular through artistic mediums. In 1525, Dürer published the first edition of his Treatise on Measurement (Underweysung der messung, mit dem Zirckel und Richtscheyt), a German manual summarizing classical sources treating geometry and perspective. In his introduction, Dürer explains that the universal nature of his subject is applicable not only to the work of painters but also to “goldsmiths, sculptors, stonemasons, carpenters, and all those for whom using measurement is useful.”

Woodcut image in black ink of two men plotting points of a lute using a measuring device.
Final woodcut image from Dürer’s Treatise on Measurement (1525) showing how to plot points on the surfaces of a geometric solid by marking its contours using a string extended from a fixed point. Rosenwald 655 (shelved as Rosenwald 657). Library of Congress, Rare Book and Special Collections Division.

The winged figure who cannot fly in Melencolia I is perhaps one such artist. Settled in a despondent state over how to render a universal ideal through the particular instruments of her craft, she may be in need of Dürer’s forthcoming artist manual. Scholars have argued about the interpretation of nearly every aspect of this engraving since it was first created in 1514, but Dürer’s emphasis on geometry and measurement is perhaps the single exception.

A block of hewn rock that has been cut into an uneasy shape rests next to the angel and dominates the middle ground of the composition. The point of what looks at first like a triangle extends toward the viewer, and it draws the viewer’s eyes up to what initially looks something like an elongated pentagon. A closer inspection of this shape begs the viewer to ask: wait, what is that?

Detail of engraving showing a polyhedron.
Detail from Albrecht Dürer’s Melencolia I. Engraving. 1514. Bradley Collection. Library of Congress, Prints and Photographs Division.

The answer to that question is a part of the perennial appeal of Melencolia I. Scholars disagree about the exact geometry of this solid. One popular theory is that Dürer rendered an unusual polyhedron sometimes called a truncated rhombohedron or a truncated triangular trapezohedron. The name is difficult to pronounce, but the shape it is even more difficult to conceptualize without the aid of a computer. This eight-sided geometrical solid is comprised of six pentagons and two triangles. A close examination of Melencolia I reveals that the top of the disconcerting block is in fact a triangle, while the sides are indeed pentagons.

Aside from a computer, the best tool for creating this particular solid is a graph of sixteen square cells. Using a pair of geometrical dividers and a simple straight edge, an artist with knowledge of geometry can draw two pentagons within this graph so that the points overlap to create a triangle. By adding vanishing lines to that central triangle, the artist can indicate the proper depth relationship of this difficult polyhedron.

A scholar named Terrence Lynch has observed that the winged figure in Melencolia I is holding the necessary dividers, which, in combination with the “magic square” on the wall behind the dejected-looking celestial figure, can be used to craft the sixteen-celled square needed to create the truncated triangular trapezohedron that is now fondly referred to as “Dürer’s solid.”

Detail of engraving showing a square with 16 interior boxes each filled with a number. A bell is placed above, an hourglass besides, and the sad angel sits below.
Detail from Albrecht Dürer’s Melencolia I. Engraving. 1514. Bradley Collection. Library of Congress, Prints and Photographs Division.

Happily, the Rare Book and Special Collections Division owns a copy of Dürer’s Treatise on Measurement. So, if you find yourself feeling melancholy over your inability to understand universal beauty and render it effectively in an artistic medium, pop by LJ-239 and have a look. We all have those days, right?

 

Image of a woodcut spiral in black ink.
Detail from Dürer’s Treatise on Measurement (1525). Rosenwald 655. Shelved as Rosenwald 657. Library of Congress, Rare Book and Special Collections Division.

 

SOURCES

Andrews, Noam. “Albrecht Dürer’s personal Underweysung der Messung.” Word & Image 32:4 (2016) 409-429.

Bubenik, Andrea. “The Shape of Things to Come: Dürer’s Polyhedron.” In The Persistence of Melancholia in Ars and Culture. New York: Routledge, 2018.

Boorsch, Suzanne and Nadine Orenstein, “The Print in the North: The Age of Albrecht Durer and Lucas van Leyden.” The Metropolitan Museum of Art Bulletin. The Metropolitan Museum of Art, vol. 54, no. 4, Spring 1997, p. 37.

Chesterton, G.K. Orthodoxy. London, John Lane, 1909.

Doorly, Patrick. “Dürer’s ‘Melencolia I’: Plato’s Abandoned Search for the Beautiful.” The Art Bulletin 86, no. 2 (2004): 255–76.

Dürer, Albrecht. Melencolia I. 1514. Metropolitan Museum of Art. 

Dürer, Albrecht. Melencolia I. 1514. National Gallery of Art Collection Highlights. 

Lynch, Terence. “The Geometric Body in Dürer’s Engraving Melencolia I.” Journal of the Warburg and Courtauld Institutes 45 (1982): 226–32.

Luber, Katherine Crawford. Albrecht Dürer and the Venetian Renaissance. Cambridge, U.K. ; New York, NY : Cambridge University Press, 2005. 

Panofsky, Erwin. The Life and Art of Albrecht Dürer. Rev. ed. Princeton: Princeton University Press, 2005.

Wisse, Jacob. “Albrecht Dürer (1471–1528).” In Heilbrunn Timeline of Art History. New York: The Metropolitan Museum of Art, 2000–.

 

Cropped detail of the sleeping dog from Dürer’s Melencolia I.
Detail from Albrecht Dürer’s Melencolia I. Engraving. 1514. Bradley Collection. Library of Congress, Prints and Photographs Division.

 

RESOURCES

Dürer’s Melencolia I is housed in the Prints and Photographs Division at the Library of Congress.

Dürer’s Treatise on Measurement is housed in the Rare Book and Special Collections Division at the Library of Congress. For more information about the Division’s collection of 15th and 16th century artist manuals, see The Art of Instruction by Emily Moore.

 

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