The Deformations of Francesco de Marchi

Caroline O’Donnell is the Richard Meier Assistant Professor at Cornell University Department of Architecture and principal of CODA, an experimental design practice working at a range of scales from the body to the city. Both her practice and teaching are rooted in issues of site and the potentially mutant relationship between the architectural organism and its niche.

The origin of topology is normally attributed to Leonhard Euler and his 1736 paper, “The Bridges of Königsberg.” In attempting to solve mathematically the popular puzzle of finding a route that crossed each of the city’s seven bridges exactly once, Euler diagrammed the city as a set of malleable relations.

This move marked a split from geometry proper, since Euler was not concerned with measurements, shapes, and distances, but instead with “qualitative properties invariant under continuous transformations.”[1] Dealing with relations rather than parts, continuity rather than form, the field has since produced such mathematical curiosities as the Klein bottle, the Möbius strip, and the coffee-cup-to-donut transformation.

The term topology, however, is not commonly found in literature until the 1920s. Its original appellations, geometria situs and analysis situs (literally analysis of position), suggest a closer relationship with the idea of site (perhaps due to its own urban origins) than the field’s content would suggest. Although immediately dissociated with any such earthly reality, the incongruous notion of place still sneaks into the modern term topology.

While contemplating Euler’s abstract and malleable graphic of his city as the origins of mathematical topology, a single urban precedent emerges: one that can be imagined as having paved the way for Euler’s own manipulations, that uniquely considered the city as deformable and malleable while maintaining continuity. Francesco de Marchi’s deformations, illustrated in his Della Architettura Militare, written between 1542 and 1565,[2] provoke fantasies of the alternative urban origins of topology: the transformations of the object by the site itself.

While his contemporaries pushed aside the dirty realities of context, designing pure geometries on tabula-rasa sites, Francesco de Marchi responded to problem sites in ways that complexified the geometry and embraced topology long before its mathematical explanation existed. In its most distilled form, De Marchi’s work demonstrates primitive operations of mutation, transformation, and reaction that are reemerging in architectural discourse today.

Although the mathematical laws on which Renaissance beliefs were based had been expressed earlier by Pythagoras and Plato, they gained new momentum in the late 15th century and entered directly into the discipline of architecture through the resurgence of the work of the classical architect Vitruvius. In his Ten Books on Architecture, he often aligned nature and architecture, writing, for example, that “in the human body there is a kind of symmetrical harmony between the forearm, foot, finger, and other small parts; and so it is with perfect buildings.”[3] Through this, and the many architectural treatises that followed, the notion that architecture should reflect the “objective” mathematical harmonies of the universe became a basic principle of the Renaissance (see image on p. 24).

More concerned with the principles than the reality of architecture, the treat­-ise was the means of communication of architectural ideals. The treatise allowed the architect to communicate the way in which a work—from the scale of the column to the scale of the city—should manifest itself under perfect conditions. Underscoring this rift that was widening between the world of the ideal and the real, Renaissance mathematician Niccolò Tartaglia, in his preface to the Latin translation of Euclid’s Elements, stressed that geometry dealt with figures in the mind rather than those imperfect forms that we see in nature with the physical eye.

The Elements to which Euclid had dedicated his book were the point, the line, and the area. The combination of these elements produced geometrical forms, and the purest of all of these was the circle. Whether attributed to the divine will of God or the indubitable laws of science, the form of the circle came to represent the geometric order of the world and the cosmos. As Alberti elucidates, “It is manifest that nature delights principally in round figures, since we find that most things which are generated, made, or directed by nature are round.”[4]

Round cities are justified by Plato using both a pragmatic and an abstract rationale: “The temples are to be placed all around the agora, and the whole city built on the heights in a circle, for the sake of defense and for the sake of purity.”[5] Vitruvius’s description propogates this belief, writing that “towns should be laid out not as an exact square, nor with salient angles, but in circular form, to give a view of the enemy from many points.”[6] Despite many such textual references to the circular city, the earliest graphic example in an architectural treatise is Filarete’s rudimentary drawing of Sforzinda.[7]

The diagram is an eight-pointed star, formed out of two squares, bounded by a circle. Although there are references to Vitruvius,[8] the mathematical order of the drawing, combined with the fact that the graphic itself had appeared previously in astrological texts representing both the disposition of the elements and the image of the world,[9] suggests that Filarete’s motivations were as much symbolic as practical.

Following this diagram, many architects, including Leonardo da Vinci and Francesco di Giorgio Martini, experimented with shape variations, including squares, triangles, circles, and rhomboids, in search of a geometry that could best enclose a city. Whether appearing in their checkerboard or radial versions, the circular city, based unfailingly on the principles of symmetry and proportion, dominated, satisfying, as Filarete’s diagram had, “all the longings of the Renaissance for an all-round harmony.”[10]

Diagram of the Ideal City of Sforzinda. Filarete’s Treatise on Architecture, Volume 2: The Facsimile (Yale University Press, 1965), Book II, folio 14R (L) and 43R (R).
Diagram of the Ideal City of Sforzinda. Filarete’s Treatise on Architecture, Volume 2: The Facsimile (Yale University Press, 1965), Book II, folio 14R (L) and 43R (R).

In the mid-16th century, a shift in focus occurred in the design of cities as a consequence of improved military technology. More sophisticated and specialized mathematical skills were required of the military engineer, who came to replace the uomo universale.[11]

In this time of extreme pragmatism, Renaissance ideas of geometry were greatly diminished by pressing military issues. Horst De la Croix claims that “the military architect was an exceedingly practical man and his appreciation of the circle was based exclusively on its functional advantages, untinged by any philosophical cogitation on its symbolic qualities.”[12] Yet, inevitably, belief in the significance of the circle remained ubiquitous, and clues from the period exist to illustrate that these concerns did seep into military design. For example, in his 1557 treatise, Giacomo Lanteri, who has been criticized for an overemphasis on the militarization of the city, describes the circle as “the perfect figure because it reflects the shape and nature of the universe.”[13]

It follows then that a nonregular circle would be eschewed by military architects on the grounds that irregularities in the defenses would create weak points (a hypothesis supported by the very few appearances made by the irregular plan in the military treatise). Yet beyond the pragmatics, a certain disdain for irregular form inevitably underlies this logic, since the deformed circle references an imperfect body and a dischordant relationship with the universe. Geometries deformed by the site and other contingencies were, on paper, at least, failures.[14]

In Filarete’s treatise, the diagrams of Sforzinda are accompanied by a third illustration—an image of that same Sforzinda diagram, now hovering over a varied landscape.

Landscape of Sforzinda. Filarete’s Treatise on Architecture, Volume 2: The Facsimile (Yale University Press, 1965), Book II, folio 12R.
Landscape of Sforzinda. Filarete’s Treatise on Architecture, Volume 2: The Facsimile (Yale University Press, 1965), Book II, folio 12R.

Although Filarete describes the site, which has been selected after much investigation, the diagram remains unchanged when inserted into its context: it hovers above the flattest part of the site, adjacent to (but unaffected by) a river.

Almost all treatises of this period contain chapters concerning the criteria for the selection of a site for these ideal cities. The argument was polarized between the flat site and the hilly site, each having its own advantages and disadvantages.[15] However, in general, the benefits of the flat site were more persuasive: attacking forces are always in full sight of defenders, horizontal fire is more effective, and the fortress can be supplied more easily, and relieved in case of a siege. More importantly, any disadvantages of the flat could be offset by the inherently greater strength of the geometric symmetry. This implied, of course, that asymmetry means weakness—the potential for breaches and cracks.

Albeit lengthy and multifarious, the discussion clearly ignores the contingencies that are inevitable in any site. While there are references to the inevitable deformation of the ideal plan in its implementation—for example, Girolamo Cataneo (Lanteri’s mathematics teacher and a geometric planner) advised that his designs must be modified by and adjusted to the local conditions of the terrain—the deformations were rarely demonstrated graphically in the hypothetical realm of the treatise. In this context of ideals, rules, and shoulds, Francesco de Marchi stands alone in his treatment of what ifs. Francesco de Marchi’s Della architettura militare presents a baffling variety of military plans.[16] While De Marchi praises the dignity of architecture, and particularly recognizes the importance of Vitruvius, he positions himself as a soldier, aiming to bring war into touch with architecture,[17] and indeed, his written style is more artillerist than theorist.

While his treatise contains many examples of geometrically ordered cities on perfect sites, what makes De Marchi’s work unique among the abundant literature of military architecture at that time is his deployment of a set of hypothetical problems arising from contextual imperfections.

In the more tentative scenarios, De Marchi hypothesizes an ideal city undisturbed by but adjacent to a contextual disruption—a harbor, in the case of City 27. This twelve-bastioned city is symmetrical down to its radial interior planning. Only the site is exerting a massive asymmetrical force, to which the city, until now, remains impervious.

City 27
City 27

A second step occurs when the city—both the exterior shell and the interior fabric—reacts to the pressure from the context. City 155 shows what reads as a once-perfect circle, conceptually deformed by the force of the sea. De Marchi writes pragmatically and undramatically about this unconventional move: “with great advantage, one can secure a maritime location, where the whole force of fleets can stand there safe from enemies, and there would be no fleet, however strong and large and well-armed it might be, neither galley nor battleships, that would dare to go amidst four towers where they could easily be hit and sunk. These towers could be constructed where there were intervals of rivers, lakes, ponds, canals, as can occur in some sites.”[18]

In the plan, what one would expect to be a convex western edge is slightly concave and frontal with the shoreline, its supposed eighth bastion being amputated to form a stumped “launch platform.” The piazza is central only to this conceptual circle and not to the actual boundary: in fact, only one block surrounds its western edge, whereas four blocks line its eastern edge. Further, the entire grid of the city has been deformed, widening both the block and street dimension toward the seawall in acknowledgment of this dominant force in the context.

City 155
City 155

Finally, the contextual element forces itself through the walls and becomes part of the interior of the city, and De Marchi’s games begin in earnest.

In City 25, the harbor gate fits perfectly between two seaside bastions, protected just as the other flanks are—by cross fire. The harbor penetrates the walls and occupies a circular area within the radial whole. At the intersection of the center of the outer polygon and the tangent of the inner circle, a new central piazza is located where it should be, both according to the outer circle and with respect to the harbor: stepping off the ship, one is immediately in the center of the city. The circular harbor acts as a cut-out in the city’s radial military (center-to-bastion) fabric, but has no effect on the street pattern itself.

City 25
City 25

In a similar operation, City 161 is interrupted by the meandering course of a river. Again, the river enters and exits cleanly between bastions, and its centerline aligns with that of the city, so that the location of the central piazza coincides with both the center of the river and the center of the city as a whole. Consequently, the axis of the city and its accompanying secondary piazzas coincide with the axis of the main bridge.

City 161
City 161

City 48 contains a distinctly larger river, widening more markedly toward the south, with three bridges (two shown) and chains replacing the flanks at each river penetration. This plan demonstrates another radical step beyond the already unconventional moves displayed by its predecessors: whereas the preceding plans overlay the natural form as a passive cut-out from an existing geometrical solution, this new strategy uses the organic form of the river to offset some of the city fabric. Thus, whereas the river-offset streets are meandering, the streets related to the city are wide and straight, as is customary. These major streets do not connect to the bastions and the bastions themselves are not aligned with each other but (presumably) are oriented according to local topography.

True to form, De Marchi justifies the entry of the river into the city on military grounds: “when one part of the land is lost, all of it is not lost, as happened at Parma, when the French once took half the land, but since the river Parma divided nearly the whole land through the middle they could not take the other part, and so they were driven back outside where each side engaged in battle across the river.”[19]

City 48
City 48

City 59 represents the most extreme struggle between nature and the artifice: here, the meandering river cuts first irregularly through a flank on entry, and on exit, intersects with a bastion. Three bridges cross the river, two on the north-south axis, which is aligned with the east wall and its respective grid-lines, and one aligned with the only east-west axis of the city, which opens to become the central piazza at the intersection of the bridge axes. The city has seven bastions, the three on the right being aligned, and those on the left producing unequal flank angles and distances seemingly related to a factor other than the river. As in City 48, not all of the axial streets connect directly to the bastions, and consequently the circulation between bastions must be managed by the ring road in the discontinuous cases.

Aware of De Marchi’s military priorities, one might attribute this meandering plan to emerge from a military logic, possibly as a result of Alberti’s statement that “the ancients used to design their streets narrow and winding, imagining that they could be defended better.”[20] However, in justifying the curve of the river embedded so completely in the city, De Marchi writes, “I would not want the river to travel in a straight line in the fortification because that would increase the velocity of the water’s course, and it seems to me more beautiful to see; that this is the truth one can see in the ancient and famous city of Pisa, for whoever wishes to praise it says that Pisa is along the Arno, which creates a curve: a perspective that is most beautiful to see.”[21] Furthermore, he continues to describe the meandering street pattern by offering variation and surprise; walking along the main streets, he writes, the citizen will be “confronted with a new and different vista at every corner he passes.”[22]

In truth, it appears that De Marchi is not guided purely by utilitarian considerations, nor, as has been suggested, is he simply performing an exercise in pattern-making.[23] Rather, he communicates the meaning of the city in relation to its context: by understanding one street, something much larger can be understood­—the push and pull that occurs between nature’s ideal order and the uncertainty of natural contingency. Despite the dryness of De Marchi’s writing, his drawings express, perhaps beyond any possible expression of words, the dynamism and reactivity that emerge from that incongruous relationship.

Today, while the “designed” city is once again being understood as a sign—from the regular geometries produced by architects, including Norman Foster’s square city Masdar, and Richard Rogers’s circular Compact City in Lu Zia Sui, Shanghai; to symbolic forms produced by developers, such as Nakheel’s the Palm, the World, and the Universe in Dubai—these urban forms reference geometric and symbolic figures that are inherently unresponsive to deformative forces. As in De Marchi’s last city (City 57), where the sea penetrates the city completely, the tabula rasa condition is reinstated, with all the symmetry and order that contextlessness implies.

And while contemporary architectural practices have translated the mathematics of topology to produce continuous surfaces and deformed geometries, the origin of the deformer has often been abstract and unmotivated. In locating Francesco de Marchi at the beginning of topology’s trajectory and in understanding topology as a discipline emerging from a deeper understanding of place, as well as from the urban condition,we may reconsider the role of topology in architecture and urbanism as a motivated deformer, where the context is a force to which our cities, even those of fiction, must react.


1. Wayne H. Chen, The Analysis of Linear Systems (New York: McGraw-Hill, 1963), 104.

2. Eventually published posthumously by Gaspare dall’Oglio in Brescia in 1599.

3. Vitruvius, The Ten Books on Architecture, trans. by Morris Hicky Morgan (Dover, MA: 1960), 14.

4. Leon Battista Alberti, De re aedificatoria, Florence in 1485, trans. Joseph Rykwert, Neil Leach, and Robert Tavernor (Cambridge, MA: MIT Press, 1988).

5. Plato, Laws, 360 BC, trans. by Benjamin Jowett,

Further, in De re aedificatoria, Alberti comments generally on fortifications, “Of all Cities, the most Capacious is the round One; and the most Secure, that which is encompassed with Walls broken here and there into Angles or Bastions jutting out at certain Distances as Tacitus informs us Jerusalem was; Because it is certain, the Enemy cannot come up to the Wall between two Angles jutting out, without exposing themselves to very great Danger; nor can their military Engines attack the Heads of those Angles with any Hope of Success.” From S. Lang, “Sforzinda, Filarete, and Filelfo,” Journal of the Warburg and Courtauld Institutes 35 (1972): 392, trans. Joseph Rykwert, Neil Leach, and Robert Tavernor (Cambridge, MA: MIT Press, 1988).

6. Vitruvius, The Ten Books on Architecture, trans. by Morris Hicky Morgan (Dover, MA: 1960), 22.

7. Filarete, also known as Antonio di Pietro Averlino, Trattato di Architettura di Antonio Filareto, Florence 1465. Reprinted as Filarete’s Treatise on Architecture, Volume 2: The Facsimile (Yale University Press), 1965.

8. And potential misinterpretations: Vitruvius mentions the eight directions of the winds but proposes that the plan of the city work with the directions of the winds, not passively accepts their abstract form. This reading of Vitruvius would have opened up another trajectory in Renaissance city design, and one that would have been more aligned with De Marchi’s thinking.

9. Lang, “Sforzinda, Filarete, and Filelfo,” Journal of the Warburg and Courtauld Institutes 35 (1972): 392.

10. S. Lang, “The Ideal City from Plato to Howard,” Architectural Review 112, no. 668 (August 1952).

11. Niccolò Tartaglia’s treatise on ballistics Nuovo Scientia (1537) and his Quesiti ed invenzione divers (1538) concerning artillery, mathematics, mechanics, and fortification marks the beginning of the shift from architect-artist to military engineer. Formerly, the architect was also a mathematician, astrologer, inventor, and artist. The new techniques for drawing pentagons, calculating distances, and shaping such things as parapets, ditches, counterscarps, ramparts, and block houses required a complete familiarity with the use of the ruler and compass. Giacomo Lanteri’s treatise of 1557, Due dialoghi … del modo di desegnare le piante delle fortezze secondo Euclide, was the first in which design of fortifications was treated as a purely abstract geometrical problem. Lanteri writes, “You must know that arithmetic is necessary to subtract or add measurements, to divide one measurement by another, to know the perimeter of the circuit to be fortified, and to learn how to report the expenses of the works of fortification. To fortify a place is impossible without this knowledge. I am amazed in many ways at the powers of geometry.”

12. Horst De la Croix, “Military Architecture and the Radial City Plan in Sixteenth Century Italy,” Art Bulletin XLII, no. 4 (1960).

13. Martha Pollak, Turin, 1564–1680: Urban Design, Military Culture, and the Creation of the Absolutist Capital (Chicago: University of Chicago Press, 1991), 22.

14. In practice, many of the ideal cities of the 15th and 16th centuries, when actually implemented, were forced to deform due to site conditions, existing city form, and civilian needs. Even Palmanova, whose final form was very similar to her paper representations thanks to the siting on a perfectly flat and open plane, was significantly altered in its eventual implementation.

15. In De La Croix’s synopsis, “Castriotto believed that mountain sites were stronger. De Marchi was undecided, but also seemed to favor the mountains, while Alghisi was a strong advocate of flat and open sites. Busca devoted nine full chapters to the problem, only to decide that a final choice between the two was most difficult.” Horst De la Croix, “Military Architecture and the Radial City Plan in Sixteenth Century Italy,” Art Bulletin XLII, no. 4 (1960).

16. In his treatise, De Marchi’s city plans do not appear in the progressive order that they appear here. Rather, these examples appear sporadically throughout the treatise, intermingled with more conventional platonic forms.

17. J.R. Hale, Renaissance Fortification, Art or Engineering (Thames and Hudson, 1977), 33.

18. Francesco De Marchi, Della architettura militare (Brescia, Italy: Gaspare dall’Oglio, 1599), chap. 57, trans. by Ashleigh Imus for the author.

19. Francesco De Marchi, Della architettura militare (Gaspare dall’Oglio, 1599), chap. 48, trans. by Ashleigh Imus for the author.

20. Leon Battista Alberti, De re aedificatoria, chap. 54, trans. Joseph Rykwert, Neil Leach, and Robert Tavernor (Cambridge, MA: MIT Press, 1988).

21. Francesco De Marchi, Della architettura militare (Brescia, Italy: Gaspare dall’Oglio, 1599), chap. 48, trans. by Ashleigh Imus for the author.
22. Horst De la Croix, “Military Architecture and the Radial City Plan in Sixteenth Century Italy,” Art Bulletin XLII, no. 4 (1960), 288.

23. In her article, The Ideal City: From Plato to Howard, Susan Lang refers to this plan as an example of the 15th century’s tendency “to draw plan as an exercise in pattern-making.” It is to this comment that Horst de la Croix responds with his article Military Architecture and the Radial City Plan in 16th Century Italy, arguing that the designs were “the products of practical men who were guided primarily by utilitarian considerations.” Their feisty argument is printed in Art Bulletin 43, no. 4 (Dec. 1961).


City plans from Francesco de Marchi, Della architettura militare (Brescia: Gaspare dall’Oglio, 1599). Courtesy Avery Architectural and Fine Arts Library, Columbia University.

Go back to 9: Mathematics