The Sacred Cut Revisited: The Pavement of the Baptistery of San Giovanni, Florence

Kim Williams

The pavement of the Baptistery of San Giovanni in Flo- rence has been metaphorically compared to a carpet that has been unrolled in a spaceJ It is true in the case of the Baptistery that the motifs in the floor were inspired by the designs in Arabic carpets. But a decorated pavement can do more than merely embellish a monument. The de- sign of a pavement may reveal an underlying ordering principle or philosophy in the architecture.

The Baptistery of San Giovanni is the oldest church in Florence. The walls of the octagonal building date from about the seventh century after Christ. Its founda- tions are even earlier. During the 1912-1915 renovation, which returned the Baptistery to its "original" state, an excavation was undertaken inside the Baptistery which revealed the older Roman substructure under the apse. Fragments of the substructure and its mosaic decora- tion may still be seen through grills in the floor. By the time she became the queen of the Renaissance, almost all of Florence's Roman and Paleo-Christian structures had been destroyed. The Baptistery serves as a reminder of Florence's ancient past. Was the Baptistery intended from its conception to be a baptistery? Probably not. Despite its central plan which is characteristic of bap- tisteries, it probably served as a chapel. San Giovanni became a baptistery officially, however, when the font was transferred from the church of Santa Reparata in 1128. The pavement is conventionally dated 1207. It is likely that the d e c i s i o n to pave San Giovanni in the late twelfth or early thirteenth century coincided with a de- sire to heighten the symbolic character of the building as a backdrop to the rite of baptism. The geometry which I h a v e f o u n d to be key in the layout of the Baptistery floor provided a particularly rich and apt symbolic base for the program of architectural ornamentation.

The pavement is a patchwork quilt of patterns. Three strips of pavement define the north-south and east axes.

1 My drawings and research on the pavement of the Baptistery of San Giovanni are supported by grants from the Anchorage Foundation of Texas and the Graham Foundation for Advanced Studies in the Fine Arts.

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The pavement of the west axis was destroyed in 1576 when Grand Duke Francesco I of Tuscany refurbished the Baptistery. A center medallion with a centripetal mo- tif was laid in 1752, but was removed during the 1912 restoration. Today, the west axis and central octagon are paved in simple brown terrazzo. The north-south and east axes radiate out from the center octagon to the three doors. The strips of pavement which define the axes sep- arate the four quadrants of pavement. Each quadrant in its turn is partitioned by decorated borders into com- partments, each filled with a different motif. Of the four quadrants, the southeast quadrant is by far the richest, presenting more complex geometrical patterns as well as figural patterns. 2 Of the axes, the east axis is the most elaborate, the east portal being on axis with the Duomo and so being the ceremonial entrance. (See Figure 1.)

2 Enrico Madoni has drawn each motif in the Baptistery pavement in his monograph, I1Pavimento del Battistero di Firenze, 1913.

Figure 1. The pavement of the Baptistery of San Giovanni in Florence, showing the central octagonal medallion and the altar on the west side. The southeast quadrant is in the lower left-hand corner. Photograph taken before the 1912 restoration of the Baptistery. (Photograph by Fratelli Alinari.)

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 2,1994 1 9

.213 ~ .707

f

~_ 1.0

Figure 2. The construction and proportions of the Sacred Cut geometry (Figures 2-10 were drawn by Kim Williams).

Initially drawn to study the wealth of geometric pat- terns in the Baptistery pavement (some 60 of them), once I began work I became more and more intrigued by the layout of the pavement as a whole. An Italian scholar named Carocci wrote in 1897, "The pavement, as far as I am concerned, wasn't conceived as a systematically arranged whole because it is so irregular in its parti- tioning and doesn't present us with a symmetric whole as does, for example, the cathedral of Siena . . . . -3 1 found the partitioning of the Baptistery floor to be very regular, especially when each individual quadrant is compared to the others. The existence of an underlying order was pointed to by the pavement having been so regularly laid out in each of the quadrants, by the layout having been so deliberately emphasized by decorated borders which separated the segments from each other, and by each segment having been filled with a different geo- metric motif. With the help of a geometrical construction called the "Sacred Cut," I found the ordering mechanism for which I was looking.

The Sacred Cut is the fundamental construction which forms the basis for Danish engineer Tons Brun4s' system of geometry, which he claims has governed the construc- tion of monuments in every period from the Egyptian to the Medieval [1]. Let us examine the construction.

To begin, one takes a reference square of a given dimension. The reference square is divided by plac- ing the compass at a corner of the square and strik- ing an arc which intersects the two adjacent sides and passes through the center of the square. Thus, each of

3 G. Carocci, I1 Pav imen to nel Battistero Fiorentino, in Arte Italiana Decorativa e Industriale, 1897, p. 34, " . . . il pav imento , secondo me, non

stato ideato come cosa organica, perch4 ~ assai irregolare neUa sua divisione e non presenta u n ass ieme s immetr ico come per esempio quello del D u o m o di Siena . . . "

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the two sides is divided into segments of proportions v~ /2 ~ .707 and 1 - (v~/2) ~ .293 of the length of the side. When similar arcs are struck from the other three corners, the operation is complete, and the refer- ence square is subdivided into a nine-module grid with modules of different proportions. The four corner mod- ules are smallest, .293 square; the middle modules are .293 x .414 rectangles; the inner module is the largest, v~ - 1 ~ .414 square, and Brun6s calls it the "sacred square." It may be seen that the diagonal of each corner square is also equal to v~ - 1 ~ .414, thus completing a regular octagon. The Sacred Cut also provides a close ap- proximation for squaring the circle: The length of the arc used to construct the Sacred Cut is equal to within 0.6% to the length of the diagonal of one-half the reference square (Figure 2).

There is no direct evidence that the Sacred Cut was actually used in antiquity. For example, the Sacred Cut is not mentioned specifically by Vitruvius, who wrote the only surviving document on architectural practice which has come down to us. However, Vitruvius does recommend the use of geometric constructions as de- vices to ensure proportional harmony in an architecture. Modern-day architectural historians have regarded it as a clue to how ancient monuments were constructed.

Tons Brun4s found the Sacred Cut to have played an important part in the proportioning of the Pantheon [2]. Professor Brun6s placed a circle within the section of the Pantheon so that it followed the curve of the dome, then circumscribed a square about the circle, which he used as the first reference square. Sacred Cuts performed on this square resulted in horizontals which coincided with the springing line for the dome and the top of the niches between the columns which encircle the main hall. Ver- ticals produced by the Cuts coincided with the upper limit of the coffers in the dome. The sacred square of the first reference square then became the second reference square, and its Sacred Cuts provided verticals which co- incided with the diameter of the oculus in the dome and horizontals which coincided with the top of the upper row of windows. The Sacred Cut was useful in analyzing the ground plan of the building as well. It again provided horizontal and vertical lines which coincided with major elements of the building, most notably the width of the porch.

More recently, the Sacred Cut has been found by Don- ald and Carol Watts to have governed the layout of a second-century A.D. housing complex called the "Gar- den Houses" in ancient Ostia [3]. The Wattses found that three successive Sacred Cuts determined the geometri- cal order of the Garden House complex. They placed a circle which touched the corner of the courtyard within the complex, and constructed a square which circum- scribed that circle. Sacred Cuts performed on the east and west sides of this reference square coincided with the position of the outer walls of the courtyard build- ings. Similar Sacred Cuts on a second reference square

Figure 3. Using the Sacred Cut to form the outer octagon.

determined by the width of the courtyard and the posi- tion of the fountains coincide with the party walls along spines of the courtyard buildings. The sacred square of the second reference square becomes the third reference square, and its Cuts coincide with innermost walls of the

�9 courtyard building. The unfolding of all the Sacred Cuts from a common center reveals an emphasis on the major east-west axis of the complex.

The success with which the Sacred Cut has been used in studying Roman architecture recommends its use in analyzing medieval architecture. The step from a Roman housing project to a medieval Tuscan sacred structure is not the large leap it may appear to be. Italian contempo- raries of Romanesque architecture believed themselves to be carrying forward Roman building techniques, and hence Roman greatness. I was drawn to study the Sa- cred Cut geometry in relation to the Baptistery not only because of the link between Roman and Romanesque architectures, but particularly because I recognized the kinship between Sacred Cut geometry and the Baptis- tery's octagonal form. Although the octagon played no part in the analyses of either the Pantheon or the Garden Houses, the Sacred Cut is the method by which one con- structs an octagon with equilateral sides from a given square.

Here is how it is applied to the Baptistery floor: We construct a first square. Using the Sacred Cut proce- dure to construct an octagon, as discussed, we obtain the perimeter walls of the space, referred to as the "outer octagon." Next, by joining alternate midpoints we construct within the outer octagon two superimposed squares, one turned 45 ~ with respect to the other. This forms an eight-point star. Sacred Cuts are next performed on each of the two superimposed squares, and the result- ing smaller eight-point star which is formed determines the octagon in the center of the pavement, referred to as the "inner octagon." (See Figures 3-6.) Originally, the baptismal font occupied the entire inner octagon. After the demolition of the font, a strongly centripetal design in black, white, and red marble took its place (see cover).

/

/ \

\

/

\ / /

Figure 4. The eight-point star found within the outer octagon.

Figure 5. Generating the inner octagon using the Sacred Cut.

Figure 6. Two superimposed Sacred Cuts form the inner oc- tagon.

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The Wattses found in the Garden Houses a series of integer dimensions which occurred as ratios of segments constructed by Sacred Cuts. Beginning with a reference square with a side of 41 and performing successive Sa- cred Cuts, using each resulting sacred square as the next reference square, one obtains approximately the follow- ing integer series:

41 29 17 12 7 5 3 2 1.

These integers and their multiples were found to figure significantly in the apartments of the Garden Houses complex. The windows in the largest rooms are based on a 7-foot module, the window itself measuring 5 Ro- man feet and the interval between windows measuring 2 Roman feet; the width of the public corridor and stair- way is 17 Roman feet; the inside width of the courtyard buildings is 58 Roman feet.

The Wattses observed that these integers can be ob- tained from a familiar Pythagorean construction. The construction begins with a square with side 1, whose di- agonal, of exact length v~ according to the Pythagorean theorem, is approximated by integer 1. Subsequent ap- proximations are obtained by recursion: if an and bn are the integers used at the nth stage, then the next stage uses

an+l = an + bn, bn+l = 2an + bn.

(See Figure 7.) The sequence so produced is

1 1 2 3 5 7 12 17 29 41 7 0 . . .

as mentioned above. It is easy to prove that the ratios bn ~an have the desired property of approaching v~.

We have obtained the sequence as two sequences in- terleaved. The first,

(an): 1 2 5 12 29 70 . . . ,

represents the successive values assigned to the sides of the squares; the second,

(bn): 1 3 7 17 41 9 9 . . . ,

represents those assigned to the diagonals. Each of these is a Pell's series, with the same recursion Cn+l = 2cn + Ca--1. The ratios cn+l/Cn approach 0 = 1 + v~. 4 This ratio comes naturally into the Sacred Cut construction: It is the exact ratio of the side of the reference square to that of the inner module or "sacred square." If the side of the reference square is taken as 0v~, the modules are simply expressed (Figure 8). 5

P. H. Scholfield points out another valuable quality of this division to the architect. A rectangle divided by 2 horizontal and 2 vertical lines generically gives 36 differ- ent shapes of subrectangle. The Sacred Cut construction limits the number of shapes to 5 [4]. The Baptistery floor contains many rectangles conforming to the shapes oc- curring in Figure 8. (See Figure 9.) Such repeated propor- tionality has been studied under the name of the "prin- ciple of repetition of ratios. "6

4 The 0 value was given its name and extensively analyzed by P. H. Scholfield in The Theory of Proportion in Architecture, Cambridge: Cam- bridge University Press (1958), 138-144. 5 This particular division of the square is illustrated in Scholfield, The Theory of Proportion, p. 133, Fig. 3f. 6 For more on the "principle of repetition of ratios," see Jay Kappraff, Connections: The Geometric Bridge between Art and Science, New York: McGraw-Hill (1991), 19.

Figure 7. The Pythagorean construction for approxi- mating the value of v~.

1 2 2 ' 3 5

5 7 12

S I

12 17 29

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I d'2 1

I t 1 I

~ e

1:1

(a) (b)

Figure 8. (a) The Sacred Cut division of the 0x/2 square. (b) The four rectangles generated by the Sacred Cut division of the 0v~ square.

Unlike the Wattses, I was unable to find within the Baptistery any linear dimensions which conformed ex- actly to the Pythagorean series or the PelFs series. 7 Rather, m y having found the propor t ions of the pave-

ment compar tmen t s to be related to 0 and v ~ points to the use of a geometr ic sys tem rather than the use of a sys tem of integer ratios to lay out the pavement . The Sacred Cut is an ideal example of such a system.

What remains enigmatic in the Baptistery pavemen t is an apparen t irregulari ty in the pentagonal pieces along the diagonal axes of the inner octagon. To mainta in radial symmetry, they should be circumscribable by a square. Instead, they are circumscribed by a rectangle. Similar irregularities in another pavemen t have been ascribed to craf tsmen who, in articulating a sys tem whose order- ing principles were u n k n o w n or lost, m a d e alterations to the pat tern as they worked it out and unintent ional ly interrupted the system. Knowing that t rade secrets in the guilds of the middle ages were jealously guarded , expos- ing them being punishable by death, this is a plausible explanation. (See Figure 10.)

7 After having taken all the measurements and produced a measured drawing of the pavement, I began to study the measure of the building. The Roman foot was most probably the standard unit of measure used in the Baptistery. Though it has been found to vary, the Roman foot is conventionally considered to be equal to 29.5 cm or 11.625 in. The value I have used to obtain the integers used in my analysis is 28.49 cm. This results in dimensions which correspond roughly to the values in the Pythagorean construction discussed earlier. It must be noted that the Baptistery is not a perfect octagon, being some 41 cm shorter on the east-west axis than on the north-south axis. There is accordingly some disagreement between dimensions in the various quadrants.

\

/

1:0~"2

1: e,'~

Figure 9. The proportions of rectangles in the Baptistery pave- ment.

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Figure 10. (a) The existing irregular trapezoids (shaded) which surround the octagonal center of the Baptistery pavement. (b) Ideal arrangement of irregular pentagons surrounding the inner octagon.

To seek the symbolism in the pavement of San Gio- vanni, we ask the significance of the geometric fig- ure produced by the Sacred Cut geometry. The super- imposed reference squares in the Baptistery create an eight-point star. The eight-point star as an ecclesiastical emblem signifies resurrection. In medieval number sym- bolism, eight signified cosmic equilibrium and immor- tality. It was considered to be particularly closely related to resurrection: Christ was resurrected eight days after he triumphantly entered Jerusalem on Palm Sunday, for example. As a symbol of baptism, the number eight, and by extension the eight-point star and the octagon, repre- sent rebirth or resurrection of the catechumen in Christ. Further, the octagon mediates between a square, which represents things terrestrial, and a circle, which repre- sents things celestial [5]. In fact, in wall intarsia in San Giovanni, the two eight-point stars on the ends have an octagonal center, whereas the two inside stars each have a circular center. I believe the unknown designer of the Baptistery pavement was making a similar state- ment when he based the layout of the floor on the same geometric figure.

The use of irrational values, or incommensurables, is linked philosophically to the symbolism of the circle and the square. A circle was indefinite, its circumference and area based on the irrational ~r, whereas the circumfer- ence and area of a square were rational values. Philo- sophically the use of irrational numbers such as ~ shows an attempt to rationalize that which is irrational, or in other words, to make sensible that which is divine or only achievable through the intellect. In this context, let's turn

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again to the Sacred Cut. According to Professor Brun6s, its special quality, its "sacredness," lies in its very nearly solving the riddle of how to square the circle. The Wattses believe that the Roman designer of the Garden Houses used the Sacred Cut to represent the squaring of the cir- cle in order to integrate sacred and profane, earthly and celestial.

I believe likewise that the designer of the pavement in San Giovanni intended to imbue his pavement with symbolism to represent this integration. The octagon me- diates between square and circle in the Baptistery, thus expressing the purpose of the baptismal rite: The cate- chumen undergoes death to sin and rebirth in Christ, coming in this way as near to the divine as a human can.

References

1. Tons Brun6s, The Secrets of Ancient Geometry and Its Use, Copenhagen: Rhodos (1967), Vol. 1, 72-80.

2. Tons Brun6s, The Secrets of Ancient Geometry and Its Use, Copenhagen: Rhodos (1967), Vol. 2, 38-56.

3. Donald J. and Carol Martin Watts, A Roman apartment com- plex, Scientific American, 255(6) (1986), 132-140.

4. P. H. Scholfield, The Theory of Proportion, Cambridge: Cam- bridge University Press, p. 132.

5. Jean Chevalier and Alain Gheerbrant, Dizionario dei Simboli, Milano: Biblioteca Universale Rizzoli (1986), Vol. 2, 175.

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