Michelangelo's Medici Chapel: The Cube, the Square and the Root-2 Rectangle

Leonardo

Michelangelo's Medici Chapel: The Cube, the Square and the Root-2 RectangleAuthor(s): Kim WilliamsSource: Leonardo, Vol. 30, No. 2 (1997), pp. 105-112Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576419 .

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TECHNICAL ARTICLE

Michelangelo's Medici Chapel:

The Cube, the Square and the Root-2

Rectangle

Kim Williams

Geometry ... is the knowledge of that which always is ...

-Plato, The Republic [1]

The Renaissance monuments of Florence are a built record of the ideals that shaped the architecture of the fifteenth and sixteenth centuries. As an architect, I am particularly at- tracted to the buildings of this period, in which clarity of con-

ception is combined with richness of detail. As part of my research for a book on pavements, I surveyed the famous Medici Chapel, or New Sacristy, Michelangelo's first architectural work (Fig. 1). An accurate groundplan enabled me to

analyze the proportions of both chapel and pavement in or-

Kim Williams (architect), Via Mazzini 7,1-50054 Fucecchio (Florence), Italy. E-mail: <[email protected]>.

der to understand the relation-

ship between the two [2]. I knew that the pavement of the Medici

Chapel was not designed by Michelangelo, as Sir John Pope- Hennessy tells us that this "ugly pavement" was laid in 1559 under the guidance of Giorgio Vasari [3], but I believed that the paving design might relate to fundamen- tal proportions that govern the space. What I found illustrates

ABSTRACT

Renaissance architects paid particular attention to systems of proportion, believing that in these lay the key to architectural excel- lence. Unfortunately, much of what was once known about these systems has been lost or forgotten in the ensuing centuries. A study of the Medici Chapel in Florence designed by Michelangelo reveals a geometric system based upon the irrational square root of two, which may have been used to generate rational proportions and architectural dimensions. The system sheds light on a chapter about geometry in the most important architectural treatises of the Renaissance, The Ten Books of Architecture by Leone Battista Alberti.

the relationship between the pavement and the architecture and reveals important proportional relationships between the groundplan and the volume, shedding light on Roman and Renaissance proportional concepts.

? l | _ ;- . Fig. 1. View of the interior of Michelangelo's Medici Chapel. (Photo: Fratelli Alinari. Alinari/Art Resource, NY. Re- printed by permis- sion.)

LEONARDO, Vol. 30, No. 2, pp. 105-112, 1997 105 ? 1997 ISAST



The Basilica of San Lorenzo in Flo- rence, located not far from the cathe- dral of that city, was rebuilt in the early fifteenth century to a design by Filippo Brunelleschi, who was commissioned by the powerful and wealthy Medici family. Tucked into the left arm of the transept is the so-called Old Sacristy, also de-

signed by Brunelleschi. (A transept is

composed of arms with axes perpendicular to the main axis of the church, resulting in a plan in the shape of a cross.) Tucked into the right arm of the

transept is the New Sacristy, commissioned of Michelangelo by the Medici in the early sixteenth century (see Fig. 2). (A sacristy is a room or small chapel located near the presbytery, where the

priests prepare for the mass.) Michelangelo designed the New Sacristy as a tomb for four outstanding members of the Medici family: Lorenzo the Mag- nificent; his brother Giuliano the Mag- nificent; Giuliano, Duke of Urbino; and Lorenzo, Duke of Nemours. The Medici

Chapel is not large; it is a one-room

space with an adjoining recess called a scarsella. Nevertheless, Michelangelo cre- ated a masterpiece. Although the architecture and sculpture of the chapel are its most discussed features, the geometry merits attention in its own right. Propor- tional studies of individual architectural components and of the individual tombs exist, but do not relate their proportions to ratios of dimensions in the architec-

I r La-jf Fig. 2. Diagram- matic plan of the Basilica of San Lorenzo, locating (a) the church; (b) the Old Sacristy; (c) the main space

J,-_IJ ~ of the New Sacristy or Medici Chapel by Michelangelo; (d) the scarsella; (e) the altar; I)9,3~ ~(f) the lavamani.

ture as a volumetric whole. However, the

presence of particular proportions in limited aspects of the chapel may indi- cate that the elements of the chapel were designed with regard to a comprehensive proportional system. This study aims to uncover that system and to so

provide the underpinning for future

proportional analyses.

KEY TO THE PROPORTIONS The pavement of the Medici Chapel is

composed of nine fields of checker- boards of gray and white marble tiles, separated by double grid lines, com-

posed of two strips of dark gray Florentine sandstone with a line of checked marble tiles between them (Fig. 3a). The length of the side of the checkerboard tiles is 29 cm; their diagonal is then 29 times the square root of 2 (29/2) or approximately 41 cm [4]. The width of the grid lines equals the length of the side of the checkerboard tiles; the width of the double grid is two times the

length of the side plus the length of the

diagonal. In the crossing of the double

grid lines, one may recognize the division of the square into nine modules, which is accomplished by a geometric construction known as the "Sacred Cut" [5] (Fig. 3b). I believed at one time that the "Sacred Cut" might have determined the division of the entire floor plane into nine modules, but this was not the case.

However, proportional relationships between the lengths of the sides and diagonal of the square proved to be fertile ground for my study, which was corrobo- rated by a study of the Medici Chapel tomb designs by Paul Joannides [6]. He derived a 1:1.46 ratio from drawings for the tombs, which he then used as a stan- dard for tracing the development of the

designs. He does not relate this ratio to the square root of two, though it deviates

by only 0.05% from that value. I found a most interesting set of rela-

tionships when I turned my attention from the pavement to the groundplan and volume of the Sacristy. Charles de

Tolnay has described the chapel as being a simple square in plan, with a rectangu- lar apse added on the north end [7]. However, if one considers the plan not as a square with a rectangle added, but as a

rectangle subdivided into smaller parts, the dimensions take on new meaning. The rectangle encompassing the ground plan of the chapel measures 11.7 by 16.49 m, deviating by 0.36% from a root- 2 rectangle [8]. Consider now the volume of the main space of the chapel, a double cube encompassing a semi-

spherical dome. The sides of the square that form the main space of the chapel measure 11.7 m; the height of the main

space, from the pavement to the top of the cornice below the dome measures 11.64 m. These measurements differ by only 0.5% from the value of a cube [9]. It is not merely by happenstance that the volume of the main space of the chapel is a double cube and its plan is a root-2

rectangle, for the root-2 rectangle is inherent in the cube. If one divides a cube in half diagonally, the resulting rectangle has a short side equal to the length of the side of the cube and a long side

equal to the diagonal of the side (Fig. 4). In the New Sacristy, then, one cube of the main space is divided diagonally in two, and the resulting root-2 rectangle is laid flat, becoming the "footprint" of the

chapel. Thus, the volume of the space and the shape of the groundplan are in-

timately related, and proportions relat-

ing to the side and diagonal of a square are prominent.

A second root-2 rectangle in the

chapel circumscribes the ensemble of the altar and the scarsella in which it is located. (Notice in Fig. 3a that the altar sits partly inside the scarsella and partly within the space of the chapel proper). This rectangle measures 4.33 by 6.03 m, differing by 1.6% from the value of a root-2 rectangle (Fig. 5).

106 Williams, Michelangelo's Medici Chapel



Fig. 3. (a, left) The pavement plan of the Medici Chapel. (The white

rectangle at the top of the figure is the altar.) (b, above) The Sacred Cut proportions in the pavement of the chapel provided the clue for

looking for proportions related to the sides and diagonal of a square in the architecture. The dotted lines in the figure are traces of four arcs struck during the operations for constructing the Sacred Cut.

A deviation of 1.6% may seem too

large to be acceptable. It is important to note that the principal rectangle of the

chapel is some 6 cm short of being a per- fect root-2 rectangle (1.4142 times the overall width of the chapel, 11.7 m, is

equal to 16.55 m, whereas the actual

length of the chapel is 16.49 m) [10]. Were these 6 cm added to the depth of the scarsella, its length would become 6.09 m, and the deviation would be re- duced to 0.67%. It must be admitted that the "missing" 6 cm creates a flaw in a system of geometry that otherwise works well in relating the dimensions of the groundplan of the chapel.

While no historical documents exist that recount the process of the design of the Medici Chapel in terms of dimensions or proportions, one curious set of historical facts suggests an explanation as to why the proportions of the chapel deviate from the ideal. Originally flank-

ing the property where the chapel was to be built were some existing houses and stables belonging to the Nelli family. The Medici bought a strip of the Nelli property, razed the existing structures, and built new ones to replace the old. This

procedure resulted in a net gain of some 30 cm in the width of the New Sacristy. The question is, why would the Medici have approved such a costly and time-

consuming step to gain a mere 30 cm? Even acknowledging that Michelangelo

faced the task of placing an important funerary monument in a space where ev-

ery centimeter was precious, one won- ders just how much difference 30 cm could have made in terms of function. However, if 30 cm might have mattered little in those terms, they could have mattered a great deal in terms of proportion: 30 cm is 2.56% of 11.7 m. There- fore, the answer may be that the 30 cm were crucial in achieving dimensions that were reasonably close to, if not ex-

actly, those demanded by the proportional system [11].

Another significant rectangle occurs in the groundplan. The distance from the front edge of the altar to the opposite wall of the chapel relates to the overall width of the chapel in a ratio of 8:9 (10.46:11.7, a deviation of 0.6% from 8:9) (see Fig. 5). The 8:9 ratio repre- sents the whole tone or tonus in musical

theory [12]. Lest it seem that in beginning a discussion of the chapel proportions with incommensurate ratios (such as ratios that contain the square root of 2) and ending it with commensurate ratios (the ratios inherent in the musical scale) we are mixing apples and or-

anges, it should be pointed out that the two systems are closely linked. Indeed, Rudolf Wittkower has written that a type of medieval geometry practiced in the Renaissance (that is, a geometric system using incommensurate ratios such as

Fig. 4. The root-2 rectangle in the cube.

Williams, Michelangelo's Medici Chapel 107

1

1



Fig. 5. A diagrammatic representation of the root-2 rectangles and the 8:9 rectangle in the Medici Chapel.

1:/2 or 1:/5) "is no more than a veneer that enables practitioners to achieve commensurable ratios without much ado" [13]. By "much ado," Wittkower may refer to an analytical method of determining dimensions (i.e. measuring and/or calculating) as opposed to a less arduous method of determining dimensions through geometric constructions. We shall presently see a concrete example of how irrationals may generate rational proportions and, specifically, how the root-2 rectangle relates to the musical ratios.

THE SQUARE ROOT OF TWO IN THE TRADITION OF ARCHITECTURE

Before proceeding to the geometric system used in the New Sacristy, let us look at how the square root of 2 has been treated by architects in antiquity and in the Renaissance. The Romans were very familiar with the square root of 2, having inherited it from the Greek tradition. Vitruvius, author of the only surviving treatise about architecture in Roman times, specifically recommended the root-2 rectangle as one of several suit- able shapes for the atrium of a house [14]. More important than theoretical references to the use of proportions based upon the square root of 2, however, is a monumental work of classic architecture which is particularly relative to the Medici Chapel: I found that the groundplan of the Pantheon is circum-

scribed by a root-2 rectangle [15]. It is defined by the side and diagonal of a

square circumscribing a reference circle

extending from face to face of the niches on the north-south axis (the diameter of which equals 175.36 Roman feet, resulting in a diagonal of 247.99 Roman feet) [16]. This deviates from the measured dimension by only 1.06% [17]. In addition, the superimposition of the root-2 rectangle on both plans allows us to recognize the similarities between the two: the rotunda of the Pantheon is analo-

gous to the main space of the Medici

Chapel; the Pantheon's pronaos, which is a small neck-like piece between the rotunda and the porch, is analogous to the scarsella; the "leftover" rectangles that flank the pronaos are analogous to the lavamani, the small rooms in the Medici

Fig. 6. The groundplan of the Pantheon and the / L root-2 rectangle. n 0 i I

Chapel that flank the scarsella (Fig. 6). Historians cite the Pantheon as the most important architectural source for Michelangelo's design for the Chapel, referring to the quality of "interiority" that characterizes both spaces and to the similarity of the coffered domes. Now it appears that it may have served as an in- spiration for the volume as well [18].

Among Michelangelo's contemporar- ies, Renaissance architects Francesco di Giorgio and Filarete both used the square root of 2 as a proportional tool [19]. But by far the most important reference to this value is found in Alberti's Ten Books of Architecture, where the root-2 proportions are described in their geometric context:

There are some other natural proportions for the use of structures, which are not borrowed from numbers, but from the roots and powers of squares. The roots are the sides of square numbers; the powers are the areas of those squares; the multiplication of the areas produce the cubes. The first of all cubes, whose root is one ... is the most stable and con- stant of all figures, and the very basis of all the rest. But if... the unite [sic] be no number ... we may then suppose the first number to be the number two. Tak- ing this number two for the root, the areas will be four, which being raised up to a height equal to its root, will produce a cube of eight; and from this cube we may gather the rules for our proportion; for here in the first place, we may consider the side of the cube, which is called the cube root, whose area will in numbers be four, and the compleat or entire cube be as eight. In the next place, we may consider the line drawn from one angle of the cube to that which is directly opposite to it, so as to divide the area of the square into two equal parts, and this is called the diagonal. What this amounts to in numbers is not known: only it appears to be the root of an area, which is as eight on every side; besides which it is the diagonal of a cube which is on every side, as twelve [20].




Let us decipher this somewhat confus-

ing paragraph step by step. Alberti begins by introducing the cube as the basis for a

system of proportions based on roots and

powers, and he defines the square root and the cube root. Next he quantifies the dimensions of the cube in order to generate ratios forming a proportional system. He chooses to begin with the number 2, because, as he says, the number 1 ("the unite") was not considered a real number

by philosophers such as Plato and

Pythagoras. So, given 2 as the length of the side of a square, the area is 4 (2 x 2) and a cube formed from that square a volume of 8 (2 x 2 x 2). Next he considers the diagonals of the cube, the first con-

necting opposite corners of the sides, the second connecting opposite corners of the whole cube, but he runs into a problem, for he cannot quantify for us dimensions which are irrational ("What this amounts to in numbers is not known"). Thus, he describes them by generating other, similar, figures: the diagonal of the

square (which we know to be the square root of 2 times 2) becomes the side of a

larger square with area 8; the diagonal of the cube (square root of 3 times 2) becomes the side of a square with area 12. Thus, beginning with 2 he has generated a number series:

2 4 8 12...

The proportions inherent in this number series belong to the musical ratios-i.e. 1:2, 1:3, 1:4 and 2:3. Thus we find the link between incommensurate and commensurate ratios, and perhaps also a rationale behind the cubic geometry of the New Sacristy.

Historian Guglielmo De Angelis D'Ossat noted the presence of the musical proportions in the designs of the doors of the Medici Chapel. In a dia-

grammatic analysis of the portal found in each of the lateral bays of the chapel, he points out the ratios 1:2, 1:3, 2:3 and 2.4:1 [21]. The first three of these ratios

may be derived from Alberti's geometry of the cube. In the 2.4:1 proportion may be recognized the proportions of the so- called theta-rectangle (theta = 1 + 1]2, or the side of a square plus its diagonal), formed by adding a square to a root-2

rectangle. Although D'Ossat does not relate his findings to the root-2 rect-

angle as such, his findings once again involve the side and diagonal of a square and the root-2 rectangle, as in the

groundplan. Joannides also noted a musical ratio in his study of the tomb de-

signs, citing a 2:3 (he calls it 0.67:1) height-to-width ratio in the tombs [22].

Fig. 7. The gnomonic division of the root-2 rectangle by halves. The continued division of the root-2 rectangle in this way produces a spiral.

Was the square root of 2 used in de-

sign because of its importance as a number? We know that, to ancient architects, geometrical forms such as the sphere and the cube were as important as sym- bols as they were for their structural

properties, and the neo-Platonism of the Renaissance revived this tradition to a certain extent. In the case of the Medici

Chapel, it seems most likely that the

square root of 2 was used because of its

particular fecundity in generating systems of proportions. In the Renaissance, the concatenation, or interrelatedness, of forms was deemed the highest ideal

of architecture, because it was believed to emulate the architecture of the cos- mos. A comprehensive series of proportions achieved this ideal by eliminating arbitrary dimensions.

A SYSTEM OF ROOT-2 RECTANGLES Let us look at some of the geometric properties of the root-2 rectangle, a rect-

angle particularly interesting in terms of gnomonic growth. Gnomonic growth is a geometric process of expansion so that an addition (called a gnomon) to the

Fig. 8. The division of the root-2 rectangle into similar rectangles, which are in the ratio of 2:3 to the original. The continued trisection of the rectangle generates musical proportions.

A




Fig. 9. The dimensions of the Chapel, which may be seen in relation to the 2:3 (root-2) rectangle construction. Refer to Table 1 for actual dimensions.

original unit results in a new form geo- metrically similar to the original. The

geometric method of dividing rect-

angles into unit and gnomon is simple: given rectangle ABCD, one draws line DE from vertex D perpendicular to di-

agonal AC, then draws line EF perpendicular to side AC, subdividing ABCD into two: the smaller (AEFD) is the unit, similar to the parent rectangle; the

larger is the gnomon. Given the root-2

rectangle, this operation results in two

equal root-2 rectangles [23]. In other words, the gnomon is equal to the original unit. If this operation is continued

infinitely, it produces a series of spiral- ing similar rectangles, as well as dividing length AB into smaller lengths that relate to AB in ratios of 1:2, 1:4, 1:8, etc.

(Fig. 7). The number series inherent in these ratios,

1 2 4 8...

is that derived from the prime integer 2. If one varies this procedure by con-

structing diagonal DE perpendicular to AC, then drawing lines EF and GE paral- lel to sides AB and AD, respectively (Fig. 8), one creates a smaller root-2 rect-

angle, each side of which is 3 the length of the analogous sides of the parent and whose area is likewise 2/ that of the

original. Continuing this procedure results in rectangles that are progressively 2/ smaller and produces a series of divi- sions of the sides into progressive thirds. If we assign to the short side of the parent rectangle the integer 27, then successive division into thirds results in ra-

tios 9:27, 18:27, 6:18, 12:18, 4:12 and 8:12. The number series inherent in these ratios is

4 6 8 9 12 18 27...

Integers 9 and 27 are generated from the prime 3, as 4 and 8 were generated from the prime 2. Together with the number series described above, all are

Fig. 10. Beginning with a given dimension from the Medici Chapel and

generating successive values by B means of the expan- t sion of the root-2 rectangle, one finds l A more than a few coincidences with the actual dimensions of the chapel.

recognizable as being part of the geometric series:

1 2 3

4 8... 6 12... 9 18 36...

27 54...

which forms the basis for Alberti's system of commensurate proportions [24]. Here, then, is a geometric system, based on the irrational square root of two, which generates rational values related to the intervals of the musical scale.

While each rectangle in the second construction relates to that immediately above or below it in the ratio of 2:3, the

relationship between those twice re- moved is 4:9. Both ratios appear in The Ten Books of Architecture [25]. The 2:3 ratio, the diapente, is also known as the

sesquialtera; the 4:9 ratio, which Alberti called the "sesquialtera doubled," is the result of what Wittkower has called a "gen- eration of ratios" [26]-that is, the com-

pound ratio 4:6:9. Wittkower notes that 4:9 is the result of calculating 2/3 times '%, but we see that it may be generated by a

geometric operation. Note also that in

Fig. 8 the successive division of line AD into thirds produces lengths measuring 9 and 8, thus providing the 8:9 ratio, the ratio of the distance between the front

edge of the altar and the opposite wall to the overall width of the chapel.




Dimension A B C D E F/3 F/2 G H J K

Measured

3.520 2.065 2.876 3.060 4.330 3.750 5.626 4.562 6.477 9.855

10.232

Generated Deviation

3.520 2.032 2.874 3.048 4.311 3.734 5.600 4.572 6.467 9.701

10.287

0.0% 1.6% 0.1% 0.4% 0.4% 0.4% 0.5% 0.2% 0.2% 1.6% 0.5%

N.B. All dimensions are given in meters.

A last important feature of this system of root-2 rectangles involves the geometric mean, which Alberti describesjust af- ter he describes cubic geometry: "The

geometrical mean is very difficult to find

by numbers but it is very clear by lines; but of those it is not my business to

speak here" [27]. We shall make it our business to speak of this. Referring to

Fig. 7, we see that the length of the di-

agonal DE is the geometric mean of the

length of sides AD and GE. The geometric mean is calculated by multiplying the two extremes and taking the square root of the result, or it may be constructed by means of our construction. Calculating the geometric means of 27 and 18 (22.045) and 18 and 12 (14.7) demon- strates that successive diagonals are related as 2:3 (14.7:22.045 = 2:3), as were successive sides and areas.

THE SYSTEM IN THE MEDICI CHAPEL Now we may examine the dimensions of the New Sacristy. I have selected a key dimension of the chapel as a starting point to generate a series of dimensions.

(Fig. 9 locates the dimensions that cor-

respond to the numbers generated by the root-2 rectangle construction illustrated in Fig. 10, while Table 1 provides a comparative analysis between measured and generated dimensions.) Let us begin with the dimension A, 3.52 m, the distance between the perimeter wall and the step up to the scarsella. This is the diagonal of a root-2 rectangle with a

long side of 2.874 m and a short side of 2.032 m. This measurement of 2.874 m

corresponds to one-quarter of dimension C (0.1% deviation), while 2.032 m

corresponds to dimension B (1.6% deviation). The next larger root-2 rect-

angle has a short side equal to /2 of 2.032 m, or 3.048, and a long side equal to 3/2 of 2.874, or 4.311. The 3.048 m

corresponds to dimension E (0.4% deviation). The system of root-2 rectangles is continued upwards, as illustrated in

Fig. 10, and more coincidences with actual chapel dimensions are noted. No less than 11 out of 15 generated dimensions correspond to actual dimensions; the largest deviation, 1.6%, equals a 7.2 cm difference over 9.77 m. The dimensions of the New Sacristy may now be seen in the context of the pattern of ex-

panding root-2 rectangles. Finding systematic proportional rela-

tionships between dimensions of the Medici Chapel is an important first step toward a new understanding of an architectural monument by an artist of gigan- tic stature. While the present study of the

proportions of the Medici Chapel has been confined primarily to plan dimensions, I believe that an examination of all architectural dimensions might reveal a

geometric system that encompasses all three dimensions of the architectural

space. This calls for a new survey of the elevations of the New Sacristy, an under-

taking that may be daunting in scope, but promises to be more than justified in terms of what may be revealed.

Acknowledgments

I would like to thank Michele Emmer for thought- ful suggestions that improved this paper, Jay Kappraff for his encouragement and insight as I ex- plored the geometry, and Eric Frank for assisting me during one of the three measurements of the chapel. This paper has been partially funded by the National Council for Research (CNR) of Italy, as part of a project coordinated by Michele Emmer.

My work on pavements has been supported by grants from the Anchorage Foundation of Texas, Houston, and the Graham Foundation for Ad- vanced Studies in the Fine Arts. The book that has resulted from this study is to be published by the Anchorage Press, Houston, in 1997.

References and Notes 1. Plato, "The Republic," P. Shorey, trans., in E. Hamilton and H. Cairns, eds., The Collected Dialogues of Plato, Bollingen Series, Vol. 71 (Princeton, NJ: Princeton Univ. Press, 1961) p. 759.

2. The problems of correctly illustrating a groundplan are demonstrated by two figures de- picting the Medici Chapel that appear in the same work. The first is a full plan, the second a plan detail; each shows the altar with slightly different proportions. The confusion is compounded by the omission of the dimensions. See P. Portoghesi and B. Zevi, eds., Michelangelo Architetto (Turin, Italy: Einaudi Editori, 1964) Figs. 155, 166.

3. J. Pope-Hennessy, Italian High Renaissance and Baroque Sculpture, An Introduction to Italian Sculp- ture, Vol. 3. (London: Phaedon, 1970) p. 18.

4. While the measurements in this paper are given in meters and centimeters, the unit of measurement used in fifteenth- and sixteenth-century Flo- rence was the braccio (plural, braccia), which was equal to 58.37 cm. In this paper, metric dimensions were not converted into their equivalents in braccia for diverse reasons. It may be verified that convert- ing the metric dimensions in Table 1 into braccia results in, if not whole numbers, at least halves or quarters of whole numbers, but these numbers in themselves do not form a series more immediately recognizable than the metric ones, and implying the presence of a simple system may mislead rather than clarify. For example, the superficial relationship between the 29-cm side of the paving square and the 58.37-cm braccio may suggest that all dimensions may be meaningfully converted, but the diagonal dimension, whether expressed in meters or braccia, remains a dimensional approximation for an irrational value. Moreover, in any system involv- ing irrationals, dimensions serve largely only to mediate the proportions. Because dimensions are to be seen as the means, and not the end, whether they are given in feet and inches, meters, or braccia is a formal matter. I have chosen the modern measurements because they are the most familiar. For a detailed discussion of the braccio, see D. Zervas, "The Florentine Braccio di Panna," Architectura 9, No. 1, 6-10 (1979).

5. For more detailed discussions of the Sacred Cut geometry, see T. Brunes, The Secrets of Ancient Geom- etry and Its Use, 2 vols. (Copenhagen: Rhodos Inter- national Science Publishers, 1967) and K. Williams, "The Sacred Cut Revisited: The Pavement of the Baptistery of Florence," The Mathematical Intelligencer 16, No. 2, 18-24 (1994).

6. P. Joannides, "Michelangelo's Medici Chapel: Some New Suggestions," The Burlington Magazine 14 (1972) pp. 541-551.

7. C. de Tolnay, The Medici Chapel (Princeton, NJ: Princeton Univ. Press, 1948) p. 29.

8. It may be argued at this point that if the 11.64 m dimension is used as a control, it forms a /2 rectangle with the overall length of the chapel to within a closer tolerance (11.64 x 1.4142 = 16.46, a deviation of only 0.08% as compared to the 0.36% deviation between 16.55 and 16.49); however, there is no evidence for citing a 11.64 dimension in the ground plan of the chapel. Thus, it appears that the 6-cm discrepancy is a shortcoming of the chapel architecture, as we shall presently see.

9. As in the application of any theory of proportion to an antique structure, the accuracy to which the theory can be proven is very important. For one discussion of acceptable percentages in the application of a mathematical theory to an existing work of architecture, see H. Saalman, "Designing the Pazzi Chapel: The Problem of Metrical Analysis," Architectura 9, No. 1, 1-5 (1979). In this context, he describes a 2.42% deviation as too large, but refers to a 0.8% deviation as too "close to the mark [to be] dismissed lightly."

10. Six centimeters, though accounting for only


Table 1. Comparative dimensions of the Medici Chapel and a constructed root-2 rectangle. Letters identify lengths named in Fig. 9. Measured dimensions are the averages of three on-site measurements of each length; generated values result from the construction in Fig. 10.



0.36% of the existing overall length of the chapel, is not a negligible dimension. The 58.37 cm braccio was subdivided into 12 crazie (4.86 cm), or 20 soldi (2.92 cm). In these terms, 5 cm is equal to 1.23 crazie or 2.05 soldi. According to Saalman, the smallest unit of convenience was one crazia. (See Saalman [9] p. 3.) Given this, 1.23 crazia is too large a dimension to be attributed to an error in workmanship.

11. Whether Michelangelo was free to decide upon the dimensions for the perimeter walls for the chapel or whether he had to work within limits set by site restrictions is a matter of debate. See C. Elam, "The Site and Early Building History of Michelangelo's New Sacristy," Mitteilungen Des Kunsthistorischen Insti- tutes in Florenz 23 (1979) pp. 155-186, and H. Saalman, "The New Sacristy of San Lorenzo before Michelangelo," The Art Bulletin 67, No. 2 199-288 (1985). For a history of Michelangelo's struggle to accommodate the tombs in the chapel, see A. Morrogh, "The Magnifici Tomb: A Key Project in Michelangelo's Architectural Career," The Art Bulletin 74, No. 4, 567-598 (1992).

12. The diapason or octave was broken down into overlapping diapente (fifth) intervals. The overlap, 8:9, was termed the tonus. Cf. L.B. Alberti, De re Aedificatoria (London, 1755), reprinted as The Ten Books of Architecture (New York: Dover, 1986) p. 197.

13. See R. Wittkower, Architectural Principles in the Age of Humanism (New York: W.W. Norton, 1971) p. 127.

14. Vitruvius, The Ten Books on Architecture, M.H.

Morgan, trans. (New York: Dover, 1960) p. 177. This translation originally published in 1914.

15. Though proportional studies of the Pantheon have appeared from time to time, this particular

feature seems not to have been recognized. For two such studies, see Brunes [5] and K. de Fine Licht, The Rotunda in Rome, Jutland Archaeological Soci- ety Series, Vol. 8 (Copenhagen: Gyldendalske Boghandel, Norkisk Forlag, 1966).

16. While the reference circle used by Fine Licht ([15] p. 195) to analyze the rotunda of the Pan- theon is defined by the masonry of the inner wall, the reference circle I have chosen may be measured without requiring any demolition or other interference with the walls. Likewise, the root-2 rectangle generated circumscribes the contiguous floor surfaces of the pronaos and the rotunda, another area that may be readily measured.

17. The dimensions used in my calculation are based upon the dimensions cited by Fine Licht ([15] p. 195)-i.e. an inner radius of 21.5 m to the finished face of the plinths of the columns in the rotunda, equal to 72.68 Roman feet (1 Rf= 0.2958 m); a depth of 13 Rf for the exedrae, and a depth of 2 Rf for the niche. The dimensions for the pronaos result from my own survey, using dimensions in Fine Licht as a check. It should be noted that this calculation is not intended as part of a comprehensive analysis of the Pantheon, but as a means of demon- strating a point of geometric similarity between the Pantheon and the Medici Chapel.

18. For studies that point out relationships between the New Sacristy and the Pantheon, see J. Wilde, "Michelangelo's Designs for the Medici Tombs," TheJournal of the Warburg and Courtauld Institutes 18, (1955) pp. 54-66; A. Schiavo, Michelangelo Nel Complesso Delle Sue Opere (Rome: Libreria dello Stato, 1990); Portoghesi and Zevi [2].

19. For the use of the diagonal of the square by Francesco di Giorgio, see Wittkower [13] p. 108, note 4. For its use by Filarete, see H. Saalman, "Early Renaissance Architectural Theory and Prac- tice in Antonio Filarete's 'Trattato Di Architettura,"' in The Art Bulletin 41, No. 4, 89-102 (1959).

20. Alberti [12] p. 199.

21. G. De Angelis D'Ossat, "La Sagrestia Nuova," in U. Baldini and B. Nardini, eds., San Lorenzo (Flo- rence: Nardini Editori, 1984) p. 189 and Fig. 190. In order to relate all his ratios to a unit 1, he cites the ratio 2:3 as 1.5:1.

22.Joannides [6] p. 545.

23. J. Kappraff, Connections: The Geometric Bridge Be- tween Art and Science (New York: McGraw-Hill, 1991) pp. 53-54.

24. For more on this particular double geometric series, see P.H. Scholfield, The Theory of Proportion in Architecture (Cambridge, U.K.: Cambridge Univ. Press, 1958) pp. 36-37.

25. Alberti [12] pp. 197-198.

26. Wittkower [13] p. 115.

27. Alberti [12] p. 200.

Manuscript received 2 October 1995.




Michelangelo's Medici Chapel: The Cube, the Square and the Root-2 Rectangle

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