Leonardo’s CAD (2)

Over the last few decades Leonardo da Vinci’s *Adoration of the Magi* and the well‑known preparatory drawing in the Uffizi have been the subject of increasingly refined perspective analyses and digital reconstructions, which have helped clarify the spatial structure of the complex architectural background.[1] In many of these studies, the grid traced by Leonardo on the floor is interpreted as a regular lattice of squares, projected according to the rules of Albertian central perspective and taken as the basis for three‑dimensional modelling of the ruins and staircases.[1][2]

However, adopting a square grid as an interpretation of Leonardo’s reticulation, and back‑projecting the drawing into a three‑dimensional CAD space, produces a markedly distorted reconstruction of the arches between the staircases. In our view this result is incompatible with Leonardo’s mathematical, artistic and architectural vision, at a time when he was still young and strongly influenced by the Florentine milieu of the later fifteenth century.[2][3]

The present work fits into this line of research but proposes a different reading of the grid underlying the Uffizi sheet. Starting from a geometric analysis of the two‑dimensional elements parallel to the picture plane, and from a subsequent verification through a perspective reconstruction of the 3D architecture in CAD superimposed on the drawing, it is argued here that Leonardo’s spatial scheme is not based on a chessboard of squares, but rather on a mesh of rectangles modularly related to the Florentine braccio, with a harmonic ratio of 4:3 between the long and the short side.

### The Florentine braccio and the 4:3 ratio

Taking the Florentine braccio, equal to 58.36 cm, as the unit of measurement (in common use in Florence around 1481), each module of the grid measures 58.36 cm by 43.77 cm. The longitudinal vanishing lines are spaced one braccio apart, while the transverse ones are spaced three‑quarters of a braccio apart.[4][5][6]

The choice of the Florentine braccio as the unit of the perspective grid is supported both by historical reasons and by the coherent overall scale of the architectural elements, the human figures and the animals represented in the drawing.[4][7][6] This choice is preliminary and, in the absence of explicit indications by Leonardo for this specific sheet, fixes the absolute scale of the reconstruction without affecting the proportional relationships between the elements.

The identification of the 3:4 ratio between the sides of the rectangles (hypothesis **x**) is the result of an a posteriori comparative verification, contrasting the reconstructions obtained by also testing the ratios 2:3 (hypothesis **y**), 1:2 (hypothesis **w**) and the golden ratio (hypothesis **z**) between the short and the long side of the grid modules.

### The generating triangle of the staircase

On this grid, clearly set by Leonardo on the floor plane on which the staircases rest (the CAD xy‑plane), a right‑angled triangle generating the nearer staircase is erected perpendicularly. This triangle lies on a plane parallel to the picture plane (CAD xz‑plane), and is therefore not subject to perspective deformation, making it easily measurable according to the rules of plane geometry.[8]

The drawing allows two possible generating right triangles for the main staircase to be identified, similar but distinct, sharing the same acute base angle of approximately 36°, which can be used for the theoretical measurement of the elements for CAD restitution:

– **Hypothesis a)** the “visible” staircase of 15 steps, with the longer cathetus measuring 7 braccia; 
– **Hypothesis b)** the “complete” staircase of 16 steps, with the longer cathetus measuring 7.5 braccia.

From the graphic analysis of the Uffizi sheet, both hypotheses must initially be considered admissible to the same degree, owing to the thickness of Leonardo’s strokes, which introduces an inevitable margin of approximation, indeterminacy and interpretation with respect to the “ideal” CAD measurements. The common acute angle at the base of the triangles measures about 36°, with only minimal uncertainty due to the line thickness.

### Measurement of the staircase, geometric hypothesis a) (15 visible steps)

In hypothesis a), the horizontal base of the visible staircase (15 steps) is assumed to correspond to seven modules, for a total length of about 408.52 cm. The calculated height of the landing at the fifteenth step, corresponding to the length of the short side of the triangle, is about 296.81 cm, divided into fifteen risers of 19.79 cm and fifteen treads of 27.23 cm.

In the preparatory drawing the generating triangle is clearly readable not only in the foreground staircase, but also in the symmetric staircase in the background, which repeats the same geometric construction within the same modular grid.

The width of the staircase equals four short sides of the grid rectangles; in our hypothesis this corresponds to 4 × 3/4 of a Florentine braccio, that is 3 braccia, or 175.08 cm. The plan projection of the staircase therefore forms a rectangle of 7 × 3 braccia (408.52 × 175.08 cm), with a dimensional ratio of 7:3.

### Measurement of the staircase, geometric hypothesis b) (16 complete steps)

In hypothesis b), the horizontal base of the “complete” staircase (16 steps) is taken to correspond to seven and a half modules, for a total length of about 437.70 cm. The calculated height of the landing at the sixteenth step, corresponding to the length of the short side of the triangle, is about 318.01 cm, divided into sixteen risers of 19.88 cm and sixteen treads of 27.36 cm.

The width of the staircase, as in hypothesis a), is equal to 3 braccia (175.08 cm). The plan projection of the staircase thus forms a rectangle of 7.5 × 3 braccia (437.70 × 175.08 cm), with a dimensional ratio of 5:2.

The other combinations with alternative grid modules (2/3 braccio, 1/φ braccio, 1/2 braccio) can then be systematically compared to these reference cases (a and b) to evaluate which modular scheme best reproduces the architectural forms without introducing visible distortions.

Citazioni:
[1] [PDF] Three Points of View for the Drawing Adoration of the Magi … – Unibo https://cris.unibo.it/retrieve/e1dcb338-c06f-7715-e053-1705fe0a6cc9/heritage-04-00123-v2_compressed.pdf
[2] [PDF] Download this PDF file – SCIRES-IT http://www.sciresit.it/article/download/13706/12029
[3] IMSS – . Scientific Analysis of the Adoration of the Magi https://brunelleschi.imss.fi.it/menteleonardo/emdl.asp?c=13419&k=1470&rif=14071&xsl=1
[4] Misure fiorentine, da perder di misura. https://www.florencecity.it/misure-fiorentine-da-perder-di-misura/
[5] Il “braccio fiorentino”: storia e restauro di un’antica unità di misura. https://firenzecittadelrestauro.it/il-braccio-fiorentino-storia-e-restauro-di-unantica-unita-di-misura/
[6] Unità di misura fiorentine e dove trovarle, in Piazza del Duomo https://duomo.firenze.it/it/opera-magazine/post/4829/unita-di-misura-fiorentine-e-dove-trovarle-in-piazza-del-duomo
[7] Curiosità fiorentine: tracce di antiche unità di misura https://arteleonardo.com/it/blog/268/curiosita-fiorentine-tracce-di-antiche-unita-di-misura
[8] squadra1.jpg https://ppl-ai-file-upload.s3.amazonaws.com/web/direct-files/attachments/images/46142614/e10eaec5-68f9-4297-b045-e8d96883d516/squadra1.jpg