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I read once (I don't remember where exactly) about an early technique (Early Renaissance) to draw a square tiling floor in perspective. The next row in the drawing is done multiplying the previous one by a constant factor. My questions:

(1) What is the value of this factor? 2/3? 3/4?

enter image description here

enter image description here (2) Are there reliable sources/references that confirm this historical fact?

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    $\begingroup$ pre-Renaissance? IMO only "empirical; the correct geometry was discovered by Brunelleschi and Alberti. $\endgroup$ Aug 24, 2022 at 12:03
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    $\begingroup$ The method for drawing square tiled floors (pavimenti) in perspective was introduced by Alberti in De pictura (1435). It was one of the first techniques of linear perspective, early Renaissance, not pre-Renaissance. The ratio depends on the location of the vanishing point, which has to be at the same height as the eye (the oculus) when the canvas is placed vertically. $\endgroup$
    – Conifold
    Aug 24, 2022 at 12:23
  • $\begingroup$ @Conifold, is there a more frequent eye position and consequently a standard ratio choice? $\endgroup$ Aug 24, 2022 at 13:43
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    $\begingroup$ I think that @user6530's answer has what you are looking for. What Alberti calls "superbipartienti” (superbiparticular = integer plus two thirds) is a pre-Renaissance rule of thumb for painting floor tilings, where the width of the front row of tiles is set arbitrarily, and then progressively reduced in each subsequent row by 1/3 of the preceding width. So the scaling ratio is 2/3. Romor's Prospectiva Pingendi describes several such rules of thumb that were in use before the establishment of modern perspective. $\endgroup$
    – Conifold
    Aug 25, 2022 at 5:54

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tl;dr

Probably, what you are looking for is the "method of superbiparticular", cited in Alberti's De pictura, that uses a fixed ratio of $3:2$.


As has already been noted in the comments, during the Middle Ages and the early Renaissance (and actually even afterwards), various rules of thumb were used. The main sources relating to these rules (besides the paintings themselves that were made following them, but in general it can be very difficult to understand which rule was used by the author, considering that different parts of the painting, such as the characters, the floor and the colonnades, could follow different rules, and that the same author could change the rules from one painting to the next) are two: Leon Battista Alberti's De pictura (written in 1435 in Tuscan with the title Della pittura, the Latin translation bu the author dated 1440 circa) and Le due regole della prospettiva prattica (sic), of Jacopo Barozzi da Vignola, published posthumously in 1583 with extensive commentary by the mathematician Ignazio Danti.

Before Alberti, or rather, before Piero della Francesca, the rules of pictorial representation were based on Peckham's Perspectiva communis, which in turn is a synthesis of Alhazen's De Aspectibus (translated in the Tuscan dialect at the end of the 14th century).

Peckham speaks of bodies arranged in a continuous and orderly manner.

The 'ordered proportion' is handed down from the translation of Euclid's Elements by Gherardo da Cremona, and consists of two series of quantities in which the terms of each series are in proportion according to an identical ratio (i.e., two series like $A,B,C$ and $D,E,F$ such that $A:D=B:E=C:F$). This proportion can be used to describe a colonnade: the apparent heights of the columns and the apparent distances between pairs of columns maintain the same ratio and are therefore in ordered proportion.

The 'continuous proportion' is given in the translation by Campano da Novara, and consists instead not in a proportion, but in a proportionality, i.e. a similarity of the form $A:B=B:C$.

Continuous proportionality is appropriate to describe apparent quantities that decrease in perspective, e.g., the heights of figures or columns and the depth of floor tiles or ceiling coffers.

Thus: ordered proportionality can describe the relationship between the relative heights and widths of the pictorial composition, e.g. the height of a man and the width of a floor; continuous proportionality describes the apparent decrease of magnitudes in the depth of the pictorial space, e.g. the decrease in depth of a floor tile.

Now the problem is to determine the constant of proportionality in a continuous proportionality. A possibility, is to consider the "metodo delle superbipartienti" ("method of superbiparticular"), described by Alberti.

'Superbipartiente' is a term from medieval mathematics and music and indicates a proportional ratio of two thirds; more precisely, it indicates the decrease of one third of one of the two quantities (e.g., as in the sequence 9, 6, 4).

Alberti harshly criticises this construction in chapter 19 of the first book of De pictura:

[...] Hic essent nonnulli qui unam ab divisa aequedistantem lineam intra quadrangulum ducerent, spatiumque, quod inter utrasque lineas adsit, in tres partes dividerent. Tum huic secundae aequedistanti lineae aliam item aequedistantem hac lege adderent, ut spatium, quod inter primam divisam et secundam aequedistantem lineam est, in tres partes divisum una parte sui excedat spatium id quod sit et tertiam lineam, ac deinceps reliquas lineas adderent ut semper sequens inter lineas esset spatium ad antecedens, ut verbo mathematicorum loquar, superbipartiens. Itaque sic illi quidem facerent, quos etsi optimam quandam pingendi viam sequi affirment, eosdem tamen non parum errare censeo, quod cum casu primam aequedistantem lineam posuerint, tametsi caeterae aequedistantes lineae ratione et modo subsequantur, non tamen habent quo sit certus cuspidis ad bene spectandum locus. Ex quo non modici in pictura errores facile succedunt.

i.e.

[...] Then some would draw a line equidistant from the diagonal of the square, and divide the space between the two lines into three parts. Taking two parts, they would draw a line at this distance from the other and proceed according to this rule, so that the space that is between the first line and the second, divided into three parts, exceeds the space between the second and the third line by one part, and in so doing, to put it as mathematicians do, the spaces would be superbipartienti. And so they would indeed do, who, although they claim to follow a certain better way of painting, nevertheless I think they are not a little mistaken, because they have placed the first line at random, and although the other lines follow according to their rule, they have no idea where the apex of the visual pyramid is, and so they make no small mistakes.

Unfortunately, Alberti does not explicitly mention those "nonnulli" who use this wrong technique.

Alberti, in order to prove that the method of superbiparticular is wrong, refers to the diagonals. If the construction were right, the diametrically opposite points on the various trapezoids representing the floor tiles would have to be aligned. Indeed, aligned points project into aligned points and must therefore be drawn on the same line (chapter 20):

[...] Qui quidem quam recte descripti sint inditio erit, si una eademque recta continuata linea in picto pavimento coadiunctorum quadrangulorum diameter sit.

i.e.

[...] And the fact that the drawing is thus correct can be seen from this: that if you join the diameters [i.e., the diagonals] of the squares together as represented, you get a single straight line.

For example, if you observe Ambrogio Lorenzetti's Annunciazione, dated 1344, which actually does not use the method of superbiparticular, you will notice the gradual curvature of the diagonals, even though the lines coming out perpendicular to the eye converge correctly, when prolonged, at the central point (try prolonging the diagonals in this blow-up).

Following Romor's Prospectiva Pingendi, Prospectiva Fingendi, in the Annunciazione, Lorenzetti uses a different method and the row widths are in arithmetic progression (and not in geometric progression, as the method of superbiparticular would imply). Nevertheless, it seems to me that in other paintings Lorenzetti uses a different technique.

Presentazione al Tempio, Ambrogio Lorenzetti

In the Presentazione al Tempio (1342) above, we observe the same problem concerning the diagonals, but now the row widths are in geometric progression (alas, I do not have the original painting available, so my calculations have no value, but it seems to me that the ratio of the row widths is about $7:5$, not far from the superbiparticular ratio).

Coming back to the method of superbiparticular, the reason why Alberti considers it wrong is that the first line is placed at random ("quod cum casu primam aequedistantem lineam posuerint"), while the fact that the row widths are in geometric progression is not a problem, on the contrary, it is acceptable to him ("tametsi caeterae aequedistantes lineae ratione et modo subsequantur"), because it follows the continuous proportionality described by Campano da Novara. To overcome Alberti's error, one will have to wait for Piero's De prospectiva pingendi where he demonstrates that the decrease in apparent magnitudes is determined neither by a proportion of those described in Book V of the Elements, nor by the distance separating the plane of representation from the eye, as Alberti claims, but is regulated by the proportion that these magnitudes have both with the distance between the eye and the plane of intersection, and with the distance between the plane of intersection and the thing seen.


Clearly, as the Lorenzetti's example shows, the method of superbiparticular was not the only "empirical" technique used by pre-Renaissance painter. For example, in Le due regole della prospettiva prattica (1583), Danti cites this wrong rule, still used in his time: fix the center point and divide the base line in parts. Draw a quarter of a circumference, and divide the arc into a number of equal parts (it is suggested to divide it into 15 equal parts), join these parts with an appropriate point and draw the parallels.

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  • $\begingroup$ @Conifold Yes, you are right, I will fix it $\endgroup$
    – user6530
    Aug 25, 2022 at 14:09

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