In 1991 Franco Grignani, at the age of 83, drew this painting with black tempera entitled “Two hyperbolic corners” (“Due angoli iperbolici”), in a 51×51 cm format on a Schoeller cardboard.
The artwork was donated to the painter Lorenzo Piemonti, well known within the artistic movement “Madì” for its plastic results derived from variations in number sequences, together with a letter relating to the visual phenomenon of undulatory plastic character.

A living signal for the geometry of the soul.

These two modular signs, mutually touch and intersect according to a common support plane.

They give rise, through a graphic system in a progression that starts from the center and goes towards the periphery of the painting, to a visual phenomenon of UNDULATORY plastic character.
It is a complex issue due to the particularly ambiguous and spatial position of the bulk and at the same time distorted signs. The applied hyperbolicity stands for multidirectional and three-dimensional exaltation, where the visual perception collected by standing side by side (to the right and to the left), makes the structure live in the sphere of a VIRTUAL DYNAMIC motion.
This work, therefore, communicates a propulsive and continuous spatial system which, for the whole duration of emotion, I call A LIVING SIGNAL FOR THE GEOMETRY OF THE SOUL.

Franco Grignani, letter relating to the visual phenomenon of undulatory plastic character, 1991
Franco Grignani, letter relating to the visual phenomenon of undulatory plastic character, 1991

But how was Franco able to create such works only with a long ruler, a sharp pencil, and black tempera? Many people still think he used a computer, but the closest tool to a computer he owned was his Olivetti Lexikon 80 typewriter … Contributing to this perception, in 1981, anticipating the times (the first Apple is from 1976), Cesare Musatti, who brought the Psychology of Gestalt to Italy, stated in an exhibition catalog from Lorenzelli Arte that in Grignani’s artworks “everything is calculated, everything is precise: and it is a work that could be the result of a well-prepared elaboration conducted by an electronic computer.”

Even now, I’m not able to ‘decrypt’ all of his artworks yet (no art critic, however, has ever made it), but this one has some clearly identifiable characteristics:

The modular sign

The modular sign at the basis of the entire process is definitely the one shown in the letter above. Franco always started from a simple sketch made on one of his squared notebooks, in order to study its geometric feasibility in the repeatability of the sign. The unit of measure was always the simplest: the single “square”. This module, therefore, has a size of 5×3:

a module by Franco Grignani, 1991 (remake 2020 with GeoGebra)
a module by Franco Grignani, 1991 (remake 2020 with GeoGebra)

The matrix

The next step was the repetition of the module in a matrix not distorted yet, in order to find the right spatial position of the elements within the future painting. There are many examples of similar modules in his notebooks; some resulted in paintings (“fatto” means “done”), but the most remained among the thousands of theoretical experiments:

The color sometimes added to the sketches was not always intended to be used, but rather to help give a three-dimensional sense to the construction of the matrix.

For what concerns the determination of the overlapping of the modules, in order to create an undulatory effect, they could NOT simply be superimposed one on top of the other but had to be offset diagonally, so as to make them touch each other properly:

WRONG overlap
WRONG overlap
right overlap
right overlap

The undistorted grid

Before proceeding with the distortion, Franco still had to make two more choices: how many squares (called “spazi“: spaces) to divide the cardboard and where to place the first module.
The size of the squares strictly depended on the usable width of the cardboard; Franco personally chose the frames for his paintings, bringing them directly to a nearby framer (the famous Nino Soldano), and therefore knew that a 51 cm square Schoeller cardboard side allowed a useful width of 50 cm:

Franco Grignani, calculation for a Schoeller cardboard
Franco Grignani, calculation for a Schoeller cardboard [*]

For this painting, Franco chose to divide it into 22×22 spaces; since the whole work is perfectly symmetrical to the center, the number had necessarily to be even (11 squares to the right and 11 to the left of each baseline of the distortion). The first module was therefore suitably positioned with respect to the center:

22×22 grid and centering (remake 2020 with GeoGebra)

The undistorted positioning

This short video (created using the opensource software GeoGebra) summarizes the process of positioning on the undistorted grid from the center, even if Franco did not need to make such precise tests, having everything clear in his mind:

The hyperbole

The next phase is undoubtedly the most complex, original, and extremely difficult to decrypt if one does not have a trained eye.
Many people think that he used some abstruse trick to create curved lines – as those immediately perceived by the eye – but in reality, he was able to create the impression of a hyperbolic distortion using only straight lines!
In a reading analysis for the artworks exhibited at the Galleria Spazia in 1982, Grignani himself wrote: “The twists are not produced by signs in the curve but by the gradual meeting of structures placed diagonally with grid coordinates“.

Franco studied Geometry at high school, and then Mathematics for two years at the Faculty in Pavia until 1929, when he moved to Turin to study Architecture, therefore he perfectly mastered mathematical-geometric tools such as conics. Even though I was just a child, I learned these techniques by observing him for hours and hours in his indefatigable work in his studio and listening carefully to his every single word, and I amused myself at primary school in trying to imitate him in my math notebook instead of listening to my teacher…

When my grandfather wanted to show me a painting he had just finished, he put it on the floor next to the bookcase and always told me: “don’t just stand in front of it, you have to move, from right to left and then the other way around, and look at it from the corner of your eye: you will always see a different picture in motion!” And so it was, as it is suggested in the letter above mentioned. So, even if one has this pic on the screen, I invite to click on the pic below and do as suggested in full screen mode:

Franco Grignani, Two hyperbolic corners, 1991

You’ll then perceive the three-dimensional movement created by the geometric distortion trough hyperbolics:

hyperbolic study on Franco Grignani, 1991 (2020 with GeoGebra)
hyperbolic study on Franco Grignani, 1991 (2020 with GeoGebra)

But, as Grignani himself wrote in 1982 about hyperbolic tensions, “the three-dimensionality is illusory and comes out from the experience that each of us has on the position of light and its relative shadow.”

The Professor of Nuclear Science Giuseppe Caglioti in the book “Simmetrie infrante” (“Broken symmetries”, cover by Grignani – 1983) well described the feeling when facing a hyperbolic by Grignani: “To a superficial observer, the work reproduced here could at first appear a two-dimensional geometric drawing, made up of many black and white parallelograms, delimited by clear and sharply defined sides. But after a few moments, these parallelograms reorganize themselves before our eyes; […] the gaze tends to identify and then to follow privileged itineraries, along which the level of attention required remains almost constant. […] After a further period of observation, new perspectives soon explode outside the surface of this painting. […] The result is an alternation of multiple current patterns of the possible structuring of space.”

In the already mentioned exhibition catalog from Lorenzelli Arte of 1981, Cesare Musatti added: “Certainly, the viewers find themselves as tough divided in two: on the one side they are dazzled by the precision of how the figural elements are repeated and only slowly and progressively change their size, inclination, and position. On the other side, they are involved in a vision of the whole that refuses to be analyzed in its details.”

In 1985 the gallerist Michele Caldarelli wrote in the magazine “Scienza 85”: “All of Grignani’s work, in its coherence, represents a continuous fusion of previous experiences with new ones. A final example (and not least to be believed) is the series of the hyperbolic structures, which almost totally summarize the previous experiments. A very complex identification of points, spatially determined on the level of each work, generates hyperbolic spaces, which fluctuate due to an illusory variation of brightness, but also due to a changed spatial interpretation and anamorphic effect if they are not observed from the front. The result is a polycentric and absolutely unstable universe of signs, yet ordered within a dynamic symmetry that reconciles, in a visual image, the illusions of time and space that are real and perceptible only when their mutual interference manifests itself in becoming of a world, whose further ‘N’ dimensions are yet to be discovered.”

The grid

So, how did Franco actually create a hyperbolic impression by using only simple straight lines?
If we were to highlight only the vertices of the many polygons in the painting above, we would get this pic:

study on Franco Grignani, 1991 - points (2020 with GeoGebra)
study on Franco Grignani, 1991 – points (2020 with GeoGebra)

Still, the curvilinear course remains vivid, but let’s just try to forget the curves and focus instead on how we might otherwise join these points with straight lines…

And that’s the key:

study on Franco Grignani, 1991 - grid (2020 with GeoGebra)
study on Franco Grignani, 1991 – grid (2020 with GeoGebra)

Franco had a series of grids (“reticoli“) which had the peculiarity of being easily adaptable to different matrices and different cardboard sizes; once again, simplicity was the basis of his process. Each grid had its own name, deriving from an Italian, European, or even World city (Pavia, Milano, Roma, Londra, Chicago, etc. but also Melitopol, in honor of the hometown of his wife Jeanne Michot), sometimes followed by a letter (A, B, C) to identify some variants.

This painting is peculiar because the sequence of the series of spaces is perfectly symmetrical on the four sides of the 50×50 cm square, but, as it can be seen from these sketches below, the solutions often included much more complex grids:

Thanks to the tracing paper superimposed on the grids he could quickly test the overall effect:

The spaces in the distorted grid of this painting, therefore, remain 22 but no longer have the same width as when they were simple equal squares, as explained above. Each type of grid had its own set of widths. Franco used, once again, the simplest solution, that is integers, which allowed him to adapt the same grid to cardboards of different sizes.
For this grid the sequence is: 14-13-12-11-10-9-8-7-6-5-4-3-2-3-4-5-6-7-8-9-10-11 for a total of 167 units. On a useful width of 50 cm, this means that one unit is 3 mm long. Therefore, for example, the first space is 14 x 3 = 42 mm long: you will need a very sharp pencil and great eyesight to accurately trace all the lines without any error!
This method approximated a vanishing point of the lines (four in this painting) that Franco could not use in practice as it was too distant from the cardboard itself.

The whole process is well summarized in this second short video:

Franco Grignani exhibited his hyperbolic works in at least five important exhibitions: in 1981 in Milan, in 1982 in Bologna, in 1985 in Como, and again in Milan in 1988 and in 1991. More hyperbolics can be seen in the Gallery.

As early as in 1971 in the magazine “Linea Grafica”, Grignani stated:

If you intervene by shifting your attention to a different, eccentric point, you automatically create a state of mind of tension that multiplies interest and communication and becomes stronger to the extent that it can cause a sort of physical discomfort.

Two years after, in an interview that appeared in 1973 again in the magazine “Linea Grafica”, he declared:

It’s not that I argue that we see wrong: we just can’t see. When one looks at a painting and understands it in its entirety, as long as he simply looks at it he forgets it because he does not participate. When does one participate? When in the object, in the painting he finds something that he lacks. Then I carry him inside my structures as in a labyrinth. And of this labyrinth, the observer wants to find the solution, so he is forced to actively participate without ever being able to find sufficient justification. […] I don’t believe in improvisation. I want to check, as much as possible, every phenomenon. In my paintings, there is an elaboration that arises precisely from this premise. Many investigations were abandoned even after long work because they did not develop the phenomenon I was looking for. They told me that mine was a substantial exhibition: but how many works have never become paintings because I had to leave on the way! Thus, what I have displayed is only the smallest part of what I have designed.

[*] courtesy of Daniela Grignani

[due to the fact that up to now no one else – not even within Franco’s family – has ever been aware of these techniques in detail, the text in this post is strictly licensed under CC BY-NC-ND 4.0: this means it can be reproduced as it is, as long as the author’s name (Emiliano Camera) and the source ( are mentioned, it is not used for commercial purposes, and it is shared with the same license: bots are used to detect breaches; pics from Franco Grignani are all © Eredi Franco Grignani; pics and videos with GeoGebra are licensed under CC BY-NC 4.0]