for Paul Eluard's book of poetry 'Repetitions', published 1922
without the help of a good drawing,
is not only half a mathematician,
but also a man without eyes. "
Lodovico Cigoli to Galileo Galilei, 1611
A number of simple, cautionary examples of the problematic relationship between Maths and diagrams exist as diagrammatic puzzles, and a famous example is the 'Missing Square Puzzle' shown in figure 1.
Figure 1: When component parts of triangle A are rearranged, an empty square appears from nowhere
When the coloured components of Triangle A are rearranged to form Triangle B, the process of their reorganisation creates an empty square from nowhere, despite the fact that both triangles appear to have identical surface areas.
The key to understanding this kind of puzzle is the idea that Mathematical objects exist in a fictional world created by mathematicians.
No matter how precisely drawn, Geometry diagrams only ever give rough approximations of the ideal objects they represent, and these objects can only ever exist in a world where points has zero dimensions and where a lines and planes have zero thickness.
Marcel Duchamp was fascinated by the idea of a parallel world of Mathematical perfection that exists alongside the chaos and imperfection of reality and daily experience. This was the subject of the blog post: 'A soggy book of diagrams as a wedding present from Marcel Duchamp', which considered one of Duchamp's less well known projects using a found book of Euclid's Geometry.
Figure 3: Mitsunobu Matsuyama's 'Paradox'
Another excellent example of this type of diagrammatic puzzle is the Mitsunobu Matsuyama Paradox, which consists of a square divided into 4 quadrilaterals. When each quadrilateral is rotated 90 degrees clockwise, the new arrangement suddenly gives rise to a small central square, as depicted in red in figure 3.
This puzzle also relies on the natural limits of accuracy to human vision. Both the height and width of the two squares (before and after rotation of their quadrilaterals) are fractionally different, but this is barely distinguishable to the naked eye, because the difference is spread around the entire perimeter of each square.
For Lodovico Cigoli, a good 17th century diagram provided a visual means of gaining a deeper insight into the mathematics of nature. However, over the course of the following two centuries, the role of the diagram shifted to the extent that it became considered more of a veil that obscured the essence of mathematics, and algebra was proposed as the only way to lift the veil.
"...it is not the figures which furnish the proof with geometers, though the style of the exposition may make you think so. The force of the demonstration is independent of the figure drawn, which is drawn only to facilitate the knowledge of our meaning, and to fix the attention; it is the universal propositions, i.e., the definitions, axioms, and theorems already demonstrated, which make the reasoning, and which would sustain it though the figure were not there." (2)
by Gottfried Leibniz (Figure 1 left) and Christiaan Huygens (figure 2 right) to Jacob Bernoulli
for publication in the Acta Eruditorum, 1691
However, the diagram remains an extremely powerful tool and a visual guide in providing an insight into the austere and pristine world mathematical geometry and topology. Leibniz's own notebooks contain an astounding array of diagrammatic sketches that accompany his mathematics, as in figure 4, and the designs and calculations for his 'Universal Calculator', some 200 years before the work of Charles Babbage.
For an interesting introduction to the notebooks of Leibniz, see Stephen Wolfram's blog: Dropping In on Gottfried Leibniz.
Man Ray described the models that he found languishing in dusty cabinets as 'so unusual, as revolutionary as anything that is being done today in painting or in sculpture', though he admitted that he understood nothing of their mathematical nature. When the Second World War came to Paris in 1940, Man Ray relocated to Hollywood, where he started work on a series of 'suggestively erotic paintings' based on his 1930's photographs. Under the title of the 'Shakespearean Equations', he later referred to the paintings as one of the pinnacles of his creative vision.
Below are images of selected paintings from the 'Shakespearean Equations' series, Juxtaposed alongside the original mathematical models they were based upon.
Note: An extensive online collection of mathematical models is available here: the Schilling Catalogue of Mathematical models
shown alongside the models they were based on, from the Institut Henri Poincaré, Paris.
As Witt points out in his own blog on this series here: Functional Surfaces I, it's said that the architect Le Corbusier kept a copy in his studio whilst designing the Phillips Pavilion. Max Ernst appropriated from the book for a series of collages and poems in the catalogue accompanying his 1949 exhibition 'Paramyths'.
Figure 6: La Fable de la Souris de Milo (The Fable of the Mouse of Milo), Max Ernst, Collage, c.1948
The graphs used by Ernst in figures 6 and 7 represent Bessel functions, prescribed solutions to differential equations found to govern a variety of wave motions, particularly those that are confined to the surface of spheres and cylinders.
The head of the venus de Milo in figure 6 also contains diagrams of electromagnetic and planetary gravitational fields.
Figure 7: Ausstellungssignet, Max Ernst, 1948
Pen and Ink on Paper, 5.6x10.5 cm.
by Eugene Jahnke and Fritz Emde, courtesy of Andrew Witt.
Dodecahedron planum vacuum, Lithograph by Leonardo da Vinci, for Luca Pacioli's 'Divina Proportione', 1509.
The dynamism and immediacy of computer rendered models has abruptly changed the way we experience the geometrical diagrams of Mathematics.
The video below was made by the Mechanical Engineer and amateur Mathematician Jos Leys using POV-Ray.
The film opens with a rotating 600 cell (a 4 dimensional analogue of an Icosahedron), before dissecting and reconstructing a 120-cell (a 4-dimensional analogue of a dodecahedron) in twelve rings of 10 dodecahedra. Two sets of six rings form 2 solid interlocked tori.
More video and information available here:
a dissection of the 120-cell in 12 rings of 10 dodecahedrons
1) Some 29 letters from Cigoli to Galileo remain, however only 2 letters from the scientist to the painter are left, as the artist's heirs chose to destroy all incriminating evidence of their association, after the papal condemnation of Galileo. (In 1610 Cigoli received from Pope Paul V the assignment to paint the dome of Santa Maggiore Maggiore with the Immaculate Conception, the Apostles and Saints.)
2) Leibniz1704, New Essays: 403