Salvador Dali: In Search of the Fourth Dimension, 1979
The five Platonic
Solids:
Fire
Earth
Air
Universe
Water
Salvador Dali: The Sacrament of the Last Supper, 1955
Is the universe a dodecahedron? J.-P. Luminet, J. Weeks et.al: Dodecahedral space topology as an explanation for weak wide-angle
temperature correlations in the cosmic microwave background, Nature 425, 2003, 593-595.
Salvador Dali: Cruxifixion (Corpus Hypercubus),
1954
"Metaphysical, transcendent cubism, it is based
entirely on the Treatise on Cubic Form by Juan de
Herrera, Philip the 2nd's architect, builder of the
Escorial Palace: it is a treatise inspired by Ars Magna
of the Catalonian philosopher and alchemist Raymond
Lulle. The cross is formed by an octahedral hypercube.
The number nine is identifiable and becomes
especially consubstantial with the body of Christ. The
extremely noble figure of Gala is the perfect union of
the develpment of the hypercubic octahedron on the
human level of the cube. She is depicted in front of the
Bay of Port Lligat. The most noble beings were painted
by Velazques and Zurbaran; I only approach nobility
while painting Gala, and noblity can only be insired by
the human being.” Salvador Dali
The cube unfolds into 2d space and is made of 6
squares. The hypercube unfolds into 3d space and is
made of 8 cubes.
Moving a square parallel in space and joining
the corresponding corners gives the perspective
of a cube.
Moving a cube parallel in space and joining
the corresponding corners gives the
perspective of a hypercube.
The cube and hypercube as distorted
in central projection
Hypercube animation
Salvador Dali: Topological Abduction of Europe - Homage to René Thom, 1983
Dali refers in the left corner to the mathematician Rene Thom with the algebraic equation for a
particular singularity, the line on the right probably marking the A9/E15 motorway between
Salses and Narbonne in France at the border to Catalunya. The “X” marks the village of Opoul-
Perrillos. Dali heard of the Lord of Perrilos in an obscure screenplay. Dali states that in one of
his hallucinations Rene Thom appeared who convinced Dali of an upcoming catastrophe and
the centre of this catastrophe, linked with the disappearance or abduction of Europe would be
in the region of Salses and Narbonne at Opoul-Perrillos. It might be significant that Dali claims
that the railway station of Perpignan, a bit further south is actually the centre of the universe…
Salvador Dali: The Swallow’s Tail (Series on Catastrophes), 1983
Salvador Dali: Untitled (Series on Catastrophes), 1983
Rene Thom (1923-2002) french mathematician, one of the most
famous mathematicians of the 20th century
• worked primarily in differential and algebraic topology
• Professor at Institut des Hautes Etudes Scientifiques, Bur-sur-Yvette,
Paris
• founder of cobordism theory (with L. S. Pontrjagin and S. P. Novikov)
and of catastrophe theory (with V. I. Arnol’d and C. Zeeman) and
mathematical morphogenesis
• groundbreaking work in differential and algebraic topology: cobordism
theory, singularity theory and catastrophe theory
• R. Thom: Structural Stability and Morphogenesis 1972
• Fields Medal 1958
The algebraic classification of the elementary
catastrophes after R. Thom
Catastrophe Theory is a branch of Bifurcation Theory
in the mathematical study of dynamical systems and
was founded by Rene Thom.
Catastrophe Theory studies and classifies
phenomena characterised by sudden shifts in
behaviour arising from small changes and
circumstances. Small changes in certain parameters
of non-linear dynamical systems can cause equilibria
to appear or disappear. Catastrophe Theory analyses
with topological and algebraic methods the
degenerate critical points of the functions describing
the system. If the function depends on only two or
fewer variables there are geometrically only seven
generic catastrophes, called the elementary
catastrophe, which locally can be described with
simple algebraic equations:
1. Fold Catastrophe
2. 2A, 2B: Cusp Catastrophe
3. Swallowtail Catastrophe
4. Butterfly Catastrophe
5. Hyperbolic Umbilic Catastrophe
6. Elliptic Umbilic Catastrophe
7. Parabolic Umbilic Catastrophe
Salvador Dali: Untitled - Head of a
Spanish Noble Man-Fashioned by the
Catastrophe Model from a Swallow
Tail and Two Halves of a Cello, 1983
Salvador Dali: El Escorial and
Catastrophe-Form Calligraphy,
1982
Salvador Dali: Catastrophe
Writing, 1982
Salvador Dali: Topological Contortion of a Female Figure Becoming a Violoncello, 1983
Man Ray: Shakesperean Equations, Twelfth Night,
1948
Man Ray: Shakespearean Equations, Julius Cesar,
1948
Man Ray: Shakespearean
Equations, Macbeth,
1948
Max Ernst: The Feast of Gods,
1948
Max Ernst: Surrealism and Painting, 1942
J. W. Alexander: Horned Sphere, 1924: a
wild embedding of the sphere into 3D
space
Man Ray: Mathematical Objects, 1934
(from: Institut Henri Poincare, Paris)
Mathematical models of complex algebraic surfaces,
1874
Algebraic surfaces: Sextic and Clebsch
Quadric