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