MATHEMATICAL IMAGES IN THE REAL AND UNREAL WORLD
ABOUT THE AUTHOR
Prof. A.T.FOMENKO (born 1945), is a Full Member of the Russian
Academy of Sci., Dr. of Sci. (Math. and Phys.), Moscow State
University (Moscow), and Head of the Department of Differential
Geometry and Applications in Moscow University.
He is a distinguished mathematician and a well-known specialist
in the fields of geometry, Hamiltonian mechanics, the calculus of
variations, computer geometry, and algorithmical problems in pattern
recognition. He was a winner of the Award of the Moscow Math.Soc.
(1974), the Award of the Presidium of USSR Acad. of Sci.
in mathematics (1987), and of the State Award of Russia
(in mathemativs) (1994).
He has obtained fundamental results in the theory of minimal
surfaces and solved the multidimensional Plateau problem -
the existence of a globally minimal surface in each spectral
bordism class. He created a theory of topological classification of
integrable Hamiltonian systems . This theory was applied to the
important problem of classification of isoenergy surfaces for
integrable dynamical systems (which arise for example in
celestial mechanics and in the theory of motion of a rigid body).
Fomenko obtained the complete classification theory forthe
bifurcation of solutions in integrable Hamiltinian systems.
He also developed a new empirico-statistical method for
the analysis of narrative texts (e.g. chronicles).
Among the well-known books written by A.T.Fomenko are:
"Variational Principles in Topology (Multidimensional Minimal
Surface Theory)", "Differential Geometry and Topology",
"Variational Problems in Topology (The Geometry of Length, Area
and Volume)", "Homotopic Topology" (together with D.B.Fuchs
and V.L.Gutenmacher), "Symplectic Geometry. Methods and
Applications", "Basic Elements of Differential Geometry and
Topology" (together with S.P.Novikov), "Modern Geometry"
(together with S.P.Novikov and B.A.Dubrovin), "A Course in
Homotopic Topology" (together with D.B.Fuchs), "Minimal
Surfaces and Plateau's Problem" (together with Dao C.T.),
"Methods of Statistical Analysis for Narrative Texts and Applications
These books were translated into English by Springer-Verlag,
Plenum, Reidel (Kluwer), Gordon and Breach, American Math.Soc.
The following books were published originally in English:
"Plateau's Problem (vol.1 - Historical Survey, vol.2 - The Present
State of the Theory" (Gordon and Breach),
"Integrability and Nonintegrability in Geometry and Mechanics"
"Integrable Systems on Lie Algebras and Symmetric Spaces" (together
with V.V.Trofimov) (Gordon and Breach).
A.T.Fomenko has also the talent for expressing abstract mathe-
matical concepts through artwork. Since the mid-1970s, Fomenko has
created more than 280 graphic works. Not only have his images
filled pages of some of his own books in geometry, but they have
also been chosen to illustrate books (of many mathematicians) on
other subjects, such as statistics, probability, number theory and
so on. In addition, his works have found their way into the
scientific and popular press and have been displayed in more than
100 exhibits in the Russia, USA, Canada, the Netherland, India and
much of Eastern Europe.
In 1990 the American Math.Soc. published the book by Fomenko:
"Mathematical Impressions" containing 84 reproductions of works by
"What's interesting about his work to me is that it shows the
impact of certain mathematical ideas", says William Thurston, Profes-
sor of mathematics at Princeton University. "It excites your imagi-
nation. It's interesting to look at and think about. It's not designed
to be just a straightforward communication of a simple idea, but to
stir up your imagination - which it does. And in that sense it is
very good and successful. I think it's a very effective way of commu-
nicating mathematics." (Insight Magazine, April 30, 1990).
Springer-Verlag published the English translation of the book:
A.T.Fomenko "Visual Geometry and Topology". This mathematical book
also contains 50 reproductions of graphic works of Fomenko.
The book ("Mathematical Images in the Real and Unreal World")
is quite different from the books listed above and contains the total
collection of Fomenko's works with 1/2-mathematical and 1/2
philosophical comments. The works are organized in chapters correspon-
ding to different branches of mathematics, history and philosophy.
The book is unique event in mathematical literature and does not have
The epilogue to this book is written by famous mathematician
The book is published in Russia, Moscow (in Russian),
Moscow University Press, 1998.
The book contains 208 reproductions of works by Fomenko (172 in
black and white and 36 in color). In the accompanying captions,
Fomenko explains the mathematical motivations behind the illustrations
as well as the emotional, historical, or mythical subtexts they evoke.
The illustrations carry the viewer through a mathematical world
consisting not of equations and dry logic, but of intuition and
inspiration. Stimulating to the imagination and to the eye, these
works can be interpreted and appreciated in various ways - methema-
tical, aesthetic, or emotional. The commentary to each graphic work
consists of two parts: 1) mathematical explanation, 2) non-mathe-
matical associations connected with the theme of the drawing (his-
torical, mythological et cetera). This second part of comments is
unique in mathematical literature and at first appears in the present
book. Each commentary has the volume approximately 1 page (type-written
manuscript). The book is oriented on the wide audience and is intented
for students, mathematicians, physists and all readers who is interested
in visualization of modern mathematical ideas and their connection
with general human concepts.
The volume of the book: 210 pages (manuscript), 172 graphic
works (black-white, half-tone, the size: 30 x 40 cm, 36 color works;
each print needs in 1 individual page. Thus, the total volume is as
follows: 210 + 208 = 418 pages.
C o n t e n t.
Introduction. Associative-visual thinking in modern mathematics.
1. Images in general and algebraic topology. Simplicial and cell
2. Images in geometry and topology of smooth manifolds.
3. Images in mathematical analysis. Functions on the manifolds,
algebraic surfaces and singular points.
4. Images in mathematical physics, mechanics, differential equations.
5. Images in the calculus of variations, differential equations,
group theory and crystallography.
6. Images in computer geometry. Algorithmical problems of recogni-
zability. Mathematical statistics and probability theory.
7. Geometrical images in the novel of M.A.Bulgakov "Master and
8. Images in the general mathematical concepts.
9. Images in color.
About graphic works of Fomenko.