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V.K. Maslyuchenko, G.-J.Ya. MaslyuchenkoЧернiвецький нацiональний унiверситет iменi Юрiя Федьковича

TOPOLOGY AND ART

International conferencededicated to 70th ann of A. Plichko

Lviv, June 26-29, 2019

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

1. Mathematics and Art

From the ancient times, Mathematics is used in different kinds of art (music, dance, pai-nting, sculpture, architecture, literature and textiles). Nowadays the study of connecti-ons between Math and Art became a subject of scientific investigations. In particular,a solid work of A.V. Voloshynov who has defended a doctorate thesis “Ontology ofthe beauty and mathematical principles of art” on the Philosophical Department ofM.V. Lomonosov Moscow State University in 1993, is devoted to such investigations.

1. Voloshynov V.V. Mathematics and Art, Prosveschenie, Moscow, 2000.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

2.1. Math and Art in Chernivtsi: abstract

Not long ago, the authors, a mathematician and an art critic, started to get interestedin the subject. They have published papers on Math and Art [2-9].

2. Маслюченко В.К., Матвiїшин Г.Я. Мистецтво i математика //VII мiжнар. наук.-пр. конф. ”Математика. Iнформацiйнi технологiї. Освiта”: Тези доповiдей (3-5 черв-ня 2018 р., Свiтязь). — Луцьк : ПП Iванюк В.П., 2018. — C.159.

3. Маслюченко В.К., Матвiїшин Г.Я. Мистецький проект ”Я формула” //VII мiж-нар. наук.-пр. конф. ”Математика. Iнформацiйнi технологiї. Освiта” : Тези допо-вiдей (3-5 червня 2018 р., Свiтязь). — Луцьк: ПП Iванюк В.П., 2018. — C.160-161.

4. Маслюченко В.К., Маслюченко Г.-Ж.Я. Математика i Казимир Малевич //Матерiали мiжнародної наукової конференцi ”Сучаснi проблеми математики та їїзастосування в природничих науках i iнформацiйних технологiях”: Тези доповiдей(17-19 вересня 2018 р., Чернiвцi). –Чернiвцi : ЧНУ, 2018. – с. 186-187.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

2.2. Math and Art in Chernivtsi: abstract

5. Маслюченко В.К., Маслюченко Г.-Ж. Я., Математика i живопис // Нелiнiйнiпроблеми аналiзу: VI Всеукраїнська мат. конф. iм. Б.В. Василишина: Тези допо-вiдей, (26-28 вересня 2018 р., Iвано-Франкiвськ – Микуличин). – Iвано-Франкiвськ:Голiней, 2018. – С. 35.

6. Маслюченко В., Маслюченко Г.-Ж. Фрактали в мистецтвi // Всеукр. наук.конф. ”Сучаснi проблеми теорiй ймовiрностей та математичного аналiзу” : Те-зи доповiдей (25 лютого-1 березня 2019 р., Ворохта). – Iвано-Франкiвськ: ДВНЗ”Прикарпатський нацiональний унiверситет iменi Василя Стефаника”, 2019. – С.5-6.

7. Маслюченко В.К., Маслюченко Г.-Ж. Я. Супрематизм i математика // VIIIмiжнар. наук.-пр. конф. ”Математика. Iнформацiйнi технологiї. Освiта” : Тези до-повiдей (2-4 червня 2019 р., Свiтязь). – Луцьк, 2019. – С. 102.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

3. Math and Art in Chernivtsi: abstract

8. Маслюченко В.К., Маслюченко Г.-Ж. Я. Математика в мистецтвi: iсторiя i су-часнiсть // Прикарпатський вiсник НТШ. Число. – 2018. – 1(45). – C.230-234.

9. Маслюченко В.К., Маслюченко Г.-Ж. Я. Про вплив математики на мистецтво// Вiсник Львiв. ун-ту. Серiя мех.-мат. – 2018. – 86. – C. 39-44.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

4. Math and Art in Chernivtsi: accepted abstracts

а). Маслюченко В.К., Маслюченко Г.-Ж. Я. Математичнi мотиви у творах суча-сних українських художникiв // Мiжнар. наук. конф., присвячена 100-рiччю вiднародження В.К. Дзядика, Свiтязь, 20-25 червня 2019 р., тези доповiдей (у друцi).

б). Маслюченко В.К., Маслюченко Г.-Ж.Я. Топологiя i мистецтво // Мiжнар. на-ук. конф., присвячена 70-рiччю вiд народження А.М. Плiчка, Львiв, 26-30 червня2019 р., тези доповiдей (у друцi).

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

5. The main directions of applications of Math inpainting

In this talk, we speak of Math and painting. We know the following directions ofapplications of Math in this branch of Art:a). Doctrine on perspective;b). Golden section;c). Mathematical motives in the art of German painter Albrecht Durer, in particular,in the painting “Melancholy” and work “A manual to measuring by compass and ruler”(1625);d). Mathematical ideas in creation of Holland painter M.Esher;e). Aesthetics of mathematical formulas in the artistic project “I am a formula” of aBukovinian sculptor Sviatoslav Virsta;f). The “Black square” of K.Malevicz and supermatism;g). Applications of fractals.h). Topology and painting (A. Fomenko).i). Mathematical motives in works of modern Ukrainian painters.

Here we deal with directions g), h).

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

6. Fractals in Math

Although the first examples of fractals (Cantor’s set, Kokh’s snowflake, Sierpinski’scarpet and sponge) have appeared in Math to the end of XIX and at the beginningof XX, and the term “fractal” itself appeared not so long ago; it was introduced byB. Mandelbrot [21, p. 5] in 1975. He considered fractals in a wide meaning, for whichthe topological dimension dim(E) differs from the Hausdorff-Besikovich domensionα0(E) [10, p. 64], as well as in a narrow meaning, for which α0(E) is not an integer[10, c.65].

10. Турбин А.Ф., Працевитий Н.В. Фрактальные множества, функции, распреде-ления. – К.: Наукова думка, (1992). – 208 с.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

7. B. Mandelbrot.

The papers by B. Mandelbrot [11, 12] are fundamental in this branch of Math. Wiki-pedia provides the notion of a fractal as a self-similar set [10, p. 67].

11. Mandelbrot B.B Fractals: Form, Chance and Dimension. – San Francisko Freeman,1977. – 346 p. (англiйський переклад з французького оригiналу виданого у 1975роцi)

12. Mandelbrot B.B The fractal geometry of natur. – New Yorc: Freeman and Co.,1983. – 540 p.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

8. Places where fractals appear

Nowadays fractals are widely used in science, in particular, in Physics and Biology(see [10] for citations). Moreover, fractals have aesthetic appeal (see [13] for illustratedinformation). So, no wonder that fractals have applications in art.

13. Peitgen H. O., Richter P. H. The beauty of fractals // Berlin etc.: Springer, 1986/– 199 p. (рос. переклад Пайтген Х. О. Рихтер П. Красота фракталов. – М.: Мир,1993. –173 С.)

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

9. Fractals in abstract art and design: Shlyk V.A.

A Belorussian scientist V.A. Shlyk [14] studies fractals in abstract art and design. Herethe notion of a fractal he interprets quite widely as a non-regular geometric figure.There are connections between fractals and works of painters-abstractionists such asF. Kupka (1871-1957), V. Kandynski (1866-1944), P. Mondrian (1872-1944) and others,and a master of graphical design, a German painter A. Shtankowski (1906-1998). Thetext is accompanied with corresponding illustrations. It is interesting to note thatfractals appeared in works of mentioned painters much earlier before B. Mandelbrot,who thought that “fractal forms are interiorly and genetically inherent to the nature”[14, c.243], and so no wonder that they find their reflection in works of art.

14. Шлык В. А. Фракталы в абстрактном исскустве и дизайне // Известия Челя-бинского научного центра, – 2004. – вып 1 (22) – С. 231–244.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.1. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.2. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.3. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.4. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.5. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.6. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.7. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.8. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.9. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.10. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.11. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.13. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

10.13. Illustrations from the paper by V.A. Shlyk

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

11. V. Kandynski and A. Zhytaru

V. Kandynski has influenced to the creation of a modern Bukovinian painterA. Zhytaru. Recently an exhibition “Good day, Kandynski” of paining by A. Zhytaruwas held in the Artist Museum in Chernivtsi.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.1. The techniques “Fractal” of O. Chornohus.

Fractals of another kind are used in tapestry by a Bukovinian artist from Vashki-vtsi O. Chornohus. She composes her beautiful works from fragments which she callsfractals, as well as the discovered by her techniques. Notice that her painting ontapestries are close to fractals as self-similar figures due to repetition of some elements.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.2. The techniques “Fractal” of O. Chornohus.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.3. The techniques “Fractal” of O. Chornohus.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.4. The techniques “Fractal” of O. Chornohus.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.5. The techniques “Fractal” of O. Chornohus.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

12.6. The techniques “Fractal” of O. Chornohus.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

13. A. Fomenko

A. Fomenko (was born at March 13, 1945, Stalino, USSR, SSSR) is a Russianmathematician of Ukrainian origin, an academician of RAN, a State Prize winner ofRussian Federation in Math (1996).

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

13.1. A. Fomenko

Main directions of scientific investigations:– Variation methods in differential geometry and topology, the theory of minimalsurfaces and Plato’s problem, harmonic mappings.– Integration of Hamiltonian systems of differential equations. Integral equations ongroups and algebras Lee, in mathematical physics. The theory of invariants of differenti-al equations. Creation of a new theory of topological classification of integral dynamicsystems.– Computer geometry, algorithm methods in topology.– Computers in three-dimensional topology and geometry.– Empirical-statistical methods for investigation of historical texts. Recognizanceproblem of depending historical texts, new statistical methods of dating. Additionsto the chronology of ancient and medieval history.The author of more than 200 publications.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

14. Graphic arts by Fomenko

15. Фоменко А.Т., Фукс Д.Б. Курс гомотопической топологии – М. : Наука, 1989.– 494 с.

16. Fomenko A.T.Mathematical impressions. American Math. Society, USA, 1990.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

15. Two-dimensional sphere

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

16. Moebius sheets and projective planes

Spheres with three identified points

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

17. An example

A locally compact Hausdorff space, which is not locally homologically connected (inthe sense of Czech) in dimension 1. This means that every open neighborhood of anendpoint (in the foreground of the picture) has non-trivial group of one-dimensionalhomologies.

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

18. A fractal with arm

An analogue of Sierpinski’s carpet with replacing of squares with discs

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

19. Wonderful numbers π and e

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART

V.K. Maslyuchenko, G.-J.Ya. Maslyuchenko TOPOLOGY AND ART