Moskva, Moscow, Russian Federation
It has long been known that there are fractals, which construction resolve into cutting out of elements from lines, curves or geometric shapes according to a certain law. If the fractal is completely self-similar, its dimensionality is reduced relative to the original object and usually becomes fractional. The whole fractal is often decomposing into a set of separate elements, organized in the space of corresponding dimension. German mathematician Georg Cantor was among the first to propose such fractal set in the late 19th century. Later in the early 20th century polish mathematician Vaclav Sierpinski described the Sierpinski carpet – one of the variants for the Cantor set generalization onto a two-dimensional space. At a later date the Austrian Karl Menger created a three-dimensional analogue of the Sierpinski fractal. Similar sets differ in a number of parameters from other fractals, and therefore must be considered separately. In this paper it has been proposed to call these fractals as i-fractals (from the Latin interfican – cut). The emphasis is on the three-dimensional i-fractals, created based on the Cantor and Sierpinski principles and other fractal dependencies. Mathematics of spatial fractal sets is very difficult to understand, therefore, were used computer models developed in the three-dimensional modeling software SolidWorks and COMPASS, the obtained data were processing using mathematical programs. Using fractal principles it is possible to create a large number of i-fractals’ three dimensional models therefore important research objectives include such objects’ classification development. In addition, were analyzed i-fractals’ geometry features, and proposed general principles for their creation.
Cantor sets, Sierpinski fractals, i-fractals, foams, sponges, tulles, one-cavity fractals, golf-fractals.
Вступление
В [9] представлено исследование пространственных фракталов, созданных путем «доращивания» мономеров (в литературе встречаются также термины «генератор» [13] и «фрагмент» [22]). Данная же статья посвящается фракталам, созданным путем вырезания мономеров (аналогам фракталов Серпинского [25] и Кантора [2], [15]).
1. Aleksandrov P.S., Pasynkov B.A. Vvedeniye v teoriyu razmernosti [Introduction to the theory of dimension]. Moscow, Nauka Publ., 1973. (in Russian)
2. Bolotov V. N. Obobshchennaya funktsiya Kantora i perekhodnoye fraktalnoye rasseyaniye [Generalized Cantor function and fractal transition scattering]. Zhurnal tehnicheskoj fiziki [Technical physics]. 2002, V. 72, I. 2, pp. 8-15. (in Russian)
3. Bushmanova G.V., Norden A.P. Vvedeniye v konformnuyu geometriyu [Introduction to conformal geometry]. Moscow, Nauka Publ., 1964. (in Russian)
4. Brylkin Y.V., Kusov A.L. Modelirovaniye struktury relyefa realnykh poverkhnostey na osnove fraktalov v aerodinamike razrezhennykh gazov [Modeling of the structure of the topography of real surfaces on the basis of fractals in the aerodynamics of rarefied gases]. Kosmonavtika i raketostroenie [Cosmonautics and rocket engineering]. 2014, I. 3, pp. 22-28. (in Russian)
5. Brylkin Yu.V. Fraktalnyy podkhod k opisaniyu poverkhnostey metallov i splavov [Fractal approach to the description of surfaces of metals and alloys]. Sbornik nauchnykh statey doktorantov i aspirantov moskovskogo gosudarstvennogo universiteta lesa. Nauchnyye trudy [Collection of scientific articles of doctoral students and post-graduates of the Moscow state forest University. Scientific works]. Moscow, MGUL Publ., 2013, I. 364, pp. 5-10. (in Russian)
6. Goncharov V.D., Ilyinova V.Y. Model poristosti na osnove obobshhennoj gubki Mengera [Model porosity based on generalized Menger sponge.] Fizicheskie ximicheskie i klimaticheskie factory produktivnosti polej [Physical, chemical and climatic factors of the productivity of the fields]. St. Petersburg, PNPI RAS Publ., 2007, pp. 44-47. (in Russian)
7. Weil H. Simmetriya: Per. s angl. B.V. Biryukova i Yu.A. Danilova pod redaktsiyey B.A. Rozenfelda [Symmetry: Tr. from English. B.V. Biryukov and Yu.A. Danilova, edited by B. A. Rosenfeld.] Moscow, Nauka Publ., 1968, p. 101. (in Russian)
8. Zhiharev L. A. Obobshcheniye na trekhmernoye prostranstvo fraktalov Pifagora i Kokha. Chast 1 [The Generalization to three-dimensional space of fractal Pythagoras and Koch. Part 1]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 3, pp. 24-37. DOI:https://doi.org/10.12737/14417. (in Russian)
9. Cardin A.N. Fraktalnoye modelirovaniye i neyronnyye elektricheskiye seti [Fractal modeling and neural electrical networks]. Voprosy sovremennoy nauki i praktiki. Universitet im. V.I. Vernadskogo [Questions modern science and practice. University. V.I. Veradskogo]. 2014, I. (52), pp. 65-70. (in Russian)
10. Cantor G. Trudy po teorii mnozhestv [Works on the theory of sets]. Moscow, Nauka Publ., 1985. (in Russian)
11. Carroll R.T. Britva Okkama [Occam's Razor]. E`ntsiklopediya zabluzhdeniy: sobraniye neveroyatnykh faktov. udivitelnykh otkrytiy i opasnykh poverij [The skeptic's dictionary: a collection of incredible facts, amazing discoveries and dangerous beliefs]. Moscow, "Dialectic" Publ., 2005, pp. 78-82. (in Russian)
12. Mamedov J.I, Nagiev A.G. Modelirovaniye nestatsionarnykh protsessov perenosa veshchestva i adsorbtsii v poristoy srede na osnove fraktala "gubka Mengera" [Modeling of non-stationary processes of mass transfer and adsorption in porous media based on fractal "Menger sponge"]. Izvestiya vysshikh uchebnykh zavedeniy. Khimiya i khimicheskaya tekhnologiya. [News of higher educational institutions. Chemistry and chemical engineering]. 2009, I. 10. (in Russian)
13. Morozov D.A. Vvedeniye v teoriyu fraktalov [Introduction to the theory of fractals]. Moscow, Institut kompyuternykh issledovaniy Publ., 2002, pp. 12-16. (in Russian)
14. Lisichkin G.V., Fadeev A.YU. Khimiya privitykh poverkhnostnykh soedinenij [Chemistry grafted surface compounds]. Moscow, FIZMALIT Publ., 2003. 592 p. (in Russian)
15. Ostashkov V.N. Dialogi o fraktalakh [Dialogues about fractals]. Tyumen: TSOGU Publ., 2011. 292 p. (in Russian)
16. Selivanov D.F. Telo geometricheskoye [Geometric body]. Entsiklopedicheskiy slovar Brokgauza i Efrona [Encyclopedic dictionary Brockhaus and Efron]. St. Petersburg, 1890- 1907. (in Russian)
17. Feder E. Fraktaly [Fractals]. Moscow, Mir Publ., 1991, pp. 30-34. (in Russian)
18. Schmidt F. K. Fraktaly v fizicheskoy khimii geterogennykh sistem i protsessov [Fractals in the physical chemistry of heterogeneous systems and processes]. Irkutsk, Irkutsk. Univ Publ., 2000. 147 p. (in Russian)
19. Avnir D. Nouv / D. Avnir, D. Farin // J Chim., 1992. V. 16. P. 439.
20. Cipriani F. Fredholm modules on pcf self-similar fractals and their conformal geometry / F. Cipriani, J.L. Sauvageot // Communications in mathematical physics. 2009. V. 286. |I. 2. Pp. 541-558.
21. Koch N.F.H. von. Na nepreryvnoy krivoy bez kasatelnykh. poluchennyye putem geometricheskogo postroyeniya [On a continuous curve without tangents, obtained by geometric construction] / N.F.H. von Koch. Stockholm., 1904. (in French)
22. Mandelbrot B.B. The Fractal Geometry of Nature / B.B. Mandelbrot, W.H. Freeman. - New York, 1982.
23. Menger K. Selected papers in logic and foundations, didactics and economics / K. Menger. Boston: Reidel, 1979.
24. Rogowski W. Ehlektricheskaya prochnost' v platinovykh kondensatorakh [Electric strength in platinum capacitors] // Arkhiv ehlektrotekhniki [Archive of electrical engineering] Bd., 1923. 12. S. 1-15. DOIhttps://doi.org/10.1007/BF01656573. (in German)
25. Sierpinski W. Na krivoj linii, kazhdaya tochka kotoroj yavlyaetsya tochkoj vetvleniya [On line curve whose every point is a branch point] // Zametki [Accounts]. Paris.: 1915. 160, 302. (in French)
26. Sierpinski W. Na krivoj linii, kazhdaya tochka kotoroj yavlyaetsya tochkoj vetvleniya schetakh [On a Cantorian curve which contains a continuous one-to-one image of any curve] // Zametki [Accounts]. Paris.: 1916. 162, 629. (in French)
27. Spider H. Z. fur. Clumie / H. Spider. 1988, V. 28, № 12. P. 426.
28. 3D pechat - budushcheye betona [3D printing - the future of concrete]. ProGrinding.ru. 2012. Available at: http:// progrinding.ru/2012/08/16/3d-pechat-budushhee-betona/ (in Russian)
29. 3D-pechat iz metalla nabirayet oboroty [3D printing metal is gaining momentum]. Top 3D Shop. 2016. Available at: https://geektimes.ru/company/top3dshop/blog/280098/ (in Russian)
