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1

Optical feedback. I static effects

Bałuka Paweł, Mielniczek Izabela, Druz Valeriia, Chabko Jakub, Kuzma Marian

Physics for Economy

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2023

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Vol. 5, No.1

71-85

EN

The feedbacks a common phenomenon. This paper gives examples of feedback in computer science (iterations). The primary aim of this paper is to observe the effects of optical feedback in a monitor-camera system. The static effects of optical feedback are presented in this paper. The problem of symmetry of the obtained images is discussed. It is explained how a multiple three-lens copying machine generates the Sierpinski triangle, which is a basic and very well-known fractal. In the next paper ( Part II ) the dynamic effects of optical feedback will be discussed.

2

Effect of soil consolidation on the fractality of the filtration law

Panahov Geylani, Museibli Parviz, Sultanov Babek

International Journal of Applied Mechanics and Engineering

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2023

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Vol. 28, no. 1

84--94

EN

In this paper, the effect of consolidation of the soil structure on the fractality of the fluid flow was evaluated. The equation of fractal law of flow in the porous medium under consolidation of two-phase, fully fluid-saturated soil was determined. Given all the simplifications, as well as the undoubted importance of the nature of the porous medium, which primarily determine the possible processes of both consolidation and fractal flow based on the results of the studies, we can conclude that a homogeneous porous reservoir at given parameters under the effect of groundwater pressure will expand its fractal structure.

3

Information Dimensions of Simple Four-Dimensional Flows

Hoover William Graham, Hoover Carol Griswold, Travis Karl Patrick

Computational Methods in Science and Technology

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2023

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Vol. 29, No. 1-4

21--26

EN

Baker Maps have long served as pedagogical tools for understanding chaos and fractal phase-space distributions. Recent work [1], following earlier efforts from 1997 [2], shows that the Kaplan-Yorke formula for information dimension disagrees with direct computation for some simple compressible Baker Maps. Here we extend this map work to simple continuous flows. We compare pointwise information dimensions to the Kaplan-Yorke dimension for a simple four-dimensional flow [3] controlling both ⟨p 4 ⟩ and ⟨p 2 ⟩: { q˙ = p ; ˙p = −q − ξp3 − ζp ; ˙ξ = p 4 − 3p 2 ; ˙ζ = p 2 − T }. Precisely similar sets of Gaussian points could be generated with Metropolis’ Monte-Carlo simulations of harmonic oscillators in Gibbs’ canonical ensemble with f(q) = e −q 2/2 / p (2π). Remarkably, we show that the dependence of the pointwise information dimension for the Gaussian distribution is linear in the inverse of the logarithm of the mesh spacing, ∝ 1/ ln(1/δ). The Hoover-Holian Gaussian oscillator problem [3] can be generalized [2–4] to some nonequilibrium steady-state problems by introducing a temperature-gradient parameter ϵ. In that case the temperature T varies from 1 − ϵ to 1 + ϵ : T = 1 + ϵ tanh(q) so that both conservative (ϵ = 0) and dissipative (ϵ > 0) flows result.

4

2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

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2023

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Vol. 29, No. 1-4

65--69

EN

In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e −p 2/2mkT and the corresponding potential distribution, ∝ e −Φ(q)/kT . (For convenience here we choose Boltzmann’s constant k and the particle mass m both equal to unity.) Soon William Hoover pointed out that Nosé’s approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e −ζ 2/2 / √ 2π, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Poznan Supercomputing ´ and Networking Center.

5

Fraktale wspierają inżynierów

Katunin Andrzej

Przegląd Techniczny : Gazeta Inżynierska

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2022

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nr 2

28--30

PL

W ostatnich dziesięcioleciach słowo fraktal na dobre weszło do naszego leksykonu i popkultury. Zapewne każdy słyszał o bardzo skomplikowanych wzorach geometrycznych, przyciągających wzrok misternymi kształtami oraz pełną paletą zmieniających się kolorów. Walory estetyczne fraktali niewątpliwie przykuwają uwagę nie tylko projektantów mody czy artystów, ale również osób, które nie są bezpośrednio związane ani z matematyką, ani z grafiką komputerową. Estetyka i piękno skomplikowania tych obiektów geometrycznych, których tworzenie opiera się na bardzo prostych wzorach rekurencyjnych, to nie wszystko, co one potrafią zaoferować. Przez swoją stosunkowo krótką historię fraktale zdążyły zaistnieć w wielu dziedzinach nauki i techniki, co więcej, stać się niezastąpionymi narzędziami w tych dziedzinach i uruchomić całe mnóstwo nowych możliwości.

EN

In recent decades the word fractal has entered our lexicon and pop culture for good. Everyone has probably heard about the highly complicated geometric patterns that catch the eye with their masterful shapes and a full palette of changing colours. The aesthetic values of fractals undoubtedly attract the attention not only of fashion designers and artists, but also of people who are not directly involved in either mathematics or computer graphics. The aesthetics and beauty of the complexity of these geometric objects, whose creation is based on very simple recursive patterns, is not all they can offer. Throughout their relatively short history, fractals have managed to appear in many fields of science and technology and, what is more, to become indispensable tools in these fields and to set in motion a whole host of new possibilities.

6

The Simplest Viscous Flow

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 3

109--119

EN

We illustrate an atomistic periodic two-dimensional stationary shear flow, ux = h x˙ i = ˙y, using the simplest possible example, the periodic shear of just two particles! We use a short-ranged “realistic” pair potential, φ(r < 2) = = (2 − r) 6 − 2(2 − r) 3 . Many body simulations with it are capable of modelling the gas, liquid, and solid states of matter. A useful mechanics generating steady shear follows from a special (“Kewpie-Doll” ∼ “qp-Doll”) Hamiltonian based on the Hamiltonian coordinates {q} and momenta {p} : H(q, p) ≡ K(p) + Φ(q) + ˙ Pqp. Choosing qp → ypx the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate (dux/dy) = ˙. The occasional x coordinate jumps associated with periodic boundary crossings in the y direction provide a Hamiltonian that is a piecewise-continuous function of time. A time-periodic isothermal steady state results when the Hamiltonian motion equations are augmented with a continuously variable thermostat generalizing Shuichi Nosé’s revolutionary ideas from 1984. The resulting distributions of coordinates and momenta are interesting multifractals, with surprising irreversible consequences from strictly time-reversible motion equations.

7

$1000 SNOOK PRIZES FOR 2021 : The Information Dimensions of a Two-Dimensional Baker Map

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 2

93--94

EN

The fractal information dimension can be computed in three ways: (1) mapping points, (2) mapping regions (two-dimensional areas here), and (3) applying the Kaplan-Yorke conjecture. For the simplest nonequilibrium Baker N2 Map these three approaches can give different results. A pedagogical exploration and explanation of this situation is the 2021 Ian Snook Prize Problem.

8

Monitoring and Biochemical Treatment of Wastewater

Dychko Alina, Remez Natalya, Kyselov Volodymyr, Kraychuk Serhii, Ostapchuk Nataliia, Kniazevych Anna

Journal of Ecological Engineering

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2020

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Vol. 21, nr 4

150--159

EN

The present paper provides the methodology for the environmental monitoring of natural and engineering wastewater systems, which involves the determination of the dichotomous fractal structure of the measuring network, the boundaries of the range and the density of pollution on the Peano and Koch curves, based on the data of the measuring network and the corresponding interpolation and smoothing algorithms, as well as determination of the dynamics of the pollution range using the Bayesian theorem. On the basis of the theory of fractals and the theory of sets, the developed algorithms for monitoring allow determining the structure of the measuring network taking into account the features of the controlled range and the sets of fractal isolines of any configuration with a given accuracy of reflection, which allows predicting the change in the composition of the effluent that comes to the reservoirs from the landscapes and improving the functioning of the equipment and environmental safety of water in general. The established dependence of biogas productivity on the different methods for destruction of the active sludge microorganisms allows determining that the maximum output of biogas occurs when applying the chemical destruction of part of the sludge.

9

Adaptation of models from determined chaos theory to short-term power forecasts for wind farms

Popławski T., Szeląg P., Bartnik R.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2020

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Vol. 68, nr 6

1491--1501

EN

The paper proposes an adaptation of mathematical models derived from the theory of deterministic chaos to short-term power forecasts of wind turbines. The operation of wind power plants and the generated power depend mainly on the wind speed at a given location. It is a stochastic process dependent on many factors and very difficult to predict. Classical forecasting models are often unable to find the existing relationships between the factors influencing wind power output. Therefore, we decided to refer to fractal geometry. Two models based on self-similar processes (M-CO) and (M-COP) and the (M-HUR) model were built. The accuracy of these models was compared with other short-term forecasting models. The modified model of power curve adjusted to local conditions (M-PC) and Canonical Distribution of the Vector of Random Variables Model (CDVRM). Examples of applications confirm the valuable properties of the proposed approaches.

10

2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 4

153--159

EN

The $1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring pairs of relatively simple, but fractal, models of nonequilibrium systems, dissipative time-reversible Baker Maps and their equivalent stochastic random walks. Two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker maps are equivalent to stochastic one-dimensional random walks. Three distinct estimates for the information dimension, {0.7897, 0.7415, 0.7337} have all been put forward for one such model. So far there is no cogent explanation for the differences among these estimates. We describe the three routes to the information dimension, DI : 1) iterated Cantor-like mappings, 2) mesh-based analyses of single-point iterations, and 3) the Kaplan-Yorke Lyapunov dimension, thought by many to be exact for these models. We encourage colleagues to address this Prize Problem by suggesting, testing, and analyzing mechanisms underlying these differing results.

11

A method of suppressing chaos in a chemical reactor and the use of chaos for identifying the reactor model

Berezowski Marek

Chemical and Process Engineering

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2019

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Vol. 40, nr 3

265–-272

EN

A method of suppressing chaotic oscillations in a tubular reactor with mass recycle is discussed. The method involves intervention in the temperature of the input flow by the recirculation flow and the temperature set from the exterior. The most advantageous solution was proved to be heat coupling elimination and maintenance of the reactor input temperature on the set level. Moreover, the reactor modelwas identified on the basis of a chaotic solution, as it provides the biggest entropy of information.

12

Development of QoS Methods in the Information Networks with Fractal Traffic

Daradkeh Y. I., Kirichenko L., Radivilova T.

International Journal of Electronics and Telecommunications

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2018

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Vol. 64, No. 1

27--32

EN

The paper discusses actual task of ensuring the quality of services in information networks with fractal traffic. The generalized approach to traffic management and quality of service based on the account of multifractal properties of the network traffic is proposed. To describe the multifractal traffic properties, it is proposed to use the Hurst exponent, the range of generalized Hurst exponent and coefficient of variation. Methods of preventing of network overload in communication node, routing cost calculation and load balancing, which based on fractal properties of traffic are presented. The results of simulation have shown that the joint use of the proposed methods can significantly improve the quality of service network.

13

Analysis of forming tread wheel sets

Ivanov I., Evseev D., Ovsyanikova I., Tarapanov A.

Transport Problems

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2017

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T. 12, z. 3

35--42

EN

This paper shows the results of a theoretical study of profile high-speed grinding (PHSG) for forming tread wheel sets during repair instead of turning and mold-milling. Significant disadvantages of these methods are low capacity to adapt to the tool and inhomogeneous structure of the wheel material. This leads to understated treatment regimens and difficulties in recovering wheel sets with thermal and mechanical defects. This study carried out modeling and analysis of emerging cutting forces. Proposed algorithms describe the random occurrence of the components of the cutting forces in the restoration profile of wheel sets with an inhomogeneous structure of the material. To identify the statistical features of randomly generated structures fractal dimension and the method of random additions were used. The multifractal spectrum formed is decomposed into monofractals by wavelet transform. The proposed method allows you to create the preconditions for controlling the parameters of the treatment process.

14

Eight city types and their interactions: the “eight-fold” model

Salingaros N. A.

Technical Transactions

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2017

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Vol. 2(114)

5--70

EN

A model for understanding the city in terms of eight characteristic “city types” is proposed. The most human cities consist of adaptive city types that act in a congruent manner. Any city can be analyzed as a particular mixture of these eight types. Competing city types combine and interact in different ways, and users feel the result as an essential quality of the environment. Some types either add to, or cancel and destroy each other, whereas others can juxtapose without interacting. This eight-fold model of city types helps us to predict the success or failure of distinct urban regions in promoting urban life. It also suggests how to repair declining or non-existent pedestrian activity, and how architectural projects could affect the city adversely or positively. One section of this paper is devoted to techniques for designing urban spaces that invite human engagement, and another to designing a campus.

PL

Zaproponowano model rozumienia miasta w kategoriach ośmiu charakterystycznych „typów miast”. Najbardziej uczłowieczone miasta składają się z adaptacyjnych typów, które funkcjonują w sposób spójny. Każde miasto można traktować jako szczególną mieszankę tych ośmiu typów. Konkurujące typy miast łączą się ze sobą i współdziałają na różne sposoby, a rezultat tych interakcji jest odbierany przez użytkowników jako integralna cecha otoczenia. Niektóre typy uzupełniają się nawzajem lub wzajemnie się znoszą bądź niszczą. Z kolei inne typy funkcjonują obok siebie, bez wchodzenia w interakcje. Ośmioaspektowy model typów miast pozwala przewidzieć sukces bądź niepowodzenie poszczególnych regionów miejskich w promowaniu życia miejskiego. Stanowi on również wskazówkę co do tego, jak można odbudować malejący lub nieistniejący ruch pieszych oraz czy dane projekty architektoniczne będą miały negatywny czy też pozytywny wpływ na miasto. Jedna z części tego artykułu jest poświęcona technikom projektowania przestrzeni miejskich zachęcających użytkowników do interakcji, a druga dotyczy projektowania kampusu.

15

Areal field and fractal based characterization of hard surfaces produced by different machining operations

Zak K.

Journal of Machine Engineering

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2016

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Vol. 16, No. 1

24--32

EN

This literature survey highlights the possible influences of surface roughness parameters on functional properties of surfaces produced by different finishing operations. The prediction of such functional properties as fatigue, sealing capacity, adhesion, friction, wear and corrosion resistance based on five groups of spatial (S) roughness parameters is overviewed. In contrast, traditional approach based on 2D roughness parameters is provided. Some real 3D surface topographies produced with desired functional properties by finishing cutting and abrasive operations are characterized. This survey confirms the vital role of machined surfaces in the functionality of machine components.

16

On a visualization of the convergence of the boundary of generalized Mandelbrot set to (n-1)-sphere

Katunin A., Fedio K.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 1

63--69

EN

In this article we analyze the generalized Mandelbrot set in higher-order hypercomplex number spaces following both the Cayley-Dickson construction algebraic spaces and the spaces defined by Clifford algebras. The particular case of the generalized 3D quasi-Mandelbrot set was also considered. In particular, we investigated the increase of power of the iterated variable and proved that when this power tends to infinity, the Mandelbrot set is convergent to the unit circle. The same is true for the generalized Mandelbrot sets in higher-dimensional hypercomplex number spaces, i.e. when the power of iterated variable tends to infinity, the generalized Mandelbrot set is convergent to the unit (n-1)- sphere. The results of our investigation were visualized for the generalized Mandelbrot set in a complex number space and the generalized quasi-Mandelbrot set in a 3D Euclidean space.

17

Fractal geometries in lateral flux capacitor design – experimental results

Kocanda P., Kos A, Gołda A.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2015

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nr 2

6--9

EN

Capacitance density is increased when lateral flux structures are used in CMOS technologies compared to classic parallel-palate capacitors. Lateral-flux capacitors where designed based on three different fractal geometries. Capacitors are designed with and without special MMC metal layer available in some CMOS technologies for capacitor design. For theoretical analysis verification a special ASIC has been designed and fabricated in UMC 0.18um technology. Presented result are obtained by measurement of 5 ICs. Some capacitor structures have much higher capacitance density than classic parallel-plates capacitor without MMC layer. Few presented structures have higher capacitance density than parallel-plate capacitor made with MMC layer. Capacitors have small process parameters spread.

PL

W porównaniu do klasycznych kondensatorów z równoległymi okładkami użycie struktur z poprzeczną pojemnością pozwala na zwiększenie gęstości pojemności przy projektowaniu kondensatorów w technologiach CMOS. Kondensatory z poprzeczną pojemnością zostały zaprojektowane na bazie trzech rożnych fraktali. Struktury kondensatorów zostały zaprojektowane z i bez użycia specjalnej warstwy metalu MMC, dostępnej w niektórych technologiach CMOS, do projektowania kondensatorów. Do sprawdzenia teoretycznych rozważań specjalny układ ASIC został zaprojektowany i wykonany w technologii UMC 0.18um. Przedstawione wyniki są efektem pomiarów 5 układów scalonych. Niektóre struktury kondensatorów mają dużo większą gęstość pojemności niż klasyczne kondensatory bez warstwy MMC. Niewiele zaprojektowanych struktur ma większą gęstość pojemności niż kondensatory klasyczne z warstwą MMC. Rozrzut parametrów kondensatorów jest niewielki.

18

From multiplication to biomimetics – playing with structure and architectural games or an actual attempt to shape objects in the 21st century

Berbesz A. M.

Czasopismo Techniczne. Architektura

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2015

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Y. 112, iss. 9-A

39--45

EN

Nature is a perfectly functioning system. Biomimetics (biomimicry) is based on analysis and transposition of processes occurring in nature to the level of technology. Architecture is a search of forms which are complementary to functional and structural solutions. There is a noticeable play with geometry, a play of shapes, and space for multiplication between these processes.

PL

Natura stanowi perfekcyjnie funkcjonujący system. Nurt biomimetyki (biomimikry) opiera się na analizie i przetransponowaniu procesów zachodzących w naturze na płaszczyznę technologii. Architektura to poszukiwanie form będących dopełnieniem rozwiązań funkcjonalnych i konstrukcyjnych. Pomiędzy tymi procesami zauważalna jest zabawa w geometrię, gra brył oraz przestrzeń multiplikacji. Architektura może stać się płaszczyzną do swoistej gry i zabawy wychodzących znacznie poza swoje podstawowe pojęcia.

19

Fractal structures and self-similar forms in the artwork of Salvador Dalí

Sławomirski M. R.

Prace Instytutu Mechaniki Górotworu PAN

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2013

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Vol. 15, no. 3-4

131--146

EN

In the paper the author discusses fractal and self-similar forms encountered in various historic and modern branches of art. The idea of fractal forms and their properties have briefly been discussed. A number of ‘non-typical’ sets (Cantor set, Fig. 2, two- and three-dimensional Cantor dust, Figs. 3, 4, Sierpiński triangle, Fig. 10), semi-natural shapes (Barnsley fern, Fig. 1, lion male mane, Fig. 6), and natural shapes (silver fi r, Fig. 5) have been presented as typical fractals shapes. The problem of the definition of fractal dimesion has been explained by means of the coastline length paradox. The intuitive idea of the Hausdorff dimension has been explained by means of covering of Britain coastline by the set of balls. Basing on the concept of the metric space understood as the pair consisting of the set and the metic attached to it, the exact definitions of the exterior measure, s-dimensional Hausdorff measure and Hausdorff dimension have been presented. The notion of the similarity dimension D has also been introduced in order to show a simple technique of determination of the fractal dimension. The author discusses fractal and self-similar shapes encountered in various branches of European, Islamic and Far East art. Referring to middle century art, the tracery in the window of the Cathedral of Milan (Fig. 11), and the façade of the Church of the Trinity, Vendôme, France (Fig. 12) have been discussed as examples of fractal and self-similar forms in the flamboyant Gothic. The interiors of Alhambra (Fig. 13) and Taj Mahal (Fig. 14) are depicted as examples of fractal forms in the islamic art. The Great Wave off Kanagawa painted by Katsushika Hokusai about 1830 and shown in Fig. 15 is the example of fractal structures present in the Japanese art. Fractal, self-similar and curvilinear motives typical for Art Nouveau are discussed and presented in sect. 3 (Figs. 16-18 and 104). The author presents the detailed analysis of two artworks of Salvador Dalí: The Persistence of Memory and The Burning Giraffe. The similarity dimension of the fragmented table contained in The Disintegration of The Persistence of Memory is evaluated. The discussion of a number of self-similar motives in The Burning Giraffe has also been presented. The purpose of the paper is to demonstrate that fractal and self-similar motives may be encountered in various branches of European and non-European art developed in various historic periods.

PL

W artykule autor omawia formy fraktalne i samopodobne spotykane w rozmaitych kierunkach sztuki, zarówno historycznych jak i współczesnych. Krótko przedstawiona została idea fraktali i form fraktalnych jako nietypowych tworów geometrycznych o specjalnych własnościach matematycznych. Jako przykłady form fraktalnych pokazano jedno- dwu- i trój-wymiarowe zbiory Cantora (Fig. 2, 3, 4), trójkąt Sierpińskiego (Fig. 10), kształty pół-naturalne, jak paproć Barnsleya (Fig. 1), lwią grzywę (Fig. 6), a także formy naturalne jak jodłę karpacką (Fig. 5). Trudności ze zdefi niowaniem wymiaru fraktala w tradycyjnym sensie pokazano na przykładzie paradoksu pomiaru długości wybrzeża. Ideę wymiaru Hausdorffa przedstawiono w sposób intuicyjny poprzez pokrycie wybrzeża Wielkiej Brytanii zbiorem kul. Wychodząc z koncepcji przestrzeni metrycznej rozumianej jako para złożona ze zbioru i rozciągniętej na nim metryki rozumianej w sensie topologicznym, podano ścisłe defi nicje miary zewnętrznej, miary Hausdorffa, wymiaru Hausdorffa i wymiaru faktalnego. Zdefi niowano również wymiar podobieństwa (niemal zawsze równy wymiarowi Hausdorffa) jako prostą metodę wyznaczania wymiaru fraktalnego. Po przedstawieniu pojęć wstępnych autor omawia formy fraktalne i kształty samopodobne spotykane w rozmaitych kierunkach sztuki europejskiej i pozaeuropejskiej. Odnosząc się do architektury średniowiecza jako przykłady form fraktalnych występujących w gotyku płomienistym przedstawiono maswerk w oknie Katedry w Mediolanie (Fig. 11) oraz fasadę Kościoła pod wezwaniem Św. Trójcy w Vendôme we Francji (Fig. 12). Fragmenty wnętrza Alhambry w Grenadzie (Fig. 13) oraz mauzoleum Taj Mahal w Agrze (Fig. 14) pokazano jako przykłady fraktalnych form w sztuce islamu. Wielka Fala w Kaganawie namalowana przez japońskiego artystę Katsushika Hokusai około 1830 roku i pokazana na Fig. 15 stanowi przykład form fraktalnych stosowanych w sztuce Dalekiego Wschodu. Fraktalne i samopodobne kształty wbudowane w systemy wymyślnie powikłanych linii krzywych typowych dla sztuki secesji omówiono w rozdziale 3, uzupełnionym Fig. 16-18 oraz Fig. 104. W następnych dwóch rozdziałach autor analizuje szczegółowo dwa słynne dzieła Salvadora Dalí: Trwałość Pamięci oraz Płonącą Żyrafę. Oszacowano wymiar fraktalny rozbitego w pył Cantora stołu występującego w obrazie Rozkład Trwałości Pamięci pochodzącego z 1952 roku i stanowiącego zmodyfi kowaną wersję Trwałości Pamięci z 1931 roku. Omówiono elementy fraktalne i samopodobne występujące w obrazie Płonąca Żyrafa. Podano też interpretację zawartości obydwu obrazów. Zasadniczym celem artykułu było pokazanie, że motywy fraktalne i samopodbne występowały i występują nadal w rozmaitych kierunkach sztuki zarówno europejskiej jak i pozaeuropejskiej odnoszących się do różnych epok historycznych.

20

Fractal dimensions of cave for exemplary gypsum cave-mazes of Western Ukraine

Andreychouk V., Błachowicz T., Domino K.

Landform Analysis

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2013

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Vol. 22

3--8

EN

Gypsum labyrinthine caves are characterized by a complex spatial structure, which can be treated as fractals and can be studied using appropriate mathematical tools. Capacitance and correlation fractal dimensions of largest gypsum caves of the Western Ukraine (as well as the World’s largest ones) were calculated. The results were used to predict findings of new, undiscovered cave mazes parts.