{"title":"On the length of arcs in labyrinth fractals.","authors":"Ligia L Cristea, Gunther Leobacher","doi":"10.1007/s00605-017-1056-8","DOIUrl":null,"url":null,"abstract":"<p><p>Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1056-8","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte fur Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00605-017-1056-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2017/5/15 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.
迷宫分形是由迷宫集合或迷宫图案定义的单位正方形中的自相似树突。当分形由水平方向和垂直方向的阻塞模式生成时，分形中任意两点之间的弧具有无限长(Cristea and Steinsky In Geom Dedicata 141(1):1-17, 2009;学报:自然科学版，2011(2):329-344。在混合迷宫分形的情况下，使用迷宫图案序列来构造树突。本文主要讨论混合迷宫分形的点间弧的长度。我们表明，根据序列中图案的选择，这两种情况都可能发生:分形的任意两点之间的弧具有有限长度，或者分形的任意两点之间的弧具有无限长度。这与自相似的情况形成鲜明对比。
期刊介绍：
The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler.
The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel).
The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.
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