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Minimal self-similar Peano curve of genus \(5 \times 5\). (English. Russian original) Zbl 1482.28012

Dokl. Math. 101, No. 2, 135-138 (2020); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 491, 68-72 (2020).
One can begin with the authors’ abstract:
“The paper presents a plane regular fractal Peano curve with a Euclidean square-to-line ratio (\(L_2\)-locality) of \(5 \frac{43}{73}\), which is minimal among all known curves of this class. The presented curve has a fractal genus of \(25\). Performed calculations allow us to state that all the other regular curves with a fractal genus not exceeding \(36\) have a strictly greater square-to-line ratio.”
In a brief survey of this paper, plane Peano curves (space-filling curves), the square-to-linear ratio (SLR) of the curve, and regular Peano fractal curves, as well as the fractal genus of a curve, the Hilbert curve, and regular Peano fractal curves, are considered. In addition, such notions as isometric curves, a \(k\)th-order junction, the reduced square-to-linear ratio of the junction, the reduced SLR of the curve, the basic transformation of a fraction, the prototype of a curve, the entrance of a curve, the transition broken line, etc., are described.
Presented investigations are given with explanations. Algorithms for estimating the square-to-linear ratio of a curve and the minimum SLR in the class of curves with given transition broken line, are described.
Finally, a special attention is given to some open directions in the topic of the present investigations.

MSC:

28A80 Fractals

Software:

Glucose; GitHub
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References:

[1] Shchepin, E. V., Proc. Steklov Inst. Math., 247, 272-280 (2004)
[2] Bader, M., Space-Filling Curves: An Introduction with Applications in Scientific Computing (2013), Berlin: Springer-Verlag, Berlin · Zbl 1283.68012
[3] Haverkort, H.; van Walderveen, F., Comput. Geom. Theory Appl., 43, 131-147 (2010) · Zbl 1203.65045
[4] Bauman, K. E., Math. Notes, 80, 609-620 (2006) · Zbl 1133.28001
[5] Bauman, K. E.; Shchepin, E. V., Proc. Steklov Inst. Math., 263, 236-256 (2008) · Zbl 1201.37061
[6] Bauman, K. E., Proc. Steklov Inst. Math., 275, 47-59 (2011) · Zbl 1302.28012
[7] Bauman, K. E., Discrete Math. Appl., 24, 123-128 (2014) · Zbl 1325.28006
[8] https://github.com/jura05/peano
[9] Biere, A.; Heule, M.; van Maaren, H., Handbook of Satisfiability (2009) · Zbl 1183.68568
[10] G. Audemard, J.-M. Lagniez, and L. Simon, “Improving glucose for incremental SAT solving with assumption: Application to MUS extraction,” in Proceedings of SAT (2013). www.labri.fr/perso/lsimon/glucose/ · Zbl 1390.68587
[11] https://pysathq.github.io/
[12] D. K. Shalyga, Preprint No. 088, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2014).
[13] H. Haverkort, “How many three-dimensional Hilbert curves are there?” E-print (2017). arXiv: 1610.00155v2 [cs.CG] · Zbl 1396.28012
[14] Korneev, A. A.; Shchepin, E. V., Proc. Steklov Inst. Math., 302, 217-249 (2018) · Zbl 1447.28008
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