## 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

Glucose; GitHub
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### References:

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