Geodesic
Multiple circle coverings of an equilateral triangle, square, and circle
Diskretnyj analiz i issledovanie operacij, Tome 22 (2015) no. 6, pp. 5-28

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We study $k$-fold coverings of an equilateral triangle, square, and circle with $n$ congruent circles of the minimum possible radius $r^*_{n,k}$. We describe mathematical models for these problems and algorithms for their solving. We also prove optimality of the constructed coverings for certain $n$ and $k$, $1$. For $n\le15$ and $1$, we present the best found (possibly, improvable) values of circles radii ensuring the $k$-fold covering of the equilateral triangle, square or a circle. Ill. 4, tab. 3, bibliogr. 39.
Keywords: multiple covering with congruent circles, equilateral triangle, square, circle, minimum covering problem. 
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Sh. I. Galiev; A. V. Khorkov. Multiple circle coverings of an equilateral triangle, square, and circle. Diskretnyj analiz i issledovanie operacij, Tome 22 (2015) no. 6, pp. 5-28. http://geodesic.mathdoc.fr/item/DA_2015_22_6_a0/