Packing up to 50 Equal Circles in a Square

The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 [Bol04]) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of vacuitas (in Latin, the gap in the triangle outside the circles). It is interesting that these packings are not always optimal in spite of the fact that they are based on hexagonal grid packings. Bolyai probably was the first author in the mathematical literature who studied the density of a series of packing circles in a bounded shape.