Maybe not asked, but you can print your paper more full of circular shapes if the shapes are placed in rectangular matrix. In the next image the unprintable edge of the paper is colored to red. The printable area is filled from edge to edge and the gaps between the shapes are in both directions the same as closely as possible.
In theory you could remove totally either the horizontal or the vertical gap between the shapes. I left some gap to both directions because otherwise the layout starts to look too dense.
Fully uniform look needs making the gap same in both directions, but that either forces to use too small shape diameter or one should use smaller printable area than the printer allows.
In my example the numbers are:
- paper size A4: 210 mm x 297 mm
- the size of the printable area 204 mm x 291 mm ; later it's called W x H
- the diameter of the shape: 22 mm ; later it's called D
- the horizontal gap between the shapes: 4 mm ; later it's called S
- the vertical gap between the shapes: 2.456 mm ; later it's called P
- the number of shapes in a row: 8 ; later it's called N
- the number of the rows: 12 ; later it's called M
I placed at first the top and the bottom shapes of the leftmost column on the white WxH rectangle. They snapped easily with smart guides. I inserted ten more copies, aligned all horizontally, distributed vertically in center and grouped them to make an evenly spaced column.
I made 7 more copies of the column, aligned all horizontally, moved one of the copies to the right edge of the white rectangle and finally distributed all horizontally in center to make the spacing even.
The shapes could be rotated (like in your own attempt) to create an illusion of tilted straight interleaved lines. The rotation angle should be in this case 43 degrees CCW. The formula for that rotation angle is arctan((D+P)/(D+S)), The calculated rotation makes the originally vertical details in the shape perpendicular with the apparent tilted line. You can rotate all selected items at once by applying Transform each > rotate. Beware groupings!
How I ended to the mentioned numbers? The dimensions W and H of the printable area are defined by the printer. I decided at first that the shape diameter D should be 20 mm or more. That's my guess of what looks big enough.
The next step was to guess the number of the rows and columns. 12 rows and 8 columns looked reasonable.
To get the exact gaps S and P I tried different diameters D to the next formulas:
S = (W - ND)/(N-1) and P = (H - MD)/(M-1). The formula for S is solved from the condition "N shape widths + (N-1) gaps S must be exactly W". Respectively M shape widths + (M-1) vertical gaps P must be exactly H"
I used Excel. D = 22 mm resulted to gap widths S = 4 mm and P = 2.456 mm. P is only 60% of S, but for me the result looks acceptable. Smaller D makes possible to have less different S and P; for example with D = 15 mm you get S and P which differ only 3 percents:
