I have just made a carrot box and I have difficulty in keeping the hexagonal lattice constant.
Eventually I figured out a little tool for making planting holes to the correct spacing. It's just a plank of the correct dimensions, in which I have put screws to make the planting holes. Maybe everyone does this, but I couldn't find it in my gardening encyclopedias.
Carrot box is 65 cm square, so plank is 62 cm long. Spacing for carrots according to Caltech is 2"-3" so I chose 6cm hence lattice is 9:10:9, to leave some edge space.
Some basic geometry and trigonometry shows perpendicular distance between rows =~ 0.85 times the spacing (= 5.1cm). Width of plank, therefore, had to be more than 10.2cm + 2cm to allow room to put screws along the edges of the plank with a bit to spare. Mine was a bit of tongued and grooved with 13cm plus the tongue.
You need three rows (or an odd number larger than three) to allow one row as a guide so you can put the second set of holes exactly the right distance from the first set.
Theoretically the screws should stick out to the planting depth of the seeds, but around a cm is ok for most vegetable seeds. I am clumsy so some screws are not vertical, but the tolerance doesn't need to be less than 3mm so a multitude of sins are possible.
Images show the result and it's sowing holes in the carrot box.
You must water before making the holes otherwise the holes fill themselves in.
Obviously, if you have good, fine soil, you can use it to make an equidistant planting lattice in open soil as well, and have different shaped and sized planks for veg with different planting distances. The 0.85 (=(√3)/2) multiple for the vertical spacing is invariant.
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