Got a 4x4 area, your arrangement fits 9, while triangles fit 8. Got an 8x8 area, squares fits 25, triangles fits 24. Got 12x12? squares 49, triangles 50. 16x16? squares 81, triangles 91 (I think).

Each time you add more people past 9 or 8, you have to position them far enough away from everybody else. As the number of people increases, triangles take less area to achieve adequate distance for each person.