Overlapping Circles Grid

Author

Eric W. Weisstein
September 1, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/PlaneGeometry/OverlappingCirclesGrid.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/OverlappingCirclesGrid.html.

Copyright 2026 Wolfram Research, Inc.

Figure

Successive hexagonal patches of an overlapping circles grid.

With[
	{
		circleCenters = Function[
			n,
			Map[
				{#[[1]] + #[[2]]/2, Sqrt[3] #[[2]]/2} &,
				Select[
					Tuples[Range[1 - n, n - 1], 2],
					Max[Abs[{#[[1]], #[[2]], Total[#]}]] <= n - 1 &
				]
			]
		],
		offsets = {0, 4, 10, 18},
		ink = LightDarkSwitched[Black, White]
	},
	Graphics[
		{
			Directive[ink, AbsoluteThickness[1.5]],
			MapThread[
				Function[
					{n, offset},
					Map[Circle[# + {offset, 0}, 1] &, circleCenters[n]]
				],
				{Range[4], offsets}
			]
		}
	]
]