A circle packing is called rigid (or "stable") if every circle is fixed by its neighbors, i.e., no circle can be translated without disturbing other circles of the packing (e.g., Niggli 1927, Niggli 1928, Fejes Tóth 1960/61). Böröczky (1964) exhibited stable systems of congruent unit circles with density 0. A rigid packing of circles can be obtained from a hexagonal tessellation by removing the centers of a hexagonal web, then replacing each remaining circle with three equal inscribed circles (appropriately oriented), as illustrated above (Meschkowski 1966, Wells 1991).
Rigid Circle Packing
See alsoCircle Packing
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ReferencesBöröczky, K. "Über stabile Kreis- und Kugelsysteme." Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 7, 79-82, 1964.Fejes Tóth, L. "On the Stability of a Circle Packing." Ann. Univ. Sci. Budapestinensis, Sect. Math. 3-4, 63-66, 1960/1961.Meschkowski, H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966.Niggli, Z. Z. für Kristallographie 65, 391-415, 1927.Niggli, Z. Z. für Kristallographie 68, 404-466, 1928.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 30-31, 1991.
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Weisstein, Eric W. "Rigid Circle Packing." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RigidCirclePacking.html