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  "body": "\n## Example\n\nStart with four [[circles|060W5CBZP6]], one enclosing the other three, that are pairwise tangent at six distinct points.\nThe three inner circles and the outer circle bound four curvilinear triangles.\nInto each such curvilinear triangle, insert the unique circle inside it tangent to its three sides.\nThis creates three new curvilinear triangles.\nRepeat forever.\nThe *Apollonian gasket* is the family of all circles obtained, or the closure of their union.\nApart from the outer circle, which encloses the others, the circles have disjoint interiors, so they form a [[circle packing|S360JBEYKD]].\nFor the curvatures, take the outer circle with negative sign.\nBy Descartes' circle theorem, four pairwise tangent circles with curvatures $k_1, k_2, k_3, k_4$ satisfy $(k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2)$.\nCheck with the start $(-1, 2, 2, 3)$: an outer circle of radius $1$, two circles of radius $\\tfrac12$ and one of radius $\\tfrac13$; then $6^2 = 36 = 2(1 + 4 + 4 + 9)$.\nEach new circle replaces one curvature $k_4$ by $2(k_1 + k_2 + k_3) - k_4$, so in this gasket all curvatures are integers.\n\n## Remarks\n\nThe leftover set, the complement in the outer disc of all open discs, has area zero.\nIts [[Hausdorff dimension|SSNFTHZ6YJ]] is about $1.3057$, strictly between $1$ and $2$, so it is a fractal.\n\n## Explanations\n\n### L1 human\n\nDraw three coins that touch each other in pairs, and a large circle around them touching all three.\nFill every curved triangular gap with the largest circle that fits, then fill the smaller gaps, and keep going forever.\nThe picture becomes a lace of ever smaller circles, the Apollonian gasket.\nWith a good start, such as curvatures $-1, 2, 2, 3$, every circle in the picture has a whole-number curvature.\n",
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