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  "body": "\n## Summary\n\nThe barycentric subdivision of a [[simplicial complex|V8GZ0W1ZZ5]] cuts every simplex into smaller simplices whose vertices are barycentres of faces.\nIt does not change the underlying space, but it makes the simplices smaller: repeated subdivision makes their diameters tend to $0$.\nThis is the key step in the simplicial approximation theorem and in the proof of excision for singular homology.\n\n## Explanations\n\n### L1 human\n\nTake a triangle and mark the midpoint of each edge and the centre of the triangle.\nJoin the centre to the three corners and the three midpoints.\nThe triangle is now cut into six smaller triangles, and each small triangle has one corner, one midpoint and the centre as its vertices.\nAn interval is cut at its midpoint into two halves in the same way.\n",
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