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  "body": "\n## Summary\n\nA box spline is a piecewise polynomial function of several variables obtained by projecting a unit cube of higher dimension onto $\\R^d$.\nIts value at a point is the volume of the slice of the cube that lands at that point.\nBox splines generalize the uniform [[B-splines|62AFX32JGT]] to several variables, and their integer translates give smooth spline spaces on regular grids and triangulations.\n\n## Explanations\n\n### L1 human\n\nShine light through a cube onto a lower dimensional screen and record how thick the cube is above each point of the screen.\nThat thickness profile is a box spline.\nMapping the square $[0,1]^2$ to the line by $(u_1, u_2) \\mapsto u_1 + u_2$ gives a tent, the hat function on $[0, 2]$.\nMore directions give smoother bumps, and in the plane one gets smooth bumps adapted to grids of squares or triangles.\n",
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