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For the space of $C^3$ quintics on the Powell-Sabin 12-split of a triangle,we determine explicitly the six symmetric simplex spline bases that reduce to aB-spline basis on each edge, have a positive partition of unity, a Marsdenidentity that splits into real linear factors, and an intuitive domain mesh.The bases are stable in the $L_\infty$ norm with a condition number independentof the geometry, have a well-conditioned Lagrange interpolant at the domainpoints, and a quasi-interpolant with local approximation order 6. We show an$h^2$ bound for the distance between the control points and the values of aspline at the corresponding domain points. For one of these bases we derive$C^0$, $C^1$, $C^2$ and $C^3$ conditions on the control points of two splineson adjacent macrotriangles.

Stable simplex spline bases for $C^3$ quintics on the Powell-Sabin 12-split