  We present a high-order surface quadrature (HOSQ) for accurately
approximating regular surface integrals on closed surfaces. The initial step of
our approach rests on exploiting square-squeezing--a homeomorphic bilinear
square-simplex transformation, re-parametrizing any surface triangulation to a
quadrilateral mesh. For each resulting quadrilateral domain we interpolate the
geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the
tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting
integral. We demonstrate efficiency, fast runtime performance, high-order
accuracy, and robustness for complex geometries.
