This article introduces a new family of quadrature rules for integrating smooth functions on 4-dimensional simplex elements (i.e. pentatopes). These quadrature rules have 1, 5, 15, 35, 70, and 126 points, and are capable of exactly integrating polynomials of degrees 1, 2, 3, 5, 6, and 8, respectively. One main advantage of these rules, is that they have a ‘pentatopic number’ of points, which means that they can exactly interpolate 4-dimensional polynomials of degrees 0 through 5. As a result, the proposed rules can be used for both quadrature and interpolation purposes. Furthermore, these rules are fully symmetric, as they remain invariant under affine transformations (rotations and reflections) of the pentatope back to itself. In addition, these rules are optimal, in the sense that the truncation error associated with each rule has been minimized via a rigorous optimization procedure. Finally, they have positive weights, and all quadrature points reside strictly within the interior of the pentatope.

本文介绍了一个新的正交规则系列，用于在4维单形元素（即，五角形）上集成平滑函数。这些正交规则具有1、5、15、35、70和126点，并且能够分别精确地积分次数为1、2、3、5、6和8的多项式。这些规则的主要优势之一是它们具有“五点数”的点，这意味着它们可以精确地插值结果为0到5的4维多项式。因此，建议的规则可用于正交和插值目的。此外，这些规则是完全对称的，因为它们在五角形笔回到自身的仿射变换（旋转和反射）下保持不变。此外，在通过严格的优化过程将与每个规则相关的截断误差最小化的意义上，这些规则是最佳的。最后，它们具有正的权重，所有正交点都严格位于五角形内部。