话说在1719，一个叫Paul Halcke的会计，也为专天文学家计算的工程师发现了三个数字，44,117, 240。如果你把这三个数任取两个，求平方和，结果仍旧是一个完全平方数。如果你把这三个数作为一个长方体的三条边，你会发现这个长方体不但所有边是整数，所有面的对角线也是整数。符合这种条件的长方体称为欧拉砖。完美立方体是啥呢：找一个欧拉砖，使体对角线也是整数！数学家至今没有找到这样的三元组，也不能证明它不存在。Saunderson的参数化欧拉砖数公式：如果u, v, w是勾股数（比如，（3，4，5），则以下公式可以产生欧拉砖数组：2009年，发现了完美平行六面体，三条边最小的长度是271, 106, and 103。6条面对角线和4条体对角线全是整数。还找到了两个面是矩形的完美平行六面体。
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