Shape complexity is a hard-to-quantify quality, mainly due to its relative nature. Biased by Euclidean thinking, circles are commonly considered as the simplest. However, their constructions as digital images are only approximations to the ideal form. Consequently, complexity orders computed in reference to circle are unstable. Unlike circles which lose their circleness in digital images, squares retain their qualities. Hence, we consider squares (hypercubes in $\mathbb Z^{n}$ ) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between $L^\infty $ norm and squares we effectively encode squareness-adapted simplification through which we obtain multi-scale complexity measure, where scale determines the level of interest to the boundary. The emergent scale above which the effect of a boundary feature (appendage) disappears is related to the ratio of the contacting width of the appendage to that of the main body. We discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity.

中文翻译：

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Zn中嵌入形状的复杂性，偏向正方形。

形状复杂度是难以量化的质量，主要是由于其相对性质。由于欧几里得思想的偏见，圈子通常被认为是最简单的。但是，它们作为数字图像的构造仅是理想形式的近似值。因此，参照圆计算出的复杂度阶数是不稳定的。不同于在数字图像中失去圆度的圆，正方形保留其品质。因此，我们考虑正方形（ $ \ mathbb Z ^ {n} $ ）是构造复杂度顺序的最简单形状。使用之间的连接 $ L ^ \ infty $ 范数和平方我们有效地编码了适应平方的简化，从而获得了多尺度复杂度度量，其中尺度决定了对边界的关注程度。超出边界特征（附件）效果消失的出现比例与附件接触宽度与主体接触宽度之比有关。我们讨论零复杂度在信息重复性和可构造性方面的含义，以及除了正方形以外的哪种形状具有零复杂度。