Abstract A Central Configuration (CC) is a special arrangement of masses in the n -body problem where the gravitational force on each body points proportionally toward the center of mass. A stacked CC is a CC that has a proper subset of the n bodies also forming a CC. In this paper, six bodies are located on two equilateral triangles Δ 123 and Δ 456 . Assume that both triangles are symmetrical about the line connecting m 3 and m 4 . Within these configurations, the six body configuration is not a central configuration if the triangle Δ 123 is above or below the triangle Δ 456 . It is also not a central configuration if more than two of the six bodies are collinear. When the two equilateral triangle configurations have a common centroid, masses on each equilateral triangles must be same respectively and the configuration can form a central configuration only if the ratio of the lengths of the sides between Δ 123 and Δ 456 falls into one of five intervals. Moreover there are some numerical evidences that, first there are exactly two nested central configurations but there may be one, two, or three twisted nested CCs for a given mass ratio; and second, there exists central configurations other than same centroid.

中文翻译：

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平面 6 体问题中的中心构型形成两个等边三角形

摘要 中心配置 (CC) 是 n 体问题中的一种特殊质量排列，其中每个物体上的重力按比例指向质心。堆叠的 CC 是具有 n 个实体的适当子集的 CC，也形成了一个 CC。在本文中，六个物体位于两个等边三角形Δ 123 和Δ 456 上。假设两个三角形关于连接 m 3 和 m 4 的线对称。在这些配置中，如果三角形Δ 123 在三角形Δ 456 之上或之下，则六体配置不是中心配置。如果六个物体中有两个以上共线，它也不是中心配置。当两个等边三角形配置有一个共同的质心时，每个等边三角形上的质量必须分别相同，只有当Δ 123 和Δ 456 之间的边长之比落入五个间隔之一时，该配置才能形成中心配置。此外，有一些数字证据表明，首先，恰好有两个嵌套的中央配置，但对于给定的质量比，可能有一个、两个或三个扭曲的嵌套 CC；其次，除了相同的质心之外，还存在中心配置。