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If $G$ is a group acting on a set $S$, then the "orbit" of a point $x$ in $S$ is defined as the set of all elements of the form $gx$ where $g \in G$. My question: why was the word "orbit" chosen for this concept? It is not obviously related to previous uses of this word, such as the path of a planet around the Sun.

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    I always thought that the name "orbit" comes from the rotations of the plane as in Qiaochu Yuan's comment. Let me add that the Latin etymology leads back to "orbita", which is the eye socket (of a vaguely elliptical shape).2011-01-29

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It seems the concept goes back further than D.W. Hall (a student of Whyburn) - at least to Kuratowski, cf. the following excerpt from Kuczma et al. Iterative functional equations, p. 14. $\quad $ One can probably find further historical details by Googling "Kuratowski-Whyburn orbit" etc.

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Note: for a simple yet enlightening example of the key role that orbits play in the solution of functional equation see this post.