A circle of radius R is the set of points in the plane that are of distance R from a given center point. A circle jerk is a set of men masturbating (themselves or each other) while facing each other, all roughly the same distance from a center point.

Since the name circle jerk certainly brings to mind an actual circle, it is fair to ask how appropriate, exactly, this name really is. We immediately run into problems: For instance, it is not hard to convince yourself that a circle contains infinitely many points, while there is no way to have infinitely many masturbators, as the human population is certainly finite. Moreover, how a circle is defined - the set of points which are all of equal distance from the center - also defines an object in the third dimension (the sphere) and, indeed, all higher dimensions (n-spheres, where n is the dimension of the ambient space; so a 2-sphere is a circle and a 3-sphere is what we normally call just a sphere)***. It is clear, at least, that the circle jerk does not generalize so easily; though a sphere is easily visualized, the "sphere jerk," if it exists, defies mental conjuration.

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The aim of the present article is to formulate a more precise definition - indeed, a family of definitions - applicable to any set of masturbating men. We will also see that these definitions generalize allow for more nuanced constructions. While the author anticipates that "circle jerk" will still see heavy usage as a colloquialism, the hope is that these definitions will open up new avenues of research into the mathematics of jacking off.

Define an n-crank to be any configuration of n men whacking off. For example, me masturbating by myself is a 1-crank; any pair of men, anywhere in the world, who are both masturbating at the same time form a 2-crank; and so on. Define a regular n-crank to be an n-crank in which all of the men are facing, and an equal distance from, a center point; this is a more precise version of the "circle jerk" definition given above.

What is the advantage of thinking in terms of cranks? For starters, we access more information; a group of five men stroking each other off in a circle is a circle jerk, to be sure, but calling it a "regular 5-crank" tells us how many men are participating. We may also begin to study just n-cranks, as opposed to regular n-cranks, to determine how more general sets of dudes beating their meat in the same room behave.

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Let us be bold and modify the definition of an n-crank to obtain a new masturbatorical object. Define an n-tug to be any set of men, all of whose penises are being stroked. Note that an n-tug may contain a man who is not himself touching any penises, but whose penis is being stroked by another man pulling double-duty. It is easy to see that the number of men in an n-tug who are not masturbating themselves or playing the five-finger fiddle with another man's instrument is at most n/2; different estimates may be obtained by forcing that an n-tug contain some number m men with only one hand.

We conclude by noting that these concepts, just like the circle jerk, have more metaphorical applications as well. For example, in the informative and entertaining conversation linked here, it is claimed that Raysism and BronzeHammer are circle-jerking; while this is true, one may also note that Raysism does not appear to be stroking BronzeHammer's broad pole in return: he does not engage with AbstrktVolcano on BronzeHammer's behalf, etc., and therefore we can say that Raysism and BronzeHammer were "2-tugging." When I chimed in, we became a 3-tug - yet one would still call it a "circle jerk," much less informatively. It is our wish that cranks and tugs will enter the common lexicon before too long.

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The author is supported by the National Science Foundation.