Garra, John Clark
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My original purpose for this paper was to establish rigorously that there are seventeen and only seventeen crystallographic groups in the plane. I had hoped that by using the elegant sorting procedure outlined by Heinrich W. Guggenheimer in Plane Geometry and its Groups it would be possible for me to rigorously determine the seventeen. However, as I neared completion of this task I discovered that the theorem by Guggenheimer which provides a basis for the enumeration is false. An alternate theorem sufficient to validate Guggenheimer's procedure has been formulated but not proved~ Its proof would no doubt be difficult, and non-rigorous determinations are available elsewhere. Consequently in this paper I will discuss the concepts necessary for the determination and understanding of crystallographic groups, describe the determination procedure Guggenheimer was aiming at, and exhibit my counterexample to his theorem.