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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In geometry, the Beckman Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that for a Euclidean space E of dimension d at least 2, any mapping f from E to itself that preserves the property of being at a unit distance apart must be an isometry. The same problem has been considered for fields other than the real numbers, where (as happens for the rational numbers) the validity of the same result may depend on the dimension.