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Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions

James R. King
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Stephen Walk
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Inspired by a Common Core State Standard referring to “definition of congruence in terms of rigid motions” and the observation that such a definition is absent from most geometry textbooks, James King gives rigid motions center stage in Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions. The “reflection axiom,” stating that for every line in the plane, there exists some nonidentity rigid motion that fixes all points of the line, is part of King’s foundation, adopted in place of SAS or another congruence principle.
The other assumptions are an incidence axiom, a plane-separation axiom, two axioms on distance and angle measure based on Birkhoff’s postulates, and an axiom on properties of dilations (which serves in place of Euclid’s fifth postulate). From this beginning, the author carefully proves the usual theorems about angles, triangle congruence, circles, parallel lines, and similarity. The other types of rigid motions are defined as, not merely shown to be, compositions of reflections, and their properties are derived from there.
Proofs throughout the book lean heavily on the properties of rigid motions; the effect on their tone, compared to corresponding arguments in other sources, is subtle. Many of them seem not only more direct, no doubt a result of the care and time taken in establishing preliminary results, but also more dynamic, as the reader imagines the rigid motion as an action rather than simply observing a static congruence relationship between two objects. It would be interesting enough to see the usual textbook theorems derived with this viewpoint and this set of axioms, but most chapters include encounters with less familiar gems such as Fagnano’s problem, the Marion Walters theorem, and Perigal’s dissection for the proof of the Pythagorean theorem.
Though the results are tightly argued, King keeps the tone conversational. There are frequent comments on the big picture of the logic—for instance, on the difference between “asserting/determining existence of” and merely “finding properties of,” and on the reasons we need a proper definition of “congruence” in the first place—as well as observations about applications of the ideas and physical analogies for visualization. The diagrams are in color, a helpful feature that becomes indispensable when tessellations are the objects of study. The exercises are an engaging mix of arguments and explorations, directing the reader to draw diagrams, trace common objects, cut shapes from cardboard, fold paper, align mirrors, and use dynamic geometry software (though no particular software is endorsed). Errors in the text are few and do not lead to confusion.
The next-to-last chapter is devoted to an investigation of symmetry groups. This is, as one might expect, the culmination of the previous explorations. There is a feeling that loose threads are being gathered:  Many of the geometric gems sown strategically throughout the book, fascinating for their own sake when we encountered them before, make unexpected and welcome reappearances as key players in the arguments of this chapter. But King has one more avenue to explore:  The final chapter is a survey of coordinate geometry, touching on topics such as complex numbers and barycentric coordinates, that concludes with a verification that \( \mathbb{R}^{2} \) is a model for the six axioms adopted at the beginning of the book.
The author notes in the introduction that this is not a high school textbook. It will nonetheless serve high school teachers well as a sourcebook of engaging explorations and interesting material for the classroom. It will also serve admirably as a textbook for a college course, whether the students are future teachers or mathematics students in general, or simply as a charming, thought-provoking read for anyone interested in a different approach and a variety of geometric results. 


Stephen Walk teaches mathematics at St. Cloud State University in St. Cloud, Minnesota. His teaching experience includes multiple semesters of the geometry course for mathematics teaching and non-teaching majors and of proof courses in general.