A Cornucopia of Quadrilaterals

 

 

Alsina and Nelson have once again produced a delightful geometric exposition comprised of historical backgrounds and clear proofs, many of which frequently use decompositions and rearrangements familiar to MAA readers from their Proofs without Words visualizations. The Cornucopia explicitly cites contemporary geometric investigations, lending it a welcome freshness.

 

Euclid explored the properties of triangles and circles almost to the omission of quadrilaterals. Yes, a limited examination is provided for simple properties of some familiar planar quadrilaterals (squares, rectangles, parallelograms, and trapezoids), but the Elements stops far short of considering the range of properties that even these figures have (or can have). Quadrilaterals have four sides, four vertices, four interior angles, two interior diagonals. The classical Euclidean treatment does not go beyond questions of pairs of interior diagonals, medians, bimedians, incircles and circumcircles, convexities, concavities, intersecting sides, or skew or complex nonplanar quadrilaterals. 

 

The Cornucopia catalogues some 27 varieties of quadrilateral families. The exposition is conducted in 11 chapters, starting with the introduction of properties of simple quadrilaterals that recur throughout the following chapters. These, the Varignon parallelogram, diagonal midpoints, the Newton line, Anne’s and Van Aubel’s theorems, and area formulas and inequalities become recurring tools in the remainder of the text. It is also here that we first see the appearance of the Pythagorean Theorem, which ends up being proven and re-proven in a variety of ways based on context. The second and third chapters, respectively, explore the properties of quadrilaterals and their circles, and quadrilaterals and their diagonals.  From here, individual classes of quadrilaterals are given their own chapters: trapezoids (three chapters), parallelograms, rectangles, squares, special quadrilaterals (concave, complex, skew, and Saccherie and Lambert). The final chapter explores properties of quadrilateral numbers (square, oblong, trapezoidal and polite) with an intriguingly unexpected discussion on sums of consecutive square numbers. The chapters end in challenge problems carefully designed to deepen the reader’s appreciation and insights.

 

The book teems with derivations of scores of calculations of areas, perimeters, ratios, and inequalities (including Cauchy-Bunyakovsky-Schwartz and Erdös-Mordell), and explorations of recreational puzzles and paradoxes (à la Henry Dudeney, Sam Loyd, Martin Gardner).  Fibonacci numbers, crossed ladders, planar dissections, pantographs all appear, as does an application of linkages to produce a straight line (sadly without giving A. B. Kempe’s proof). The book also includes interesting trivia.  For example, we learn that Congressman James A. Garfield had published a short paper on the properties of a right trapezoid having a side a, a side b and having a top fitted from an equilateral right triangle of side c. In his paper, printed shortly prior to his presidency and assassination, Garfield algebraically derives the Pythagorean Theorem.

 

 

In reading the early chapters, I found myself harkening back to Euclid’s Elements to refresh my memory of the classical triangle and circle theorems that form the basis on which many of the book’s simpler diagrams depend. Almost playfully, Alsina and Nelson produce diagrams in which rotations and translations provide insights that go far beyond the context of a derivation they discuss. It is such rewards that transformed this Cornucopia into a gift that continues to give delights to this reader.

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Marvin Schaefer served as Chief Scientist at the National Computer Security Center at NSA as part of his career in formal modeling and theory of computer operating systems. His primary interests have been number theory, algebra, and mathematical logic. In retirement he and his wife operated an antiquarian bookshop and he has continued trying to improve his skill on the Bavarian zither. He joined the MAA in 1961.