Quadrilaterals and their properties

This lesson asks you to list the geometric properties of quadrilaterals, including diagonals and symmetry. The museum offers one record, on a quadrilateral from a book of geometry; it does not describe the diagonals or the symmetry, so that analysis is your class's work. ¹

Draw the shape the record describes. List what you can say about its sides, its angles, its diagonals and any line symmetry, and mark which of these the record states and which you found from your drawing.

Khayyam's quadrilateral is a geometrical method, not a metal object: two equal legs rise perpendicular to a base, and their upper ends are joined. The investigation asks whether the two summit angles are right, acute or obtuse. In Vahabzadeh's analysis, Khayyam first establishes that the two angles are equal, so he does not have to examine nine combinations of two independently varying angles ². He uses the figure in his Commentary on Difficulties in Euclid's Postulates, organised in eight propositions to replace the twenty-ninth proposition of the first book ^{3 2}.

Vahabzadeh's analysis reports that Khayyam objected that earlier attempts used assumptions no clearer than the postulate being explained, and that he especially rejected introducing motion into geometry, in the analysis of his criticism of Ibn al-Haytham. He wanted a starting point grounded in the concepts of line and angle, with premises he regarded as philosophically immediate, rather than a moving drawing that looked convincing to the eye ⁴.

Khayyam's quadrilateral can be mistaken for an independent proof of the fifth postulate. The record says instead that it concerns proof organisation and starting assumptions, and does not prove the fifth postulate from Euclid's other assumptions alone ^{3 2}. The figure is now called the Khayyam–Saccheri quadrilateral because Saccheri investigated it in his book of 1733; the record says this modern name compares mathematical structures and does not by itself establish a route to Italy ³.

Khayyam wanted a starting point that did not rest on a moving drawing that merely looked convincing. Draw his quadrilateral on grid paper, with two equal legs perpendicular to the base. Is it enough that the two top angles look like right angles for you to know that they are? Why might Khayyam say no?

• Khayyam's quadrilateral analysis of parallels: the figure, its three cases for the summit angles, and the limits of what it shows.
• Commentary on Difficulties in Euclid's Postulates: the three parts of the book, and what the museum has and has not read of it.
• Umar Khayyam: the record of the man who set out the figure.

Read the record on Khayyam's quadrilateral. Sketch it, list its properties, and then compare your list with a classmate's.