7.6 Outline

A. Golden rectangles
      1. definition
      2. divine proportion
      3. tau
B. Mathematics and art
      1. golden ratios in art
      2. proportions of the human body
      3. spiral constructed using a golden rectangle
C. Projective geometry
      1. Duccio's Last Supper
      2. Hogarth's Perspective Absurdities
      3. false perspective
      4. Masaccio's The Holy Trinity
      5. Durer's Designer of the Lying Woman
D.  Non-Euclidean geometry
      1. Euclid's fifth postulate
      2. Saccheri quadrilateral
      3. Lobachevskian postulate
      4. hyperbolic geometry
      5. pseudosphere
      6. great circle
      7. elliptic geometry
      8. table showing comparisons of major two-dimensional geometries

7.6 Essential Ideas

A Saccheri quadrilateral has right angles as base angles and sides of equal length. The summit angles may or may not be right angles.

The Lobachevskian Postulate: The summit angles of a Saccheri quadrilateral are acute.

This section discusses projective geometry and its relationship to three-dimensional representation in art. Next, non-Euclidean geometries are investigated with the idea in mind that Euclidean geometry is not the only possible geometry. The principle non-Euclidean geometries are hyperbolic and elliptic geometries.



A representative line in each geometry is shown in color for each models, and the shaded portion showing a Saccheri quadrilateral is shown directly below the respective models.
Geometry is on a plane:	Geometry is on a pseudosphere:	Geometry is on a sphere:
The sum of the angles of a triangle is 180 degrees. Lines are infinitely long.	The sum of the angles of a triangle is less than 180 degrees. Lines are infinitely long.	The sum of the angles of a triangle is more than 180 degrees. Lines are finite in length.

The essential idea in classifying the correct geometry is the Lobachevskian postulate:
The summit angles of a Saccheri quadrilateral are acute.