## 7.6 Outline

- Golden rectangles
- definition
- divine proportion
- tau

- Mathematics and art
- golden ratios in art
- proportions of the human body
- spiral constructed using a golden rectangle

- Projective geometry
- Duccio’s Last Supper
- Hogarth’s Perspective Absurdities
- false perspective
- Masaccio’s The Holy Trinity
- Durer’s Designer of the Lying Woman

- Non-Euclidean geometry
- Euclid’s fifth postulate
- Saccheri quadrilateral
- Lobachevskian postulate
- hyperbolic geometry
- pseudosphere
- great circle
- elliptic geometry
- 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.

The essential idea in classifying the correct geometry is the Lobachevskian postulate:

The summit angles of a Saccheri quadrilateral are acute.