Gerard Desargues and Projective Geometry Quiz

Projective Geometry

• Gerard Desargues is considered the father of projective geometry.
• He revisited Euclid's second postulate and considered parallel lines intersecting in a plane, calling this the ideal point.
• Theorem of Desargues: If two triangles are perspective from a point, then they are perspective from a line.
• Projective geometry includes the study of vanishing points and perspective.

Topology

• Jules-Henri Poincaré published "Vorstudien zur Topologie" (Introductory Studies on Topology) in 1895.
• Topology is about discovering and analyzing the essential similarities and differences between sets and figures.
• Topological transformations include stretching, shrinking, twisting, or bending a figure without merging points.

Geometry

• Geometry comes from the Greek words "geo" (earth) and "metron" (measurement).
• Geometry has contributions from ancient Greeks, including the discovery of irrational numbers like √2, pi, and the golden ratio (phi).

Symmetry

• Symmetry is prevalent in mathematics, nature, art, and design.
• Reflection symmetry: points of a figure are equally positioned about a line (axis of symmetry).
• Radial symmetry: points on a figure are uniformly situated around a central point.
• Angle of rotation: smallest angle that preserves the figure when rotated.
• Order of rotation: a rotationally symmetric object with an angle of rotation θ has an order of rotation 360/θ.
• Translation symmetry: a figure can be drawn in its entirety by taking a portion and shifting it along an axis.

Frieze Patterns

• There are seven frieze patterns:
  • F1 (Hop): Translation symmetry only.
  • F2 (Step): Translation and glide reflection symmetry.
  • F3 (Sidle): Translation and vertical reflection symmetry.
  • F4 (Spinning Hop): Translation and rotation symmetry.
  • F5 (Spinning Sidle): Translation, glide reflection, and rotation symmetry.
  • F6 (Jump): Translation, horizontal reflection symmetry, and rotation.
  • F7 (Spinning Jump): Translation, horizontal, vertical reflection, and rotation symmetry.

Hyperbolic and Elliptic Geometry

• Nikolai Ivanovich Lobachevsky and Janos Bolyai were key contributors to hyperbolic geometry.
• Johann Carl Friedrich Gauss contributed to many fields of mathematics.
• Hyperbolic geometry: assumes the negation of Euclid's 5th Postulate, considering a circular disk instead of a plane.
• Characteristic Postulate of Hyperbolic Geometry: Through a point P not on a line l, there exists at least two lines passing through point P parallel to line l.
• Elliptic geometry: there are no parallel lines, and any two lines must intersect; the sum of the interior angles of an elliptic triangle is more than 180.
• Characteristic Postulate of Elliptic Geometry: Through a point P not on a line l, there exists no line passing through point P parallel to line l.