Geometry

The study of shape, size, and space.

Euclidean Geometry

• Euclidean Axioms — Hilbert's axioms
• Point — Undefined term
• Line — Undefined term
• Plane — Undefined term
• Betweenness — Order on a line
• Congruence — Same size and shape
• Triangle — Three segments
• Triangle Inequality — |a+b| ≤ |a| + |b|
• Angle — Union of two rays
• Angle Sum of Triangle — 180 degrees
• Polygon — Closed broken line
• Regular Polygon — Equal sides and angles
• Circle — Equidistant from center
• Circumference — Perimeter of circle
• Pi — Circumference/diameter
• Area of Circle — πr²
• Chord — Segment connecting two points on circle
• Tangent — Line touching at one point
• Secant — Line intersecting at two points
• Arc — Part of circumference
• Sector — Wedge-shaped region
• Pythagorean Theorem — a² + b² = c²
• Pythagorean Proof (Rearrangement) — Proof by moving triangles
• Pythagorean Proof (Similar Triangles) — Proof using similarity
• Pythagorean Proof (Algebraic) — Algebraic proof
• Similarity — Same shape, different size
• Similar Triangles — AAA criterion
• Sine Law — a/sin(A) = 2R
• Cosine Law — c² = a² + b² - 2ab cos(C)
• Trigonometric Functions — Sine, cosine, tangent
• Polygon Angle Sum — (n-2) × 180°
• Parallelogram — Two pairs of parallel sides
• Rectangle — Right angles
• Rhombus — Equal sides
• Square — Equal sides and right angles
• Trapezoid — One pair of parallel sides

Non-Euclidean Geometry

• Parallel Postulate — Fifth postulate
• Hyperbolic Geometry — Negate parallel postulate
• Elliptic Geometry — No parallel lines
• Great Circle — "Lines" on a sphere

Topology

• Metric Space — Distance function
• Open Set — Union of open balls
• Closed Set — Complement of open
• Boundary — Points on the edge
• Interior — Points inside
• Closure — Set plus boundary
• Limit Point — Every neighborhood intersects
• Compactness — Every open cover has finite subcover
• Heine-Borel Theorem — Compact iff closed and bounded in R^n
• Connectedness — Cannot split into two open sets
• Path-Connected — Can draw path between any two points
• Continuity (Topology) — Preimage of open is open