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