△Geometry
Plane, solid, analytic, and differential geometry
Subfields
Plane Geometry
Points, lines, planes, angles, triangles, quadrilaterals, circles, congruence, similarity
Solid Geometry
Polyhedra, cylinders, cones, spheres, volume, surface area
Analytic Geometry
Coordinate systems, lines, circles, conic sections
Non-Euclidean Geometry
Spherical geometry, hyperbolic geometry, parallel postulate
Differential Geometry
Curve theory, surface theory, manifolds
Riemannian Geometry
Riemannian metric, geodesics, curvature, relativity applications
Symplectic Geometry
Symplectic manifolds, Hamiltonian mechanics, classical mechanics
Algebraic Geometry
Algebraic varieties, schemes, number theory applications
Projective Geometry
Projective spaces, homogeneous coordinates, perspective
Concepts
Point, Line, Plane
A point has position but no size. A line is the shortest path between two points. A plane is a 2-dimensional surface with length and width.
Angles
An angle is formed when two lines meet at a point. The measure of an angle indicates how far apart the lines are.
Triangle
A polygon with three sides. The sum of interior angles is always 180°.
Pythagorean Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Area of Triangle
The area of a triangle is half the product of its base and height.
Similar Triangles
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
Congruent Triangles
Two triangles are congruent if they have exactly the same shape and size. All corresponding sides and angles are equal.
Quadrilaterals
A polygon with four sides. The sum of interior angles is 360°.
Circle
The set of all points in a plane that are at a fixed distance from a center point.
Arc and Sector
An arc is a portion of the circle between two points. A sector is the region bounded by two radii and an arc.
Distance Formula
The straight-line distance between two points in a coordinate plane, derived from the Pythagorean theorem.
Midpoint Formula
A formula to find the coordinates of the midpoint of a line segment between two points.
Slope
A measure of how steep a line is, calculated as the ratio of vertical change to horizontal change.
Conic Sections
Curves formed by intersecting a cone with a plane: circle, ellipse, parabola, and hyperbola.
Solid Figures
Three-dimensional shapes with volume and surface area.
Metric Tensor
A symmetric (0,2)-tensor defining distances and angles on a manifold. Used to raise and lower indices.
Christoffel Symbols
Coordinate expression of the Levi-Civita connection. Used to define geodesic equations and covariant derivatives.
Covariant Derivative
A method of differentiating tensor fields in curved spaces. Related to parallel transport, providing coordinate-independent differentiation.
Riemann Curvature Tensor
A (1,3)-tensor measuring intrinsic curvature of a manifold. Defined by the non-commutativity of covariant derivatives.
Ricci Tensor
A symmetric (0,2)-tensor obtained by contracting the Riemann tensor. Central component of Einstein field equations.