VisualCalc
visual-mathematicscalculusgeometryeducation
Abstraction: Mamikon's visual calculus solves calculus problems without formulas
Key points:
- Mamikon's Theorem (conceived 1959): the area of a tangent sweep equals the area of its tangent cluster, regardless of the shape of the original curve
- Tangent cluster is formed by translating all tangent segments to a common point; for constant-length tangents it is always a circular sector
- Proves Archimedes' result (parabolic segment = 1/3 rectangle area) and extends to all integer and real powers of x without integration
- Solves the bicycle tire problem (area between front and rear wheel paths) without needing path equations
- Exponential curve areas, cycloidal arches, and oval rings all tractable via tangent sweep geometry
- Method also implies the Pythagorean Theorem and relates tractrix and exponential as special cases of a family at + bs = constant
Connections: Mamikon Mnatsakanian · Visual Mathematics · Calculus · Geometry