Math Blog

Perpendicular circles in four-space

High-Dimensional Geometry

Geometry and Coordinates At the Differential Geometry math art shop we prefer shape to symbolic description. Sometimes, however, symbolic description can guide us where geometric intuition may falter. For example,...

High-Dimensional Geometry

Geometry and Coordinates At the Differential Geometry math art shop we prefer shape to symbolic description. Sometimes, however, symbolic description can guide us where geometric intuition may falter. For example,...

Complex addition and multiplication acting on the Riemann sphere

Complex Numbers

Formally, a complex number is an expression \(a + bi\), with \(a\) and \(b\) real numbers and \(i\) an entity satisfying \(i^{2} = -1\). The square of an arbitrary real number...

Complex Numbers

Formally, a complex number is an expression \(a + bi\), with \(a\) and \(b\) real numbers and \(i\) an entity satisfying \(i^{2} = -1\). The square of an arbitrary real number...

Circles, Rotations, and Groups

Circles, Rotations, and Groups

Circles In a Euclidean plane, fix a point \(c\) and a positive real number \(r\). The circle with center \(c\) and radius \(r\) is the set of points at distance \(r\) from \(c\). To describe a...

Circles, Rotations, and Groups

Circles In a Euclidean plane, fix a point \(c\) and a positive real number \(r\). The circle with center \(c\) and radius \(r\) is the set of points at distance \(r\) from \(c\). To describe a...

Algebra to Geometry: Functions, Cartesian Coordinates, and Graphs

Algebra to Geometry: Functions, Cartesian Coord...

Functions In mathematics, a function represents a causal relationship between “input” and “output.” For example, the area \(A\) of a square of side length \(s\) is given by the formula...

Algebra to Geometry: Functions, Cartesian Coord...

Functions In mathematics, a function represents a causal relationship between “input” and “output.” For example, the area \(A\) of a square of side length \(s\) is given by the formula...