Math Blog

Conformal Torus Knots

Conformal Torus Knots

In this post we'll meet a staple at the Differential Geometry math art shop: Flat tori in four-space, and "conformal torus knots" lying on these surfaces. Although we generally strive...

Conformal Torus Knots

In this post we'll meet a staple at the Differential Geometry math art shop: Flat tori in four-space, and "conformal torus knots" lying on these surfaces. Although we generally strive...

A collection of hypocycloid curves with 15-fold symmetry, represented as green and blue dots on a black background

Compound Circular Motion

Circles and Parametrization Recall that a circle is the set of points in a plane lying at fixed radius \(r\) from a point, the center. In Cartesian coordinates, if \(c...

Compound Circular Motion

Circles and Parametrization Recall that a circle is the set of points in a plane lying at fixed radius \(r\) from a point, the center. In Cartesian coordinates, if \(c...

The complex squaring map between the complex plane at top and a surface obtained by wrapping the complex plane around itself twice at bottom.

Square roots with a complex twist

In this post we'll build a bridge starting with the parabola familiar from plane geometry and arriving at the “Riemann surface of the complex square root,” which often mystifies complex...

Square roots with a complex twist

In this post we'll build a bridge starting with the parabola familiar from plane geometry and arriving at the “Riemann surface of the complex square root,” which often mystifies complex...

Stereographic projection from a punctured sphere to the plane

Stereographic Projection and Möbius Transformat...

Perspective “Point projection” is a simple model for representing a three-dimensional scene on a planar canvas in perspective. Fix a plane \(P\) in Euclidean three-space and a point \(p\) not on \(P\). For...

Stereographic Projection and Möbius Transformat...

Perspective “Point projection” is a simple model for representing a three-dimensional scene on a planar canvas in perspective. Fix a plane \(P\) in Euclidean three-space and a point \(p\) not on \(P\). For...

Loxodromes on the sphere, close relatives of logarithmic spirals in the plane

Polar Coordinates and Polar Graphs

Polar Coordinates Mathematicians translate between geometry and algebra using “coordinates”, assignments of ordered lists of numbers to geometric locations. To define Cartesian coordinates in the Euclidean plane, we fix perpendicular...

Polar Coordinates and Polar Graphs

Polar Coordinates Mathematicians translate between geometry and algebra using “coordinates”, assignments of ordered lists of numbers to geometric locations. To define Cartesian coordinates in the Euclidean plane, we fix perpendicular...

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,...