Symmetry Natural forms such as snowflakes, sea shells, water droplets, mineral crystals, and faces exhibit “symmetry.” Often, symmetry correlates with beauty: a visual stimulus translated directly to sensations of pleasantness without words or conscious thought. In this post we’ll explore what mathematicians mean by symmetry and see how symmetry inspires work at Differential Geometry. Our…
Category: About Math
Projective Geometry
Rays and Perspective Imagine a flat floor extending to infinity in all directions and covered with square tiles, creating a rectangular coordinate system. A parabola, such as the graph \(y = x^{2}\), is painted on the floor. If we stand below the origin and look at the parabola in perspective, what kind of curve do…
Compound Circular Motion
Circles and Parametrization The blog post about circles, trigonometry, and groups defined a circle to be the set of points in a plane lying at fixed radius \(r\) from a point, the center. In rectangular coordinates, if \(c = (x_{0}, y_{0})\) is the center, then a point \((x, y)\) lies on the circle of radius…
Conformal Torus Knots
In this post we’ll meet a staple at Differential Geometry: Flat tori in four-space, and “conformal torus knots” lying on these surfaces. Although we generally strive in these posts to avoid coordinates, we’ll refer to rectangular coordinates throughout. Our story starts with circles in the plane. If \(t\) is real, the point \(c(t) = (\cos…
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 analysis students. We’ll refer to complex numbers, coordinate geometry, and mappings. We’ll also draw pictures guided by coordinate geometry in more than three dimensions. Concretely,…
Stereographic Projection and Möbius Transformations
Perspective “Point projection” is a simple model for representing a three-dimensional scene on a planar canvas in perspective. Fix a plane \(P\) in three-space and a point \(p\) not on \(P\). For each point \(x\) distinct from \(p\), there is a unique line through \(p\) and \(x\). If this line is not parallel to \(P\), it intersects \(P\) in a unique point,…
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 rectangular coordinates in the plane, see our blog post on functions, coordinates, and graphs, we fix perpendicular “axes”: a horizontal and a vertical number line meeting at a point called the “origin”, let \(x\) denote…
High-Dimensional Geometry
Geometry and Coordinates At Differential Geometry, we prefer shape to symbolic description. Sometimes, however, symbolic description can guide us where geometric intuition may falter. For example, rectangular coordinates in the plane may be viewed as a “perfect dictionary” between points of the plane and “addresses,” ordered pairs \((x, y)\) of real numbers: Each location has…
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 is non-negative, so whatever \(i\) may be, it is not a real number. Complex Addition and Multiplication Suppose \(z = 3 + 4i\) and \(w =…
Circles, Trigonometry, and Groups
Circles In a flat 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 circle algebraically, we might use rectangular coordinates. Write \(c = (x_{0}, y_{0})\), and let \((x, y)\) denote the coordinates of a general point on the…