martes, 6 de octubre de 2015

Pythagoras trees

The Pythagoras tree is a plane fractal constructed from squares.The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, each scaled down by a linear factor of $\frac12\sqrt{2}$, such that the corners of the squares coincide pairwise. The same procedure is then applied recursively to the two smaller squares, ad infinitum

GeoGebra Applet:

sábado, 4 de abril de 2015

Vito Volterra's work on pathologic functions

In the 1880s the research on the theory of integration was focused mainly on the properties of infinite sets. The development of nowhere dense sets with positive outer content, known nowadays as nowhere dense sets with positive measure, allowed the construction of general functions with the purpose of extending Riemann's definition of integral. Vito Volterra provided, in 1881, an example of a differentiable function $F$ whose derivative $F'$ is bounded but not Riemann integrable. In this article we present and discuss Volterra's example. To read more about this follow the link below:

Link to journal: Vito Volterra's function

domingo, 29 de marzo de 2015

Möbius transformations with stereographic projections

A Möbius transformation of the plane is a rational function of the form
$$f(z) = \frac{a z + b}{c z + d}$$
of one complex variable $z$. Here the coefficients $a, b, c, d$ are complex numbers satisfying $ad - bc\neq 0.$

Geometrically, a Möbius transformation can be obtained by stereographic projection of the complex plane onto an admissible sphere in $\mathbb R^3$, followed by a rigid motion of the sphere in $\mathbb R^3$ which maps it to another admissible sphere, followed by stereographic projection back to the plane. 


Inversion in the complex plane

Inversion using stereographic projection

A Möbius transformation is a combination of dilatation, inversion, translation, and rotation.

Rotation, inversion and translation in the plane
Rotation, inversion, and translation using stereographic projection
The following applet shows the stereographic projection representing different Möbius transformations. Move the sliders to see what happens.

Made with GeoGebra, link here: This applet was made based on the work of D. N. Arnold and J. Rogness.

Further reading:

Arnold, D. N. & Rogness, J. (2008).  Möbius transformations revealed. Notices of the AMS. 55, 10: pp. 1226-1231. 

sábado, 28 de febrero de 2015

Roots of complex numbers

Consider $z=a+ib$ a nonzero complex number. The number $z$ can be written in polar form as
\[z=r(\cos \theta +i \sin \theta)\]
where $r=\sqrt{a^2+b^2}$ and $\theta$ is the angle, in radians, from the positive $x$-axis to the ray connecting the origin to the point $z$.

Now, de Moivre's formula establishes that if $z=r(\cos \theta +i\sin \theta)$ and $n$ is a positive integer, then
\[z^n=r^n(\cos n\theta+i\sin n\theta).\]
Let $w$ be a complex number. Using de Moivre's formula will help us to solve the equation $z^n=w$ for $z$ when $w$ is given. Suppose that $w=r(\cos \theta +i\sin \theta)$ and $z=\rho (\cos \psi +i\sin \psi)$. Then de Moivre's formula gives $z^n=\rho^n(\cos n\psi+i\sin n\psi)$. It follows that $\rho^n=r=|w|$ by uniqueness of the polar representation and $n\psi = \theta +k(2\pi)$, where $k$ is some integer. Thus
\[z=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right) \right]\]
Each value of $k=0,1,2,\ldots ,n-1$ gives a different value of $z$. Any other value of $k$ merely repeats one of the values of $z$ corresponding to $k=0,1,2,\ldots ,n-1$. Thus there are exactly $n$th roots of a nonzero complex number.

The complex number $z$ can also be written in exponential form as
because $e^{i\theta}=\cos \theta +i \sin \theta$.

Thus, the $n$th roots of a nonzero complex number $z$ can also be expressed as
where $k=0, 1, 2, \ldots , n-1$.

In the following applet you can see a geometrical representation of the $n$th roots for a family of complex numbers. Change the values of the real and imaginary parts of $z$ and the $n$th root. Some examples are:
  1. Re(z)=1/2, Im(z)=1.72, n=3; 
  2. Re(z)=sqrt(2), Im(z)=pi, n=7.

External link to GeoGebra applet:

viernes, 2 de enero de 2015

Foundations of mathematics, Supplementary notes


This notes are meant to be a supplementary material for the  course Mathematical Foundations and are constantly updated. If you've found a typo or a factual error, by all means let me know:

Supplementary Material by Juancarlos Ponce