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Showing posts from 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:

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

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. 

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

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=…