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© Cordon Art-Baarn-the Netherlands

Many of you must be aware of the works of the famous Dutch artist Maurits Cornelis Escher who explored the notions of symmetry and  infinity and depicted visual paradoxes and impossible worlds through his art. He worked extensively on regular divisions of the plane, also called tessellations, which are arrangements of closed, interlocking planar shapes which cover the whole plane without any gaps. Escher distorted basic planar figures such as triangles, squares and other polygons into organic forms to construct his tessellations. Even though he was not mathematically-trained, Escher displayed a keen intuition and creativity which appeals to mathematicians and non-mathematicians alike. You can visit Escher in the Classroom, a fantastic site which walks you through some of Escher’s constructions.

Inspired by a drawing by H. S. M. Coxeter, Escher created his Circle Limit series, which uses the Poincare disk model of Hyperbolic space

Math and the art of M. C. Escher and   the mathematical art of M. C. Escher are also wonderful sites which talks in detail about the mathematical nature of Escher’s work.

One of Escher’s most fascinating works is the Print Gallery. It shows a young man standing in an exhibition gallery, viewing a print of a Mediterranean seaport. As his eyes follow the buildings shown on the print from left to right and then down, he discovers among them thevery same gallery in which he is standing. A circular white patch in the middle of the lithograph contains Escher’s monogram and signature. Artists and mathematicians have wondered if at all this white patch could  be filled. It was only in 2002 that Henrik Lenstra, a professor of Mathematics, came up with a mathematical explanation for how the Print Gallery can be constructed and his solution provided him with a way to fill the mysterious hole in the center. For more details you can refer to the excellent article titled Artful Mathematics: The heritage of M. C. Escher in the Notices of the American Mathematical Society, and visit the website of this project titled Escher and the Droste Effect, in which a step-by-step process at arriving the solution is explained.

Achieving the Unachievable is an award-winning film that unravels the mystery behind the Print Gallery. It has been screened in many universities across the world and is a must-watch for everyone. You can enjoy the film at the screening organized by Singularity at 11:00 a.m. on Saturday, March 26 in L1. For now, here’s a trailer:

Relationship between real and Complex differentiability The most commonly used definition of differentiability of a function ${f: \mathbb{R}\rightarrow \mathbb{R}}$ is as follows. The function ${f}$ is (real) differentiable at ${x_0 \in \mathbb{R}}$ if the limit

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(x_0+ h)- f(x_0)}{h} \end{array}$

exists. Equivalently, We say that ${f:\mathbb{R}\rightarrow \mathbb{R}}$ is differentiable at ${x_0}$ if there exists a linear transformation ${A: \mathbb{R}\rightarrow \mathbb{R}}$ satisfying

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(x_0+ h)- f(x_0)- Ah}{h}= 0. \end{array}$

Such a linear transformation, if it exists, is unique and is called the derivative of ${f}$ at ${x_0}$. We then write ${A= f^{\prime}(x_0)}$. More generally, a function ${f: U \subset \mathbb{R}^n\rightarrow \mathbb{R}^m}$ is differentiable at ${\mathbf{x_0} \in \mathbb{R}^n}$ if there exists a linear transformation ${A: \mathbb{R}^n\rightarrow \mathbb{R}^m}$ such that

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{||f(x_0+ h)- f(x_0)- Ah||}{|h|}= 0, \end{array}$

where the norms ${||\cdot||}$ and ${|\cdot |}$ are the standard norms on ${\mathbb{R}^n}$ and ${\mathbb{R}^m}$ respectively. The linear transformation, if it exists, is unique and is called the Jacobian of ${f}$ at ${x_0}$ and is denoted by ${Df(x_0)}$. If we write ${f= (f_1, f_2, \cdots, f_m)}$ and ${x= (x-1, x_2, \cdots, x_n0 \in \mathbb{R}^n}$, then we can write the Jacobian in matrix form as

$\displaystyle \begin{array}{rcl} \left[Df(x_0)\right]_{ij}= \left[\frac{\partial f_i}{\partial x_j}(x_0)\right] \end{array}$

Before looking at complex differentiability we recall Proposition: Let ${A: \mathbb{C}\rightarrow \mathbb{C}}$ be${\mathbb{R}}$-linear. The following statements are equivalent:

1. There exists ${\alpha \in \mathbb{C}}$ suchthat ${Az= \alpha z}$;
2. ${A}$ is ${\mathbb{C}}$-linear;
3. ${A(i)= iA(1)}$;
4. The matrix with respect to the canonical basis ${\{1+ 0i, 0+ i\}}$ has the form$\displaystyle \begin{array}{rcl} \left(\begin{array}{cc}a & -b\\b & a\end{array}\right), \quad a, b \in\mathbb{R}. \end{array}$

Now, a function ${V \subset \mathbb{C} \rightarrow \mathbb{C}}$ is complex-differentiable at ${z_0 \in \mathbb{C}}$ if the limit

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{f(z_0+ h)- f(z_0)}{h}=: f^{\prime}(z_0) \end{array}$

exists. This can also be reformulated as follows. We say that ${f}$ is complex-differentiable at ${z_0}$ if there exists ${\alpha \in \mathbb{C}}$ such that

$\displaystyle \begin{array}{rcl} \lim_{h\rightarrow 0}\frac{|f(z_0+ h)- f(z_0)|}{|h|}= 0, \end{array}$

and we have ${\alpha= f^{\prime}(z_0)}$.

From the above discussion, it is clear that if ${f: V \subset \mathbb{C}\rightarrow \mathbb{C}}$ is complex-differentiable at ${z_0= x_0+ iy_0}$, then it is real-differentiable at ${(x_0, y_0)}$ and the Jacobian ${Df(x_0, y_0)}$ is ${\mathbb{C}}$-linear.