# Mathematics Today…

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When we talk about doing mathematics, what comes to  our mind is Blackboard-chalk, notebook-pen and books. No doubt that these are and will remain one one of the most important instruments leading to elegant mathematical discoveries.  But, the evolution of technology we use has also affected the way we do, learn and share mathematics. In ancient time, mathematics was shared in form of books and letters. Then in 17th Century people started publishing academic journals periodically, which has today become one the most profitable business (like pharmaceuticals).  In 1960s computer algebra systems were invented (called MATHLAB). Then in 1970s books were digitized and today we have dedicated ebook readers.  Another major challenge  of publishing mathematical knowledge was to be able to typeset weird symbols, and this problem was fully solved using computers in 1978 by Donald Knuth. (to know more about this transition read this discussion).

Now it’s 21st century and the shape of sharing mathematical knowledge has changed significantly in past decade.  To begin with, in 2003 Poincaré conjecture’s solution was not published in any journal but was rather posted on arXiv. Today we have people on social networking sites like Facebook, Twitter, Google+, Tumblr, Weblogs...  who let you know the results just as they are being cooked up. For example, Live-tweet of Babai’s first Graph Isomorphism talk, in this talk one of the most interesting theorem of 2015 was proved. Many big shots announce their big results directly on their Weblogs, for example Terence Tao announced his proof of Erdős Discrepancy Problem on his blog. Today we can have interactive textbooks (like this), articles (like this) and assignments (like this) with advent of MathJax, SageMath

So far I have been concerned about “print” mathematics, but with advent of cheap internet, whole new methods of mathematical ideas sharing have come into picture. Today almost every reputed research organization maintain video lecture archives (IAS, CIRM,  IHÉSIHPInstitut Fourier, MatScience). Apart from mainstream mathematics, popularization of mathematics has become much more interesting today. We have lots of interesting mathematics popularization channels on YouTube like ViHartNumberphile, Mathologer, 3Blue1Brown, The Global Math Project,… and SoundCloud like BBC Radio 4: More or Less, ACMEScience ,… For a big-list of online mathematics videos see this and for big-list of mathematics podcasts see this.

Before this internet era, there were similar mathematics popularization attempts. Like my favourite: “Donald Duck in Mathmagic Land” (1959) [updated the link on 28 Dec’18]

But I’m not aware of existence of mathematical radio programs back then. So, if you know about such radio programs,o please let me know about them as comments below.

# 135th Carnival of Mathematics

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Welcome to the carnival…

Before we begin, let’s know our number-friend of this month. 135 is the smallest three-digit number that is the sum of its first digit and the square of its second digit and the cube of its third digit: $\displaystyle{135 = 1 + 3^2 + 5^3}$.

Now, get hold of pen and paper, the carnival begins…

Cross-Number Puzzle – Numbrcise.com

This is a “medium” brainteaser to help you warm up. To play, you must fill in all the blank squares in the grid with numbers ranging from 1 to 9 in order to find a synchronized solution to a series of horizontal and vertical equations – all at once.

Three dimensional tessellation of crosses – circlesandtriangles (Dan)

Day after tomorrow is 118th birthday of Maurits Cornelis Escher and will be celebrated as first “Tessellation Day” [details]. And in this article, Dan investigates the three dimensional analogue of Greek Cross tessellation using animations. He elegantly explains the mathematics involved behind tessellation by reducing  the 3D problem to a 2D one.

Snell and Escher – Joshua Bowman

When we study the concept of refraction, one of the central law is “Snell’s Law”. Motivated by Johann Bernoulli’s (a mathematician) clever application of Snell’s Law to solve brachistochrone problem,  Joshua introduces reader to hyperbolic geometry via works of Maurits Cornelis Escher  (an artist) using Snell’s Law! I have many times discussed hyperbolic geometry with young  minds, but this introduction is really illuminating.

5 Mistakes to Avoid when Drawing a Soccer Ball – David Swart

At some point of time, you must have tried your hand at drawing various objects around you. Moreover, soccer is claimed to be most played sports in the world. But still many of us can’t draw a soccer ball properly! In this lovely article, David helps us to mathematically identify and correct these mistakes…

Boolean Algebra – mathsbyagirl

Banach-Tarski Paradox  –  Austin Lawson

If you try to take a stroll through “mathematical logic”, you will encounter the statements that contradict themselves and yet might be true.  Such statements are called paradoxes. In this article, Austin gives a sketch of the Banach-Tarski theorem and paradox.  It is intended for mathematicians of intermediate skills though the first few paragraphs are accessible to more people.

Programming is a piece of cake  – Paula Rowinska

This is an article discussing the importance of computing in the modern maths, especially the applied areas. There’s a common misconception that mathematicians work only on very pure and useless for the society problems – Paula explains that it’s a myth.

Many of us would have appeared for exams in May, and hence this article by a gifted mathematician becomes oddly relevant! In this article Prof. Terence does a serious analysis of grading system using techniques from Probability and Statistics. In case you happen to be a person associated with academia, this is worth reading.

Float like butterfly and Sting like a Mathematician!  – Nira Chamberlain

Muhammed Ali has been regarded as the most influential sportsman of the century. A week ago, he died aged 74. This is intriguing story about how a heavyweight boxing champion influenced Dr. Nira to become a mathematician.

Interview with a mathematician: Maria Chudnovsky  – Anthony Bonato

Graph Theory is one of those branches of Mathematics which consist of simple to state but difficult to solve problems. In case you are not familiar with it, just spend some time on this “Math for seven-year-olds” kit. Prof. Anthony interviewed Maria Chudnovsky, a leading mathematician specializing in graph theory. Maria is famous for proving the “Strong Perfect Graph theorem”, which was open for forty years. Maria is engaging and gives great advice to young mathematicians. She also talks about her upbringing, including facts you can’t find on her wiki page or other interviews.

Prime After Prime  – Brian Hayes

There’s lots about the prime numbers that seems random; you can even play a good game of dice with them. But in March Robert J. Lemke Oliver and Kannan Soundararajan discovered some remarkable biases or correlations between pairs of consecutive primes. Brain explores this discovery in computer code and “very illuminating” pictures.

This article builds upon Brian Hayes‘ article about the distribution of primes, which we just discussed.  Motivated by the argument that “Primes aren’t random, but sometimes it can be useful to treat them as if they were.”, John writes about the connection between statistics and number theory. Mathematically mature audience will surely enjoy reading this.

Maximal density subsquare-free arrangements – Peter Karpov

I will end this Carnival with an open problem. In this article, Peter discusses some computational results for the “no subsquares problem”. The statement of problem is as follows

What is the largest number of points that can be placed on a N × N grid so that no four of them form a square?

Get your hands dirty and try to find an efficient algorithm to calculate answer for N>16!

But, before we say good bye to this edition of monthly roundup, let me remind you to visit the previous edition which was hosted by Kartik at Comfortably Numbered. Join us next time for the 136th edition, hosted by Manan at Math Misery. You can find all the other Carnivals, and submit articles for future carnivals at: http://aperiodical.com/carnival-of-mathematics/

# Numbers and Logic

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I am a big fan of number theory. I find the answer to Hilbert’s Tenth Problem fascinating. I was introduced to this problem, a couple of years ago, via the documentary titled : “Julia Robinson and Hilbert’s Tenth Problem“, here is the trailer:

You can read more about it here. Also for the sake of completeness, let me state Hilbert’s Tenth Problem:

Does there exist an algorithm to determine whether a given Diophantine equation has a solution in rational integers?

In 1970, Yuri Matiyasevich completed the solution of this problem by using the concept of Turing Machine. This short video provides a nice overview about Turing Machines in general

The answer to Hilbert’s Tenth Problem problem is

No such algorithm exists.

This interplay of number theory and logic is really interesting, isn’t it? But I can’t discuss solution of Hilbert’s Tenth Problem here, since I have never read it. But there is nice overview at Wikipedia.

I will rather discuss a puzzle from Boris A. Kordemsky’s book which illustrates the idea of this interplay.

Ask a friend to pick a number from 1 through 1000. After asking him/her ten questions that can be answered yes or no, you tell him/her the number. What kind of question?

The key to the solution is that 2 to the tenth power is 1024 (that is, over 1000). With each question you knock out half the remaining numbers, and after ten questions only the thought number is left.

I welcome you to think of a number and write the corresponding yes/no questions as a comment below.

# Call for Submissions

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I will be hosting 135th Carnival of Mathematics on 15 June 2016.

The Carnival of Mathematics is a monthly blogging round up hosted by a different blog each month. The Aperiodical is responsible for organising a host each month.

The Carnival of Mathematics accepts any mathematics-related blog posts: explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable.

Last date for submission is 10 June 2016

# Revision 1: Why I love Mathematics?

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Unlike Mathematics itself, which is forever, philosophy of mathematics keeps on changing. Philosophy of mathematics includes basic definitions and ideologies. About one and a half years ago I wrote about “Why I love Mathematics?” and in these one and a half years my ideologies changed.

Now, for me love means

a feeling of awesomeness for something/ someone.

I believe that this definition captures the general idea. Now let me update the answer to the question.

Mathematics is not a human being, so it can’t accept or reject me. I like logical, concrete and unambiguous statements which are provided by mathematics. Hence I study it.

Unfortunately, contrary to what I believed earlier, mathematics doesn’t have its own language.

The language of mathematics is the language being spoken by the citizens of the “world center(s) of mathematics” of that time.

Let me illustrate my point:

• The ancient mathematics was communicated in Greek, Arabic and Sanskrit (symbolic languages of Babylonians and Mayans are yet to be fully deciphered).
• The medieval mathematics (14th century to 18th Century) was communicated in Italic languages like Latin, Italian, French etc. A good supporter of this argument is the fact that Carl Friedrich Gauss being German wrote his most celebrated book, Disquisitiones Arithmeticae, in Latin.
• The before-my-birth mathematics (18th Century to 20th Century) was communicated in Germanic languages like English and German, since then Cambridge (UK) and Göttingen were the “world centers of mathematics”. A good supporter of this argument is the fact that Paul Erdős being Hungarian wrote his first paper in German (though, he and his friends also published in Hungarian). Interestingly, Russian was the language of many beautiful olympiad problem books until disintegration of USSR.
• The after-my-birth mathematics (21st Century onwards) is communicated in English (Germanic Language) and French (Italic Language) since today’s world centers of mathematics are USA and France. German lost its position as scientific language because of Adolf Hitler‘s dictatorship.

I am not claiming that today mathematics doesn’t exist in any languages other than English or French, but these are the languages in which we today consider publishing our work.  For example, there are more than 550 million people speaking Spanish so it is obvious that their mathematics textbooks are written in their languages (some spanish books). Even in  India, though we have more than 260 million people speaking Hindi and high school mathematics textbooks in Hindi (see: KhanAcademy in Hindi), but still at college level English is only official mode of instruction so that the students have access to the latest discoveries.

# Integration & Summation

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A few months ago I wrote a series of blog posts on “rigorous”  definitions of integration [Part 1, Part 2]. Last week I identified an interesting flaw in my “imagination” of integration in terms of “limiting summation” and it lead to an interesting investigation.

While defining integration as area under curve, we consider rectangles of equal width and let that width approach zero. Hence I used to imagine integration as summation of individual heights, since width approaches zero in limiting case. It was just like extending summation over integers to summation over real numbers.

My Thought Process..

But as per my above imagination, since width of line segment is zero,  I am considering rectangles of zero width. Then each rectangle is of zero area (I proved it recently). So the area under curve will be zero! Paradox!

I realized that, just like ancient greeks, I am using very bad imagination of limiting process!

The Insight

But, as it turns out, my imagination is NOT completely wrong.  I googled and stumbled upon this stack exchange post:

There is the answer by Jonathan to this question which captures my imagination:

The idea is that since $\int_0^n f(x)dx$ can be approximated by the Riemann sum, thus $\displaystyle{\sum_{i=0}^n f(i) = \int_{0}^n f(x)dx + \text{higher order corrections}}$

The generalization of above idea gives us the Euler–Maclaurin formula

$\displaystyle{\sum_{i=m+1}^n f(i) = \int^n_m f(x)\,dx + B_1 \left(f(n) - f(m)\right) + \sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R}$

where $m,n,p$ are natural numbers, $f (x)$ is a real valued continuous function, $B_k$ are the Bernoulli numbers and $R$ is an error term which is normally small for suitable values of $p$ (depends on $n, m, p$ and $f$).

Proof of above formula is by principle of mathematical induction. For more details, see this beautiful paper: Apostol, T. M.. (1999). An Elementary View of Euler’s Summation Formula. The American Mathematical Monthly, 106(5), 409–418. http://doi.org/10.2307/2589145

# Colourful complex functions

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Recently I became curious about functions defined from $\mathbb{C}$ to $\mathbb{C}$ and I asked myself following question:

How would the complex functions look like if we try to plot them?

Graphs of complex functions lie in $\mathbb{C}^2$, which can be identified in a natural way with $\mathbb{R}^4$, real four-dimensional space.

So I jumped to SageMath  and plotted $z^2$


sage: f(z) = z^2
sage: complex_plot(f, (-5, 5), (-5, 5))



graph of z^2 plotted using SageMath Version 7.0

Now this looked like an enigma to me. What do the colours stand for? As usual, there is an interesting entry about this on Wikipedia, Colour wheel graphs of complex functions.

I digged further and discovered that these are called “2D colour maps” and is one of many other ways of visualizing complex functions, like 3D models, 2D vector plots, 4D perspective projection,  conformal maps…

HLS Cylinder (By SharkDderivative [CC BY-SA 3.0 or GFDL], via Wikimedia Commons)

But what is the colour map? The colour map uses the HLS colour system (“hue-lightness-saturation”). HLS is a cylindrical-coordinate representations of points in an RGB color model.  In cylinder, the angle around the central vertical axis corresponds to “hue”(i.e. shade of a colour) , the distance from the axis corresponds to “saturation”, and the distance along the axis corresponds to “lightness”.

The argument φ and modulus r locate a point on an Argand diagram i.e. complex plane.(By Kan8eDie [CC BY-SA 3.0, CC BY-SA 3.0 or GFDL], via Wikimedia Commons)

The hue represents the argument (also called phase angle) of the complex number z. The absolute value (also called magnitude or modulus) is given by the lightness of the colour. All colours of the colour map have the maximal saturation (with respect to the given lightness).

HLS Colour Wheel (source: iliasky.com)

Positive real numbers always appear red. The primary colours appear at phase angles  $\frac{2 \pi}{3}$ (green) and $\frac{4\pi}{3}$ (blue). The subtractive colours yellow, cyan, and magenta have the phases $\frac{\pi}{3}$, $\pi$, and $\frac{5\pi}{3}$.

The poles of a complex function are white, the zeros are black.

Finally, to conclude [from : Visual quantum mechanics : selected topics with computer-generated animations of quantum-mechanical phenomena by Bernd Thaller.]

This colour map is obtained by a stereographic projection from the surface of the three-dimensional colour space (in the hue-lightness-saturation system) onto the complex plane.

An appropriately colored surface graphics or a density graphics can give a useful graphical representation of a complex valued function. Another example of complex valued function, a wave function, is given here:

Visualizations of a wave function in two dimensions. The left graphic shows the function as a “density plot” with additional contour lines for the absolute value. In the three-dimensional surface plot the height of the surface gives the absolute value of the wave function. (By Bernd Thaller, created using Mathematica. © 2000 Springer-Verlag New York, Inc.)

For more such graphs, visit Bernd Thaller’s Gallery of complex functions .