Spiral Determinants

We consider Spiral Determinants of the kind
$$\text{det}\left(
\begin{matrix}
{16}&{15}&{14}&{13}\\
{5}&{4}&{3}&{12}\\
{6}&1&{2}&{11}\\
{7}&{8}&{9}&{10}
\end{matrix}
\right)
$$
and
$$\text{det}
\left(
\begin{matrix}
{17}&{16}&{15}&{14}&{13}\\
{18}&{5}&{4}&{3}&{12}\\
{19}&{6}&1&{2}&{11}\\
{20}&{7}&{8}&{9}&{10}\\
{21}&{22}&{23}&{24}&{25}
\end{matrix}
\right)
$$

where the entries spiral out from the center. Christian Krattenthaler, who is one of the greatest experts on determinants, tells the story of how he came across such determinants and how he went about discovering the formulas for such determinants. The preprint is available on arxiv.

I have wanted to work with Christian ever since my Ph.D. days, when I tried to generalize a matrix inversion due to him. Finally, we have a joint paper. This also means that my Erdos number has come down from 4 to 3.

The picture above is from Christian's course on "Bijections" which I had an opportunity to attend in the University of Vienna during the period October 2016 to January 2017.

Update: April 26, 2017  The paper has been accepted and will appear in Linear Algebra and its Applications. Here is a preprint on arxiv.
Update: May 10, 2017. The paper is published online. The reference is:
G. Bhatnagar and C. Krattenthaler, Spiral Determinants, Linear Algebra Appl., 529 (2017) 374-390.
Here is a link to the publisher's site: https://www.sciencedirect.com/science/article/pii/S0024379517302719

Analogues of a Fibonacci-Lucas Identity

Recently, in 2014, Sury published a Fibonacci-Lucas  identity in the Monthly. It turned out that the identity had appeared earlier (as Identity 236 in Benjamin and Quinn's book: Proofs that count: The art of combinatorial proof). When I tried to prove it using my usual telescoping method, I found its connection with one of the oldest Fibonacci identities due to Lucas in 1876. I also found many generalizations and analogous identities for other Fibonacci type sequences and polynomials. This  small paper has been accepted in the Fibonacci Quarterly.

Here is a link to a preprint: Analogues of a Fibonacci-Lucas Identity

Update: Its has appeared. The ref is: Analogues of a Fibonacci-Lucas identity, Fibonacci Quart., 54 (no. 2),  166-171,  (2016)

I use the approach of my earlier paper on Telescoping: In praise of an elementary identity of Euler.

I am pleased, because I have thought of getting a paper in the Fibonacci Quarterly since I was in high school, and feel lucky I found something they found acceptable!

How to Discover the Rogers-Ramanujan Identities

Dec 22, 2012: It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

Here is a link:  How to Discover the Rogers-Ramanujan Identities.

This was presented to a some high school math teachers in a conference. I tried to write it in a way that it could be understood by a motivated high school student.

Update (May 26, 2015): The article has been published. Here is a reference. Resonance, 20 (no. 5), 416-430, (May 2015).

Update (January 18, 2014): This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.

Of Art and Math: A series of articles with Punya Mishra

Right Angle: One of the many ambigrams made by Punya Mishra that appear in this series of articles appearing in "At Right Angles". All ambigrams are copyright Punya Mishra and cannot be used without permission. 

Punya and I are writing a series of articles on the subject of ambigrams. All the ambigrams are made by Punya. For this series, he has been making many new ambigrams, which communicate mathematical ideas. Already, in the space of working on a few articles, it looks like he has made the largest number of mathematical ambigrams.

Here is a longer blog entry from Punya's blog, about this series of articles. His blog has further links to his amazing ambigrams.

Updates

Dec, 2015. I presented Punya's and my work in TIME 2015, in Baramati, Maharashtra in my talk: On Punya Mishra's Mathematical Ambigrams. This was the seventh edition of TIME, which stands for 'Technology and innovation in Math education'.

July, 2015. The fifth article is Part 2 of 2 on the subject of paradoxes. It covers self-reference, Russell's Paradox and visual paradoxes. This article includes a 'new paradox', a version of Jourdain's card paradox by Punya. 

Mar, 2015. The fourth article is on Paradoxes. It is part 1 of 2 articles on this topic. Here we consider what TRUE and FALSE mean in the context of mathematics. Its an introduction to math philosophy. Again, it has many interesting ambigrams.

Feb 2015. The Michigan State Museum has launched an exhibit entitled "Deep Play: Creativity in Math and Art through Visual Wordplay." Check out: the exhibitions web-page

July 2014. The third article on Self-similarity. This one has some amazing ambigrams, and a graphic of the binary pascal's triangle I made many years ago.

Mar, 2014. The second article is on Introducing Symmetry. I think Punya outdid himself in some of the ambigrams here. The ambigram for sin (which is periodic, a sin wave, an odd function) and inverse (modeled on a hyperbola) and exp-log were my favorites. But this month's  puzzle ambigram is mind-blowing too.

Nov, 2013. The first article has come out. It is: Introducing Ambigrams. There is a hidden message in the article. See if you can find it.