In 2010 we had the "Like" button.  Perhaps the most significant examples were facebook and youtube like buttons, but there were many variations, some of them invented earlier, like the tweet button, the digg button and the "insert social web service"-button.  The central idea is that you can share your feelings about web content, with your friends or the content provider's own metrics, without interruption.

Turn around and you'll see an elephant in the room; How do content providers and producers make money?  The question has been around for as long as there have been artists and entertainers, and various solutions have supplemented our past relationships with them.  If you ask an internet savvy individual what we do now, you are likely to hear the word "advertising".

But there is a problem with the principle of ad-driven content.  The other day I was notified by Google that my little 10,000 view video has made $27 in advertising revenue.  This means that each of the 2000 or so viewers that I had while the ads were running, paid a tiny bit to see my video by looking at the advertisement, whether they liked it or not.  Worse still, those who thought it was very helpful or interesting were unable to contribute anything more.

The Internet needs a "Support" button.  The Support service can be provided by an Internet payment company like paypal.com, and content providers can put the Support button on their site, allowing viewers to donate a tiny amount of money with each click.  I can hear you preparing your counter arguments as I write this, so bear with me while I try to address them.

Don't we already have that option with paypal.com and donate buttons?  Yes, but you are interrupted by having to login to paypal.com.  No one is going to take 3 minutes to donate 10 cents.  I'm aiming for 1-click payments and one login per month.

Won't people be unwilling to participate for fear of overspending, or abuse by other users on their computers?  I have thought of two ways of solving this.  The first is that users can login to the Support service provider and nullify any payments before they go through at the end of the month.  The second is that they can give themselves a spending cap.  If the sum of their support clicks are less than the cap, then 9.5 cents per click goes to the providers (maybe 0.5 cents to paypal).  Otherwise, a lesser amount goes to each one, based on an easy calculation, dividing the spending cap by the total number of clicks.

What will people gain from this and how will the initiative spread?  By making the Support button a social sharing device, you are able to say, "I liked this enough to give a bit of cash" (incidentally, some websites might want to use an anonymous support button).  It will be driven, initially, by some people's good will, and eventually social infection.  If a friend gives 10 cents for a video, you are more likely to do so as well.  In addition to this, the Support button will only work for those logged into the service (like the facebook like button on websites).  Those who are not logged into the service will see advertisements (including for the service itself), motivating them to join.

The Support service will give users a way of opting out of the ad-driven business model.

Recently, I was introduced to a wonderful object constructed of sticks and elastics, called the hexastix. These mind boggling structures were last seen in a book called The Symmetries of Things by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss.

Naturally, I wanted to build some of my own, so here is a brief account of those escapades.

First I made the biggest hexastix I've seen.  I keep looking hungrily at wooden doweling and bamboo though.

If you build your own, you'll observe that the inner cage of the above shape is approximately a rhombic dodecahedron.  More on inner cages in a moment.

Hmm, better yet, why don't we just snip off the extra bits and reveal the rhombic dodecahedron inside.

Realizing that a bunch of sticks (I'm calling each set of parallel sticks a bunch so that hexastix have four bunches) can be used in more than one hexastix, I started another structure - slightly wilder this time.  It still needs work (more sticks and nicer elastics, actually).

I was so excited by this time that I made a video about making them and colouring your own toothpicks (highly recommendable.  both the video and the colouring of the toothpicks).

Wanting to take the torch on the road with my I made a little educational kit so that I could show people (at the bar, no less) how to build their own hexastix.  I think I've educated about six so far.  It's a great party trick if you are a quirky guy like me.

One of my more contemptuous bar pupils made rhombic bunches instead of hexagonal ones.  I was shocked at first and didn't know what to make of them.  Then I made one like that at home and discovered that it has a different cage!  It appears to be an octahedron (thanks to Luis Goddyn for noticing the error) that has had two opposite vertices squashed together somewhat, and then two other opposing vertices sliced off (in a natural way), to reveal rhombuses.  Hopefully a good drawing or computer rendition will be forthcoming from me (or you?).

Stay tuned because I have one more surprise that I hope to build and show here soon :)

update 1: In the interim, I made a Fatsostix.  It's a very full hexastix.

update 2:  Thousands of viewers are on the edge of their seats, waiting for the final Hexastix project as the Hexastix champion (I am championing them to you right now) brings us one step closer with ... a Hexastix in a jar!  I call it a Jarofstix.

This poster appeared at the 2010 CMS Winter Meeting.  Download the pdf for a larger version.

This page will be updated as new publications appear.

A careful search reveals the following publications and mentions of tatami tilings in mathematics:

Mitsuyoshi Yoshida, Japan 1641, Jinkōki ( translated as “Inalterable Treatise”, first published in 1627 according to britannica.com).

• Knuth reprints a 6 x 5 dimer (1 x 2) tiling from Yoshida’s text in The Art of Computer Programming

David Wolfe, Tom Rodgers, 2002, Puzzlers’ Tribute: A Feast for the Mind.  Puzzle by Yoshiyuki Kotani, Y2K problem of dominoes and tatami carpeting.

• Some of this puzzle can be seen in the Amazon.com preview for the hardcover, page 413.  The puzzle seen in the preview is: Find a shape that has exactly 2000 domino tatami tilings.

Dean Hickerson, 2002: OEIS http://oeis.org/A068920/a068920.txt

• Treats fixed height dimer (1 x 2) tilings and finds recurrences for them.  References to several related OEIS sequences.

• As of October 24, 2011, searching oeis.org for “tatami” returns 34 sequences related to tatami tilings.

Knuth, 2009: The Art of Computer Programming, volume 4, fascicle 1B, Binary decision diagrams, exercise 215 and errata.

• Exercise on dimer (1 x 2) tilings:  “Find all domino coverings of a chessboard that are also tatami tilings”

Frank Ruskey and Jennifer Woodcock, 2009: Counting Fixed-Height Tatami Tilings, Electronic J. of Combinatorics.

• Generating function for fixed height dimer tilings, stemming from Knuth’s exercise, above.

Alhazov, Morita, and Iwamoto, 2009: A note on tatami tilings, Proceedings of 2009 LA Winter Symposium.

• Following Ruskey and Woodcock, this paper treated tilings with exactly one monomer (1 x 1).  It motivates further the study of arbitrary monomers but was written prior to our discovery of their structure.

A mention by Don Knuth at theory lunch, 2010

Alejandro Erickson, Frank Ruskey, Mark Schurch, Jennifer Woodcock, 2010: Auspicious Tatami Mat Arrangements, 16th COCOON Conference, LNCS 6196, pp. 288–297.

• Full characterization of the structure monomer-dimer (1x1 and 1x2) tatami tilings of rectangular regions.
• Maximum number of monomers in an $r x c$ tiling is max(r,c) or max(r,c)+1 and has parity the same as rc.
• The number of $n\times n$ tilings with $n$ monomers is $n2^{n-1}$.
• The generating function for fixed height monomer-dimer tilings is a rational generating function.

Alejandro Erickson, Frank Ruskey, Mark Schurch, Jennifer Woodcock, 2011. Monomer-Dimer Tatami Tilings of Rectangular Regions. Electronic Journal of Combinatorics, 18(1) (2011) P109, 24 pages.

• Expanded journal version of the above

A. Erickson, M. Schurch, 2011: Enumerating tatami mat arrangements of square grids, in 22nd International Workshop on Combinatorial Algorithms, University of Victoria, June 20-22, volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages.

• Abstract:     We prove that the number of monomer-dimer tilings of an $n\times n$
  square grid, with $m<n$ monomers in which no four tiles meet at any
  point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity.
  In addition, we present a new proof of the result that there are
  $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings
  into $n$ classes of size $2^{n-1}$.  The sum of these over all $m\le
  n$ has the closed form $2^{n-1}(3n-4)+2$ and, curiously, this is equal
  to the sum of the squares of all parts in all compositions of $n$.

Tatamibari, Nikoli

• We are given a rectangular grid with certain symbols.  The goal is to find a certain subdivision of the grid into rectangles which satisfies the tatami condition.  My own post about this puzzle is here.

2010 Frank Ruskey, Alejandro Erickson

• Players take turns adding tiles to a rectangular grid.  They must not violate the tatami condition and the first player with no available move loses.

Tomoku! 2010, Martín Matamala, Alejandro Erickson

• The player must reconstruct a tatami tiling given its row and column projections.
• A website, web game and book are dedicated to this at http://tomokupuzzle.com

Tatami dots and boxes, 2010, Alejandro Erickson, Gwen Temmel

• Played just like dots and boxes except that no four lines may touch at any point.

Water striders, Frank Ruskey, Alejandro Erickson, Brian Wyvill

• Players place x shaped pieces which must not overlap.  The first player with no available move loses.  This is inspired by the rays formed by the tatami configurations called bidimers and vortices and the desire to pack them.

There is another puzzle due to Knuth I think, with 1x3 tiles, but I don’t remember where I saw it.