This month's topics:

• "Alice's adventures in algebra"
• The wizard's clock
• Packing tetrahedra into a quasicrystal
• A weekly math column in the New York Times online

Image from Victorian Web. According to Melanie Bayley, the Mad Hatter's tea party is a farcical allegory of Hamilton's quaternion theory.

The New Scientist for December 16, 2009 ran "Alice's adventures in algebra: Wonderland solved," by Melanie Bayley. Bayley's DPhil research in Victorian literature led her to analyze the differences between Lewis Carroll's first, hand-lettered, version of "Alice's Adventures Under Ground" (1864) and the better known "Alice's Adventures in Wonderland" (now usually "Alice in Wonderland") published a year later. The newer version is almost twice as long as the original, which lacks some of the best-known episodes: the Cheshire Cat, the Duchess's Baby and the Mad Hatter's Tea Party. Bayley's thesis is that Carroll took advantage of the opportunity to poke additional fun at some of the new mathematics of the period. In particular she mentions "the Mad Hatter's tea party, which explores the work of the Irish mathematician William Rowan Hamilton."
    Hamilton had hypothesized that the fourth coordinate in his quaternions was connected with time (Lectures on Quaternions, 1853); and in fact Time is the missing guest at this party. "the members of the Hatter's tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers. Their movement around the table is reminiscent of Hamilton's early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix." There is some play on non-commutativity and then "When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms." Bayley's piece is available online.

The "wizard's clock" gives a convenient way of organizing the 52 cards in a standard deck into three 17-card supersuits (plus the Ace of Spades); they correspond to the three concentric rings.

One of Colm Mulcahy's mathematical card tricks was featured in John Tierney's New York Times blog for December 24, 2009. The item was written by Pradeep Mutalik. The trick involves three people: the victim V, the accomplice A and the wizard. V selects four cards from a deck; A examines them, returns one to V and places the other 3 in a row on the table, some face-up, some face-down. He then telephones the wizard and tells him exactly what is showing on the table. Based on this information alone, the wizard correctly identifies the card held by V. The explanation, which involves binary numbers, modular arithmetic and the pigeonhole principle, is given in full on the blog. For other card tricks with interesting mathematical structure see Colm's October 2000 feature column on this site.

Analysis of the quasi-crystalline structure of tetrahedron packing at density 0.8324. In this slice, the edges of the turquoise pentagons join centers of nearest-neighbor tetrahedra. The pentagons and their interstices can be grouped into squares, triangles and rhombs (yellow lines) forming part of a non-periodic tiling of the plane. The diffraction pattern generated by the packed solid (inset) has 12-fold rotational symmetry, known to be incompatible with spatial periodicity. σ is the edge length of a tetrahedron. Image from Nature 462 775, used with permission.

Tetrahedra, unlike cubes, cannot fit together perfectly to fill 3-dimensional space; but there has been an ongoing competition to find the most efficient way of packing them. Last September's column picked up a Nature article in which Sal Torquato and Yang Jiao described inter alia a packing of tetrahedra with density φ = 0.7820, a record. Later work by the same authors (Phys. Rev. E 80 041104) boosted the best density to 0.8230. Tetrahedron packings reported in the December 10 2009 Nature by Sharon Glotzer and collaborators break this record by achieving densities of 0.8324 and 0.8503. The packings, and the method by which they were discovered, are of interest themselves. In a Monte-Carlo simulation using a suspension of of a large number (8000) of tetrahedra, a phase change occurs as pressure is increased, leading to the packing with φ = 0.8324 represented above; it is part of a quasi-crystal (the diffraction pattern it generates has 12-fold symmetry, ruling out spatial periodicity). The packing with φ = 0.8503 (see below) has the same local structure as the quasi-crystal but is periodic; its slices correspond to the Archimedean snub square tiling 43433.
    Late-breaking news: the record has been broken again. Kenneth Chang in the January 4 2010 New York Times ("Packing Tetrahedrons, and Closing In on a Perfect Fit") reports on preprints raising φ to 0.8555 (Torquato and Jiao, posted on December 21) and then to 0.8563 (Elizabeth R. Chen, Michael Engel, Sharon Glotzer, posted January 5).

The periodic approximant to the quasi-crystal shown above has density φ = 0.8503. Here part of the 43433 Archimedean tiling of the plane (yellow) has been superimposed on the image of 2 slices of the packing. Adapted from Nature 462 775.