Update: See this newer essay whether the amplituhedron is revolutionaryIn December 2012, Nima Arkani-Hamed along with Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka released a very interesting paper
previous TRF articles on that concept). This development arose from the twistor minirevolution sparked by Witten's 2003 paper although the twistors turned into a small part of a greater story with many more important mathematical structures.
The tesseract isn't quite the same as the objects talked about here (the amplituhedron should be a jewel in an infinite-dimensional space, in fact) but it's close enough and I just liked the animation.
While the locality in the four-dimensional spacetime is being obscured by the new formalism (even the existence of time: the mathematical structure is "timeless"), and so is the unitarity, in fact, the new formalism unmasks many other fascinated symmetries (associated with the integrability of such models: the Yangian) and related mathematical properties of the scattering amplitude which are behind the physicists' ability to calculate amplitudes previously uncalcalculable even with the best computers (due to their complexity) on a sheet of paper.
Nima was preparing me for a looming publication of a new, even cooler paper (now it seems that it will be several papers) about these issues that will be co-authored by him and his Czech PhD student (now a postdoc) Mr Jaroslav Trnka [Yuh-rho-sluff trrrrn-kuh; please treat the ringing sound "rrrr" as a good enough ersatz vowel – the same thing with "llll" in "Motl" – and avoid bullshit claims that the Czech language suffers from a shortage of vowels]. :-)
This paper should reorganize the previous new method to calculate the scattering amplitudes in yet another, deeper way. The publication seems imminent because Natalie Wolchover just wrote a wonderful report about the advance for the Simons Foundation:
I recommend you to read the piece. You will learn about the technical content just a little bit – it wasn't enough for me to learn much more than what is already clear from the previous papers. (Update: I actually have learned new things from that, especially when I checked Nima's and Jaroslav's talks linked at the bottom and compared them with the Simons Foundation popular piece.)
However, you will also notice some excited comments about the simplicity and originality of the new pictures from some very well-known mathematicians and theoretical physicists including the co-authors of the previous papers mentioned above as well as Hodges, Skinner, Deligne (who helped them), Witten, and others.
So I am surely looking forward to the new paper. Unless the famous mathematicians etc. are lying, it should allow us to look at the amplitudes in yet another new way. We're being promised that in the new picture, the physical notions will follow "just" from the amplituhedron's geometry and other physical assumptions may be erased – a vowed progress relatively to the positive Grassmannians. The amplitudes should "literally" be the volume of the jewels again (but the volume form isn't constant, it has factors like the inverse distance from the faces) and the shape of the jewel should be determined by some brand new principles – a convex envelope of a sort. Faces of the jewel should correspond to the cuts.
Nima's and Jaroslav's new calculations should increase the efficiency and "covariance" relatively to the BCFW-like twistor prescriptions which had to divide the jewel to many pieces and sum them; they can sometimes find the full result without summing anything. An amplitude exact up to 6 loops was calculated as a volume of a shape only and verified to agre with laborious multipiece old calculations. They also claim to have found a "master amplituhedron" with infinitely many faces in infinitely many dimensions which should now be as important as the circle in two dimensions. ;-) Its volume counts the "total amplitude" (?) of all processes; faces of this master jewel harbor the amplitudes for processes with finite collections of particles.
To be honest, I am probably going to look at the slides of Trnka's 1-month-old talk in Utrecht, NL and Nima's talk in Italy a week later. ;-) Other links are available via the Wikipedia's entry for the amplituhedron and via Google.