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This month's topics:

"Cuddly Topology"

was the caption chosen by Nature's Emma Maris for this image, part of its 2008 Gallery of "the year's most eye-catching science" (December 17, 2008).

 The hyperbolic plane rendered in crocheted yarn. Image from Nature 456 856, used with permission. The textile sculpture is the work of Daina Taimina (Cornell) and was exhibited in the "Beyond Measure" show at Kettle's Yard in Cambridge, UK. More of Taimina's work can be seen online at the Institute for Figuring: look for "A gallery of Crocheted Hyperbolic Models;" other hyperbolic activity there under "Crocheted Hyperbolics with Cabinet Magazine." Another online show is in the Eleven Eleven Sculpture Space, which features an artist's statement: "Once I was participating in geometry workshop led by Professor David Henderson. He was showing a paper model of hyperbolic plane that was made using William Thurston's idea of annuli. And then it came in my mind - if one can make it out of a paper, then I should be able to crochet it."

Hiding behind curvature

Imagine a device that could bend light rays around you and make you invisible. This is what Ulf Leondardt (Singapore) and Tomás Tyc (St. Andrews) are proposing in a January 2 2009 Science report, "Broadband Invisibility by Non-Euclidean Cloaking." The physical device that will actually carry out the transformation is not spelled out in detail, but the authors describe how the geometry of a region of space could be changed, keeping the topology intact, to create a pocket of invisibility. To keep things simple I will only describe the 2-dimensional analogue they give to motivate their construction; it proceeds in two steps. First a 2-dimensional spherical surface is spliced into the Euclidean plane: the sphere is sliced open along a proper segment of a meridian line, and the two edges of the cut are identified with the two edges of a same-length cut in the plane. In this illustration the plane is bent to be tangent to the sphere along the splice.

The round sphere and the flat plane are spliced together along a proper segment of a meridian line. The yellow curve represents the path of a light ray in the resulting space: a straight line in the plane crosses the first suture, loops once around a great circle, crosses the second suture and continues in the plane as a straight line. There is such a ray going through any point on the sphere except those on the meridian line and not on the splicing segment (and their antipodal images). Part of the forbidden region is shown in red. Image courtesy of Ulf Leonhardt.

Second step: another sphere is spliced to the first one along a sub-segment of the forbidden region. Light rays coming from the new sphere cannot reach the plane: points on the new sphere are invisible from the outside.

In this image the sliced-open spheres are flattened out into lens shapes bounded by two copies of their slicing segment (this shows that the constructed surface is topologically a plane). The inner one becomes the blank lens shape; it is bordered by two copies of a red (forbidden) segment. The outer one is bordered by two copies of the black segment where it joins the plane. The blue circle (in the plane) and the yellow light ray appear in both images. Note that the coordinate lines in this image match those above, and are not the standard latitudes-meridians on a round sphere. Image courtesy of Ulf Leonhardt.

 One end of the splicing segment. A light ray from 0 through the center. [An invisible region does not contradict the existence of a shortest path (geodesic) between any two points in the space. The discrepancy is caused, in this model, by the presence of points where the metric is singular. These are the ends of the splicing segments. A small circle of radius r about one of those points has circumference close to 4πr. In the images on the right, points on that circle are labeled by their angular distance from an arbitrary fixed point 0. If two points on that circumference are at angular distance greater than π, the shortest path between them on the surface goes through the singularity, and their distance in the surface is exactly 2r. A light ray sent along the shortest path from one of those points toward the other disperses as it passes through the singularity, and shines evenly on all the points at angular distance greater than π from the emitter (this is illustrated in the lower image, in which angles have been compressed so as to give a planar image of the whole circle and the enclosed disc); since there are infinitely many of them, any one of those points gets zero illumination. So any two points in the space are joined by a geodesic, but if that geodesic goes through one of the singularities (as must be the case if one point is in the hidden region and one is in the outside plane), the points are mutually invisible. Thanks to Dennis Sullivan for insight into this seeming paradox. -TP]

Tony Phillips
Stony Brook University
tony at math.sunysb.edu