Images in Christmas Balls

We describe light-reflection properties of spherically curved mirrors, like balls in the Christmas tree. In particular, we study the position of the image which is formed somewhere beyond the surface of a spherical mirror, when an eye observes the image of a pointlike light source. The considered problem, originally posed by Abu Ali Hasan Ibn al-Haitham -- alias Alhazen -- more than a millennium ago, turned out to have the now well known analytic solution of a biquadratic equation, being still of great relevance, e.g. for the aberration-free construction of telescopes. We do not attempt to perform an exhaustive survey of the rich historical and engineering literature on the subject, but develop a simple pedagogical approach to the issue, which we believe to be of continuing interest in view of its maltreating in many high-school textbooks.Comment: 13 pages, 7 figures plain LaTeX; Also see http://cft.fis.uc.pt/eef/mirrors.htm, revised version has simplified formulas, more transparent for a wider audience, one reference adde

Carbon Fiber Filters

Disclosed is a filter comprised of activated carbon fibers, wherein said filter has a Virus Removal Index (hereafter “VRI”) of at least about 99%, as measured in accordance with the test method described in the specification. The filter may comprise unbound fibers, or the fibers may be bound with a binder to form a composite of fibers. Also disclosed is a method of removing viruses from a liquid, the method comprising contacting the liquid with a filter comprising activated carbon fibers wherein said filter has a VRI of at least about 99%. Also described is an article of manufacture comprising: (a) a filter comprising activated carbon fibers, wherein said filter has a VRI of at least about 99%; and (b) instructions which inform a user that the filter may be used to remove viruses from a liquid