Space (i.e. our universe and beyond) is supposedly of one of three types: flat (no curvature), hyperbolical (negative curvature) or spherical (positive curvature). Now, based on recent reports by a team of French cosmologists and mathematician Jeff Weeks, a new shape is being proposed. (Note: My thanks to a student for quietly pointing this new theory out to me, after I had just made a presentation to a geometry class on why the shape of space was hyperbolic!)
The proposed new shape of space is that of a dodecahedron, as described in J. Luminet et al's article "Dodecahedral space topology as an explanation for wek wide-angle temperature correlations in the cosmic microwave background" (Nature (October 9, 2003). The new theory is traced back to a 1998 discovery of something known as "dark energy," a cosmic pressure that remains "little-understood."
The authors' abstract in Nature includes an overview of why their measurements no longer "fit" theories about space as being infinitely flat or hyperbolic: "Temperature correlations across the microwave sky match expectations on angular scales narrower than 60° but, contrary to predictions, vanish on scales wider than 60°. Several explanations have been proposed. One natural approach questions the underlying geometry of space—namely, its curvature and topology. In an infinite flat space, waves from the Big Bang would fill the universe on all length scales. The observed lack of temperature correlations on scales beyond 60° means that the broadest waves are missing, perhaps because space itself is not big enough to support them."
Now, I am not sure what all of this means, but perhaps you do. If not, consider Jeff Weeks' more readable, understandable explanation in a National Geographic News article "Universe is Finite, 'Soccer Ball'-Shaped, Study Hints" or the fascinating discussion of space as being a "hall of mirrors" in NewScientist.com's explanation in "Tantalising evidence hints Universe is finite."
Despite the expected obfuscation due to scientific language, this new theory is an example of things you should be discussing with seconday students...it involves the use of real mathematics...and is an example of mathematicians working with other scientists. The suggested web sites are readable and are good examples of the use of mathematics terminology in a meaningful context (i.e. in contrast to the contrived word problems in today's textbooks).