Hyperbolic architecture would be a very different process from boring euclidian architecture. So much so, that selling real estate in hyperbolic space would run into many problems. In the sequel, I assume a hyperbolic space with fixed curvature. This would ideally be true if we lived in a hyperbolic space (except that little things like mass in the general theory of relativity would break it); physicists in particular would like this model as it assures them of a homogeneous space. Without fixed curvature, you can get very different results, but then the most important real estate maxim undoubtably becomes "location is everything".

The most important thing to remember when doing hyperbolic architecture (and especially interior design) is the Gauss-Bonnet theorem. For every polygon, the sum of the angles is "too little" (compared to euclidian space), but you can calculate the area directly from what's missing! For space with fixed curvature, Gauss-Bonnet says that the area of a polygon is proportional to the "missing angle" between the sum of its angles and the sum of such a polygon in euclidian space. For instance, we can have squares with all angles 60° and with all angles 75°. 4×60°=240°, so 120° are "missing" compared to a "flat" quadrilateral in the 60°-square, while 360°-4×75°=60°, so the 75°-square is missing only 60°. Thus, *the 75°-square has half the area of the 60°-square*!

The profoundness of this statement goes well beyond being a sleazy real-estate salesperson, but that's material for another node. It's enough for you, Mr. Prospective Hyper Salesperson, to know that you can compute the area of a house just by looking at its corners! Stun your sellers and buyers with your knowledge of geometry! Run all surveyors out of business!!

Ahem. An illustration. A 17-gon with all right angles would be *big*! It's missing 15×180°-17×90°=1170°; compared to a 17-gon with all angles 120° (itself not too small, given that we have 17-gons going almost all the way up to 159°!), which misses 15×180°-17×120°=660°, it's almost twice the area.

There's a fairly intuitive reason for all this. Small patches of hyperbolic space look a lot like flat space (just like the small patches of the Earth we build out houses on are spherical but look a lot like flat space, and we use euclidian geometry on them). To get the "missing angle" effect of hyperbolic geometry, we need to look at bigger patches.

The wily hyperbolic salesperson won't wax elliptic about this. A client walking in to buy a right 17-gon is going to be spending *BIG HYPERBUCKS*!

"But then again, if you've gone hyperbolic, you should buy a house big enough to show it!"

The Gauss-Bonnet theorem also applies to spherical geometry, but "in reverse" (spheres have positive curvature, while hyperbolic space has negative curvature): the area of a polygon is proportional to the *excess* of the sum of its angles over the sum of angles of a similar polygon in flat space. Of course, spheres are compact, so the amount of real estate you can sell is severely limited; this will raise prices above $0/m^{2}, if you believe in free markets and other fairy tales, but will reduce you chances of finding someone to sell. Still, some real-estate salespeople here on Earth bill themselves "euclidian", when in fact they're quite spherical. And the corners of a room in my house don't add up to 360°, but that's more due to builders' incompetence than to spherical geometry.

Don't forget your commission!

Thanks to unperson for pointing out we want fixed curvature for homogeneity, not for anisotropy. You expect

*me*to know about

**physics**?