Selling real estate in hyperbolic space would be a great thing to do. In the mathematical movie "Selling Real Estate in Hyperbolic Space", we see why. Our hero Mel Slugbait has some convincing arguments for why we should buy land in the lucrative market of hyperbolic space. Why? Well, in hyperbolic geometry, Euclid's fifth postulate gets thrown right out the window. The fifth postulate of Euclidean geometry says that given any line and any point not on that line, there is exactly one line through that point that is parallel to the original line. But in hyperbolic geometry, there can be an infinite number of "lines" that satisfy the condition. (You may have trouble visualizing this until you realize that "lines" in the hyperbolic geometry sense may look like arcs of circles in the Euclidean sense.) Another neat fact about hyperbolic geometry, is that the sum of angles in a triangle add up to less than 180 degrees.

And this is part of what makes real estate in hyperbolic space so attractive. In the geodesic model of hyperbolic geometry, which is used by M. C. Escher in some of his works, the plane is a circle and the lines are arcs that intersect this circle at 90 degree angles. Because the sum of angles of polygons (remember, you can break any polygon into triangles) is always less than what it would be in Euclidean geometry, you can have pentagons, hexagons, and even 17-gons with all right angled sides. And Mel argues that this is just what people want. Wouldn't you want a room with seven or eight right-angled corners? No more worries about where to put that beautiful corner desk or that rectangular prism of a dresser. Another advantage to land in hyperbolic space (aside from it being cheap, which it is, because very few people realize it is for sale) is that there is just so much of it, because the metric used to measure distance is different from the one used in Euclidean geometry.

The movie was humerous and enjoyable. Mel Slugbait had some convincing arguments, but he just reminded me a bit too much of a used-car salesman. He isn't very rigorous, and everything he says should be taken with a grain of salt.

If you find hyperbolic geometry to be intriguing, you might also wish to consider spherical geometry, which better describes the Earth we live on. In spherical geometry, it is possible to have a line and a point such that no parallel (non-intersecting) lines can be drawn through the point. Also, the sum of the degrees in a triangle is greater than 180 degrees.

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/m2, 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.