# What is the proof that rectangles do not exist in hyperbolic geometry?

I am in need of help figuring this out-- If the only straight lines in hyperbolic geometry are those that pass through the center, then isn't there a right angle? (horizontal and vertical) Which fulfills that requirement of the definition of a rectangle. That leaves the other two sides as hyperbolic lines with negative curve and extending infinitely, resulting in three acute angles, right? Aren't the two straight lines passing through the center parallel to their opposite hyperbolic line? Which also fulfills the definition of a rectangle.
I'm sure I am overlooking something or have gotten myself very confused. Please help.

• Since the sum of the angles in a triangle is less than $\pi$, the sum of the angles in a convex quadrilateral is less than $2\pi$. – egreg Jun 11 '18 at 16:07
• Just to clear up some obvioius confusions. First, there is no center in the hyperbolic plane just as the euclidean plane has no center. Second, the lines in the hyperbolic plane are just as straignt as those in the euclidean plane. Neither of them are curved, otherwise they would be called "curved lines" or curves. You are probably thinking of a model of the hyperbolic plane in the interior of a disk in the euclidean plane. The hyperbolic model lines may or may not "look" straight. In any model, all hyperbolic lines are regarded by definition as staright. – Somos Jan 19 at 0:37

## 2 Answers

It's not clear to me what you mean by "parallel" geodesic in the hyperbolic plane. Those that meet at an infinitely distant point?

Also, it is not clear what definition of rectangle you are using. One possible definition involves having four right angles. This cannot happen in the hyperbolic plane. Indeed, the sum of interior angles in every geodesic triangle is strictly less than $\pi$. Since any quadrilateral can be divided into two triangles, the sum of its interior angles is strictly less than $2\pi$.

A very illuminating approach to this problem would be to look at Saccheri and Lambert quadrilaterals in the Hyperbolic plane. Depending on your background, you might consider using NonEuclid (http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html) as a way to help you visualize the constructions and lines in the Hyperbolic plane.