# Can we design a geometry where the angle between two lines can increase infinitely?

As a contrast, the maximum angle between two lines is 90° in Euclidean geometry.

I am just thinking people living in the universe with unlimited line angles will be curious that there exists a world where line angles are limited to a certain value, just like people are curious about the fact that in Minkowski geometry there is a maximum speed, i.e., the speed of light.

• How is this a physics and not a Mathematics question? Nov 10, 2022 at 8:16
• @ACuriousMind Because I believe it might be useful to help people intuitively understand the speed of light . Nov 10, 2022 at 8:19

In Special Relativity, one could define the [rapidity] angle between two lightlike directions to infinite since the area of the sector is infinite: $$\int_0^\infty\frac{1}{x}dx\quad\mbox{diverges}$$ https://www.wolframalpha.com/input?i=integral+1%2Fx+from+0+to+inf

The [rapidity] angle between two future-timelike directions has no limit.

So, Minkowski spacetime geometry has this feature.

• All I can say is "Wow" but the comment has to be longer. Question: Is there any practical use for Minkowski spacetime geometry ? Nov 10, 2022 at 18:38
• @JosephDoggie The special theory of relativity and quantum field theory. Nov 10, 2022 at 18:54
1. The exact analogue of angle from Euclidean geometry, in Lorentz geometry, is the rapidity. Its formula is $$\tanh ^{-1} \frac{v}{c}$$, or $$\tanh ^{-1} v$$ with $$c=1$$

2. The analogue of $$v$$ in Euclidean geometry is not the angle. It's the slope $$m$$. $$v$$ and $$m$$ literally equal the slope in the space/spacetime diagrams. Angle is related to slope by the formula $$tan ^{-1} m$$.

3. Euclidean geometry with the metric $$x^2+y^2$$, is the $$c=1$$ version of the general Euclidean geometry . In general, the Euclidean geometry metric can be $$c^2x^2+y^2$$. The equivalent of rotations here would be "going in an Ellipse".

4. The $$c\rightarrow \infty$$ limit, of both Euclidean and Lorentz geometries, is the Galilean geometry. Basically, the invariant metric of both geometries can be written as $$x^2 \pm \frac{1}{c^2}y^2$$. Only $$x^2$$ remains as the invariant quantity in the $$c\rightarrow \infty$$ limit, which corresponds to an "absolute $$x$$ co-ordinate" (like absolute time).

So, a Galilean geometry is sort of what you are looking for. But Galilean transformations do not allow for greater than 90 degree angles. Angles, defined according to $$\tan ^{-1} \frac{v}{c}$$, become 0 in the $$c\rightarrow \infty$$ limit. Your mistake was in identifying the speed limit from relativity, with the 90 degree limit from Euclidean geometry. $$c$$ from relativity is instead identified with the $$c$$ in $$c^2x^2+y^2$$. But this $$c$$ from Euclidean geometry does not represent any "speed limit".