In Texas kids learn about the number “pi” in 6th grade. I do not know how it is in Kansas. But we are not in Kansas anyway, so we can learn it now.
We are not in Texas, and we are not in Kansas. We are on the unit disk with hyperbolic geometry defined by the transitive action of SU(1,1) – as it was discussed in the recent series of posts. We know the line element:
We can now ask this pressing question: what is the ratio of the circumference of a circle to its radius?
Any point of the disk is as good as any other, but in the standard coordinates that we are using the origin of the disk is most convenient. Let us introduce polar coordinates
We need to express our line element in terms of We calculate
Consider now a circle with and running from to What is the length of its circumference? We have to integrate Going around circumference is constant, so Therefore
What is the length of its radius? Along the radius is constant, so , so
From the last equation we get
Substituting into the formula for :
Looking at the graph of the function
we see that the ratio circumference to radius on the disk is always greater than In fact the ratio is growing faster and faster with increasing .