An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian.

A full circle corresponds to an angle of 2π radians.

3.14159265358979323846264338327950288419716939937510

1. Pi is a constant.
2. Pi is an irrational number.
3. Pi is a transcendental number.
4. Pi is a non-repeating number – no pattern has been identified using computer analysis within over twelve trillion places.
5. Pi ( π ) is the exact ratio of the circumference of a circle to its diameter.   It is that simple.

Thank you, Wikipedia, for the graphics (above) that demonstrate this simple definition.  There are over 45 Wikipedia articles about pi.

So, what do you make of it?  What is going on?

Perhaps a few more questions and comments would help.

1. What is it about a circle and sphere that pi is always-always- always true?
2. How does a number become a constant, irrational and transcendental all at the same time?
3. Let us compare pi to other unique numbers that have a special role among all numbers.  These are e, 0, 1, and I. They are all magical, but π stands out. So, let’s ask, “What are the shared qualities of these numbers?” Let’s study them to see if we can find any necessary relations.
4. We have the ratio between a circle and a line. Perhaps this is the fundamental transformation between the finite and infinite? Are circles and spheres always implicating or imputing the infinite?

That is a big question and enough to ponder for awhile.

Notwithstanding, there are many more questions to ask.

• Is this ratio between a line and a sphere the simple tension that is the foundation for the unknown, the creative, the unique and the new? 
• Why not?  Come on now, loosen up a little. 
• Does everything have to be overly complicated?  Doesn’t everything start simply?

Some speculations: Pi may be the key to unlock the small-scale universe within the big Board-little universe
1.   To get to the application of pi  within the Planck Units, we’ll need to emerge from the singularity of the Planck Units.  Is the radian a key to understanding this process?  First, a radius is extended from the singularity.  A radius extends into the preconditions for space and time, a now emergent small-scale universe. It makes that first arc equal to its own length.  It does it again and again and again and again and again (six radians) and then makes that last leap, 2 pi, to complete the circle. Is this a reasonable scenario? Why? Why not?

2. We need to run through dozens of scenarios, often, and slowly and carefully.  What scenarios are perfect and obvious?

3. We are at the singularity of the Planck Units.  We are establishing the foundations for the physical world.  If all things start simply, this must be the place to start.  It doesn’t get more simple and more mysterious. Nothing is a mistake, everything comes from a perfection to a space-time moment, so what could possibly happen?

What happens within the first six doublings?    (to be continued)

For further discussion:
1.  Is the Small-scale Universe the basis for the homogeneity and isotropy of space and time?
2.  Does everything in the universe share some part of the Small-Scale universe?
3.  How is Planck Temperature calculated?  Does it begin with the other Planck Units and expand from that figure at the first notation?

Note:  All of human history has occurred in the last doubling.  Yet, all doublings remain active and current and dynamic.  Continuity trumps time. Symmetries trump space.

What does sleep have to do with anything?  If all time is current, within the moment, we particularize by the day and uniquely within a given waking day.  Sleep seems to bring us into the infinite.  Dreams seem to be the helter-skelter bridge between the finite and infinite.  It seems that these naïve thoughts are worth exploring further.