[Bill S (OP)]

Is there a point at which no further division of an angle is practically possible?  In the case of angular measurement, one might be tempted to suggest that a limit would be reached when the segment of a circle subtended by the angle in question became so small that it reached the Planck length.  Obviously, this would not work, as, by the simple expedient of lengthening the radius of the circle, the length of the segment would be increased without any change in the angle.  My feeling is that quantization of an angle might not be meaningful, but I’m open to correction.

[alancalverd]

Beware! The Planck length has no special physical significance, it is merely a convenient normalisation

In 1899 Max Planck suggested that there existed some fundamental natural units for length, mass, time and energy.[5][6] These he derived using dimensional analysis, using only the Newton gravitational constant, the speed of light and the "unit of action", which later became the Planck constant. The natural units he further derived became known as the "Planck length", the "Planck mass", the "Planck time" and the "Planck energy".

and whilst physics gets interesting around 10^-35m, it doesn't represent anything like a limit.

[jeffreyH]

If you look carefully at what Alan posted you will see the circular nature of the definition. Planck used G to derive his units, which is only determined by direct measurement. You can derive G from the Planck units, which themselves can be used in deriving G, etc, etc ad infinitum. You have no fixed reference. As far as angles go you summed the issue there up nicely. It is always the radius that makes the difference.

[Bored chemist]

An angle is, mathematically, a ratio- as such it's dimensionless and I can't see how that will work as an analogy to length or time.