tpi or the gap between each teeth on knurl is used to calculate the diameters you can use your knurl.

is a good practice the division [(3.14*part diameter)/knurl pitch] be zero. it produce better looking knurls.

mm or inch will procuce same results.

since the pich on knurls wheels are always very fine, there a lot of diameters options we should make our parts. they vary for few 0.anything

There's a little magic when it comes to knurls and on the rare instances I need to do it, I always cut a practice piece. Getting the numbers to line-up as mentioned above is a guideline but, I think cascao misspoke and intended to say the relation should result in an integer number -not zero. I believe the intention was to say, the fractional portion of the result should be zero. In theory, if the number of grooves does not align naturally with the diameter of the part, you'll have an oddball groove that does not overlay on the surface evenly. So here's the problem. Pi (3.141592654...) is not a natural number and it will make the numbers a little hard to come out without a fractional part. That's OK because knurling is not an exact process....

As you clamp down on the part, the metal deforms and the diameter of the part is effectively changing. In addition, the TPI of the knurl becomes less important and the diametral pitch becomes more important. Given all this, even if you adjust a part diameter and knurl TPI so that circumference divided by TPI is an integer (not even, but integer such as 1, 2, 3... ) it will go out the window as soon as you start sinking the knurl into the piece.

This is the same theory as a bolt and the threaded hole it goes in. A 1" bolt and corresponding hole do not have any physical feature that is 1 inch.

If you want ideal knurls, you're probably better off thinking about diametral pitches of both the knurl and the "imaginary" one that gets formed on the part.

All that said, the last couple times I tried to make a pretty knurl I did a couple practice pieces of different diameters. On some, it was easier to establish and maintain a nice knurl. Sadly, I forgot to measure which diameters worked best. I would not be surprised if the correct way of doing this is really about determining the volume of metal that has to be displaced into the die-teeth of the knurl. I could make such calculations if so desired... I doubt it's worth my time.

Regards

Ray C.