Offset Hoop Technique, first look

Do not read this in order.  Go look at all the pictures, then come back and read the words, then go look at the pictures.  Repeat as needed.

Hoops, like the 210/180 from the Full Page gear set, have an interesting property.  There are only some points on the hoop where there teeth on the inside and on the outside of the hoop line up.  Lets call this an alignment point.

On the 210/180 there is an alignment point every 7 outer teeth, or 6 inner teeth.  This is because on a 210 gear each tooth is 1.714... degrees and on the 180 gear each tooth is 2 degrees.  7*1.714... = 12, and 6*2=12
If you look at each side of the hoop and cancel the common prime factors that will tell you the periodicity of the alignment points.

When you put the 210/180 hoop in the 210 ring in the frame you can impact the orientation of the 180 by where you place the alignment point.  I'll explain.
Think of the 3 o'clock position on the 210 frame as our default starting position.  It has a dot marking it.  For many gear sets that position in the ring has a dart making it easy to find.
With the Alignment Point at the default starting position the 180 side of the 210/180 hoop is also in precise alignment with the default starting position.
Now imagine lifting the 210/180 ring and rotating it one tooth counter clockwise (CCW).  There will be a tooth in the 210 side of the ring at the default starting position, but the 180 inside of the 210/180 ring won't quite align anymore.  It is 0.286 degrees clockwise from the default starting position.  If you repeat these steps 7 times you will be back at a position where there is an alignment point in line with the default starting position.

You can use each of these intermediate positions to make a design with a gear that is at a reproducible and different starting point.  This means that there are 1260 starting points for a design when using the 210/180 hoop like this.

Visualizing the steps from one alignment point to the next.  Starting at teal.  Then next step takes you to red, where you can see the difference between the 210 (no boarder) and the 180 (black boarder) lines that show the beginning of the offset of the two sets of gears.  Progressing through Yellow, Blue, Purple, Green, Black, and finally Teal.  Each colour step show a widening gap between the 210 and 180 alignment until the 180 black line aligns with the 210 teal line.

If you use 7 of the adjacent steps, which is the same space as making one design and counting one tooth over and repeating it, you can fit all these lines into it.  At this density there is no white gap between the lines even at their maximum separation.

If you use every second step, effectively 3 lines per tooth, then you get this.  Still very dense but showing faint whisps of white between the lines.

I made a magnet with an arrow on it to help me keep track of where the current position on the 210 ring was.
In this photo set I use every second step to make the design.
Design 1 (black), no offset.  Hoop is at the alignment point and the default start position. 
Design 2 (red), the hoop has been stepped 2 steps CCW.  The lines on the hoop help me keep track of which 180 tooth corresponds to which 210 tooth.  The 60 gear is put on the tooth that lines up with the default start position (marked by the magnet with the arrow).
Design 3 (green), the hoop has stepped another 2 steps CCW.  The 60 gear is still positioned to align with the default start position, but is now shifted most of a gear tooth clockwise (CW)
Design 4 (black), the arrow magnet has been moved CW 1 step and the hoop has reset the alignment point back to starting and is aligned with the arrow magnet.

Different hoops have different ratios.  By combining (with tape) the 210/180 and the 180/140 I made a 210/140 where every 3 teeth on the 210 are 2 teeth on the 140.  This means that it can only make one extra step between teeth.  In the pictures above the black lines are a regular design repeated 5 times each with one step of the 60 CW.  The red lines are two repetitions of the design halfway between the black ones.  Effectively at 1.5, and 2.5 teeth CW.