If you're using the 3d printed rings, we'll want the wires to pass through the holes in the rings and get strung with beads or pearls before you connect both ends of the wire. The hanging crystal wires will need to pass through the lamp base before being connected together.
For now we'll work from the bottom up, and prep the three wires that go OUT of each part to the next one in the series. We'll solder the three IN wires to the rings and connect the individual pixels together during final assembly.
More about soldering NeoPixel rings can be found in the Make It Glow: How to Solder NeoPixels Guide.
Solder a red wire to VOUT, a white or yellow wire to A1, and a black wire to G. These wires will eventually connect to the smaller 24 pixel ring.
Solder a red wire to one of the + pads, a black wire to one of the G pads, and a yellow wire to DOUT. You can find detailed instructions on soldering to this ring on the How To Solder NeoPixels guide.
These wires will go out from this ring to the 60 pixel ring. Don't solder an IN wire just yet - we'll use the wires we just soldered to the Circuit Playground for that later on.
This ring comes in 4 segments that you'll need to assemble into a ring. The technique is almost identical to this technique for soldering two NeoPixel strips together.
Tape the rings to your table and match up the pads. Bridge 3 of the 4 segments with solder. If you want extra stability, you can add a small piece of solid core wire (or paperclip, or a staple, or whatever) across the segments.
For the fourth segment, tin all the pads but do NOT bridge the solder together. We'll use this fourth segment as our data input and output section.
Notice also that there are small circular pads marked G and + on the back of each segment. Solder a red wire to one of the + pads and a black wire to one of the G pads. It's helpful to spread your wires out so they're as equidistant as you can make them.
Again, don't solder the IN wires yet. We'll use the wires we just soldered to the 24 pixel ring to bring power, ground, and data to this ring.
I'm using 4 conductor stranded telephone wire for the individual pixels. I cut about 3 feet of wire for each hanging crystal, even though my crystals will mostly hang a lot closer than that. It's easy to cut the wires shorter and much more challenging to lengthen them.
This wire can be tricky to strip without breaking the inner wires. There are purpose-made tools for this, but I didn't have any of those so used my flush-cutters for the outer coating and a pair of good wire strippers for the tiny inner wires.
Strip off about an inch of the outer shielding. Strip 1/4" from each of the internal wires. Solder the yellow wire to IN, the green wire to OUT, the red wire to + and the black wire to G.
Repeat for each pixel. My chandelier has 8 crystals so I wired up 8 pixels. Then I wired a couple extras just to have as a backup since these wires are so tiny and break so easily.