I finally completed the massive amount of soldering for this project. I decided that the best way to connect each group of 3 RGB LEDs was to make 28 small wires, 4 for each connection, and solder them from one group of 3 to the other. This task was extremely tedious because I had to cut the wires to the right size, then strip both ends off of each individual wire, then solder it to the correct groups. Here is a picture of how I connected each group of 3 RGB LEDs.
I tested them as I soldered another group to the end to find any shorts in my circuit. I eventually got to the end and tested it out.
Once I was done with the final test, I soldered some breadboard wires to the end of the last group so I can run them out to my arduino board.
Once I uploaded my new code to my arduino board, the RGB LEDs light up blue then flash red in a heartbeat pattern.
My only two problems left are I have a small short in my circuit that causes the red to glow only when the wire is in a specific position and the casing is now too small for my diffusor with all of the extra wires I had to add to connect the RGB LEDs. I will be printing new casing and coils to hold the diffusor and the wires and checking out that short soon.