I did a bit of thinking about the connection of the HC and the 340. I don't have anything concrete, but maybe someone can spin this further.
1. The HC contains the letter F (in sign language) and the number 14. So, why not treat the 340 like a chessboard (with rows numbered 1,2,... and columns A,B,...) and see what field F 14 is?
Depending on how you rotate the 340 and in which direction you count you have eight possibilities:
Obviously, because of the inherent symmetry, you get intersecting lines. Interestingly, the corners of the "L" shapes lie on those lines.
2. The HC contains a symbol with four dots arranged around angular shapes. The 340 also contains dots and those prominent "L" shapes. Those shapes define two diagonals.
If you reflect the dots along the diagonals you get the letters R & G (Richard Gaikowsky?), the Zodiac symbol, and three other symbols I can't make anything of.
It's also interesting to note that only four of the six dots can be reflected to letters/symbols, because two of the dots are reflections of each other.
There are only four dots in the HC.
I hope this sparks something in someones brain.