Lp.pdb

# These are the vectors in Schiemann's S2 lattice of length <= sqrt(8).

# This is (loosely) a Protein Data Bank format file.
# <https://en.wikipedia.org/wiki/Protein_Data_Bank_(file_format)>

# I tried to find an embedding with rational coordinates, but I don't think
# it's possible. Here I use (1 + sqrt(61)) / 4 = 2.2026 and
# (1 - sqrt(61)) / 4 = -1.7026.


# To verify correctness, note that the Gram matrix can be generated in PARI/GP
# following command based on this script:
# <https://gist.githubusercontent.com/maresb/4ccc43ce07bc9e44e031aa9837a5104f/raw/schiemann.gp>

#  colvectors_ALp~ * ALp * colvectors_ALp


#     [ 8 -1  0]
#     [-1  8 -1]
#     [ 0 -1  4]

# It can be verified that the Gram matrix is correct with respect to the coordinates
# of atoms 6, 4, 2.


ATOM      1  A   HIS A   1      0.0000  0.0000  0.0000  1.00 25.00           C
ATOM      2  D   HIS A   1      0.0000  0.0000  2.0000  1.00 25.00           S
ATOM      3  E   HIS A   1      0.0000  0.0000 -2.0000  1.00 25.00           S
ATOM      4  F   HIS A   1      1.7026 -2.2026 -0.5000  1.00 25.00           O
ATOM      5  G   HIS A   1     -1.7026  2.2026  0.5000  1.00 25.00           O
ATOM      6  B   HIS A   1      2.0000  2.0000  0.0000  1.00 25.00           O
ATOM      7  C   HIS A   1     -2.0000 -2.0000  0.0000  1.00 25.00           O

CONECT 1    2    3    4
CONECT 1    5    6    7