Element 7, the carbon of this world: five bonds, its handedness, and its diamond.
Carbon bonds by mixing its one s orbital with its three p orbitals into four equal sp³ hybrids, each a quarter s. They point to a tetrahedron's corners, 109.47° apart: methane. Darbon has four p orbitals, along x, y, z and w, and five outer electrons. Mixed, they make five sp⁴ hybrids, each a fifth s, pointing to the corners of a 5-cell 104.48° apart. Its hydride is DH₅, and a chain of n darbons carries 3n + 2 hydrogens where ours carries 2n + 2 (D₂H₈ is darbon's ethane).
A carbon with four different neighbours is chiral: it and its mirror image can't be turned into each other. Darbon with five different neighbours is too, and for the same reason: a 5-cell's turns reach only the even half of the ways to shuffle its corners. But every chiral molecule we know is a 3D object sitting in one slice of 4D, and half a turn through w carries it into its mirror image. In a 4D world our handedness would vanish, and darbon's would be new.
Diamond is two copies of a lattice (face-centred cubic), one bond apart, every atom with four neighbours. The same recipe makes graphene's honeycomb in 2D and, with the 4D lattice A₄, darbon's diamond: five neighbours each at 104.48°. Diamond sits easily in the cube, because a tetrahedron's corners are alternate corners of a cube. A 5-cell's corners are no tesseract's corners. A regular simplex fits a hypercube only when a Hadamard matrix of its size exists, and none has size 5. So darbon's crystals don't come from the hypercube. They have a five-fold symmetry instead (turning its five bond directions round), which no 3D crystal can have but a 4D one can. In 4D the allowed turns are of order 1, 2, 3, 4, 5, 6, 8, 10 and 12.
These are the geometry of hybrid orbitals and of the nets they make, not computed bonding: the solver above handles one atom at a time. Whether darbon would really choose this diamond, or graphite's 4D cousin (3D diamond-like sheets stacked along w, with the fifth bond across), needs molecules: the motion chapter has the laws of motion and the three twists about each bond, and the molecular-dynamics view is next.
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