The structural and electronic properties of self-lacing atomic chains on Pt modified Ge(001) surfaces have been studied using low-temperature scanning tunnelling microscopy and spectroscopy. The self-lacing chains have a cross section of only one atom, are perfectly straight, thousands of atoms long and virtually defect free. The atomic chains are composed of dimers that have their bonds aligned in a direction parallel to the chain direction. At low temperatures the atomic chains undergo a Peierls transition: the periodicity of the chains doubles from a 2 × to a 4 × periodicity and an energy gap opens up. Furthermore, at low temperatures (T<80 K) novel quasi-one-dimensional electronic states are found. These quasi-one-dimensional electronic states originate from an electronic state of the underlying terrace that is confined between the atomic chains.