  We present a real-space view of one-dimensional (1D) to three-dimensional
(3D) topological materials with 13 representative samples selected from each
class, including 1D trans-polyacetylene, two-dimensional (2D) graphene, and 3D
topological insulators, Dirac semimetals, Weyl semimetals, and nodal-line
semimetals. This review is not intended to present a complete up-to-date list
of publications on topological materials, nor to provide a progress report on
the theoretical concepts and experimental advances, but rather to focus on an
analysis based on the valence-bond model to help the readers gain a more
balanced view of the real-space bonding electron characteristics at the
molecular level versus the reciprocal-space band picture of topological
materials. Starting from a brief review of low-dimensional magnetism with `toy
models' for a 1D Heisenberg antiferromagnetic (HAF) chain, 1D
trans-polyacetylene and 2D graphene are found to have similar conjugated
(\pi)-bond systems, and the Dirac cone is correlated to their unconventional 1D
and 2D conduction mechanisms. Strain-driven and symmetry-protected topological
insulators are introduced from the perspective of material preparation and
valence-electron sharing in the valence-bond model analysis. The valence-bond
models for the newly developed Dirac semimetals, Weyl semimetals, and nodal
line semimetals are examined with more emphasis on the bond length and electron
sharing, which is found consistent with the band picture.
