  In a cold-dark-matter universe, cosmological structure formation proceeds in
rough analogy to origami folding. Dark matter occupies a three-dimensional
'sheet' of free- fall observers, non-intersecting in six-dimensional
velocity-position phase space. At early times, the sheet was flat like an
origami sheet, i.e. velocities were essentially zero, but as time passes, the
sheet folds up to form cosmic structure. The present paper further illustrates
this analogy, and clarifies a Lagrangian definition of caustics and streams:
caustics are two-dimensional surfaces in this initial sheet along which it
folds, tessellating Lagrangian space into a set of three-dimensional regions,
i.e. streams. The main scientific result of the paper is that streams may be
colored by only two colors, with no two neighbouring streams (i.e. streams on
either side of a caustic surface) colored the same. The two colors correspond
to positive and negative parities of local Lagrangian volumes. This is a severe
restriction on the connectivity and therefore arrangement of streams in
Lagrangian space, since arbitrarily many colors can be necessary to color a
general arrangement of three-dimensional regions. This stream two-colorability
has consequences from graph theory, which we explain. Then, using N-body
simulations, we test how these caustics correspond in Lagrangian space to the
boundaries of haloes, filaments and walls. We also test how well outer caustics
correspond to a Zel'dovich-approximation prediction.
