  We calculate the spectrum of photons resulting from electromagnetic cascades
through thermal radiation, and examine the consequences of including triplet
production in these cascades. We assume that the cascade is one-dimensional,
and we find that this approximation is justified in the present work for
thermal radiation with temperature less than $10^{-3} mc^2$. Results are
obtained for both monoenergetic and power-law primary spectra, and for a
variety of path lengths. We find that triplet production is particularly
important in electron--photon cascades through thermal radiation when the
primary energy exceeds $10^5 m^2c^4/kT$ for propagation over small path
lengths. The importance of triplet production decreases as the path length
increases, and it has no effect on saturated cascades.
