Rapid cooling and collapse in a thermally unstable medium can lead to filamentary structures that are comparable to those observed in old supernova remnants. Here, simple formulae are derived for the shape of a filament forming in this manner. The cross-sectional shape of a filament will flatten and the ratio of major to minor axes (aspect ratio) increases in proportion to the density as it becomes sheet-like. Thus the aspect ratio of observed filaments is expected to be of the same order as the temperature change during the collapse, T1/T0. In very old remnants the filaments may appear to be tubes or sheets depending on perspective with aspect ratios less than 10. This model predicts a correlation between optical morphology and the temperature of the shock-heated gas. For T1 between 105K and 106K, the thin sheet that forms is likely to split up through asymmetric warping disturbances, leading to several parallel striations, consistent with observations in the Cygnus Loop. Above 106K, the instability may cause the sheet to degenerate before it has cooled to 104K. Hence, an absence of well-defined filaments is expected near X-ray emitting regions.