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We compare a very deep X-ray image of M87, at the center of the VirgoCluster, to high-quality optical images of the low excitation emission-line gasin the same region. There are striking coincidences of detail between the two.We explore the possiblity that this represents a thermal interaction betweenhot gas at 10^7 K and warm gas at 10^4 K. We find two temperatures are presentin the X-ray gas, with the lower more prevelant in the vicinity of the opticalfilaments. Electron conduction from the hot phase to the cooler one provides aquantitatively acceptable energy source for the optical filaments, and we showadditionally that it can do so for the brightest X-ray cluster, Perseus. Ifoperative, conduction in the presence of gas-rich galaxy mergers, may explainthe presence of "cool cores" in clusters of galaxies.Comment: 18 pages, 6 figures, 2 table

X‐Ray and Optical Filaments in M87