Back in February, Nick Richardson wrote at the LRB blog:
Lenin’s values were eternal so should be represented as a cube, Stalin’s appointed architect Alexey Shchusev suggested, because ‘in architecture the cube is eternal’. The decision damaged the visual language of communism for ever.
What damaged the “visual language of communism” was the decision to install a shrine in Red Square with Lenin’s embalmed body inside. A prehistoric temple of the dead at the heart of a revolutionary, forward-looking nation, complemented with prominent Communists buried by or in the Kremlin wall. [To keep it simple, I’m ignoring the Fedorov-Bogdanov resurrection follies.]
The first mausoleum was indeed a large wooden cube with a tetrahedron on top. Who came up with “cube as eternity” first? I have no idea, but Malevich endorsed the concept, a 3D version of his black square. Yet the mausoleum owes no more to Malevich than to ancient burial pyramids of Mesopotamia, Egypt and Mexico. Shchusev, a major Art Nouveau master, was particularly adept at pastiching architectural styles of the past. See the churches he designed under the old regime, and the Kazan Station in Moscow.