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Yeah that’s not correct. Facebook bought into the oculus project in a share swap/cash deal with Lucky Palmer. Carmack joined the project later because he believes in VR in a big way and he contributed very important parts of the rendering methods to greatly improve performance.
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Carmack was interested in making his own VR headset prototype to show off at e3. He found Palmer Luckey on mtbs3d, showing interest in Palmer’s headset prototypes. Carmack duct taped some hardware together with Luckey’s headset and made a demo to show off at e3 2012. Carmack was absolutely responsible for kick-starting VR to where it is today, with the help of Palmer.