The event horizon of a Kerr black hole is often depicted as being spherical, but this seems to be a reference to the horizon as defined in Boyer-Lindquist coordinates, where horizons appear at a constant value of r. However, Thorne describes the "black hole's horizon bulg[ing] out at its equator in the manner depicted in Figure 7.9" (text and figure on page 293 of my edition of Black Holes And Time Warps). Similar diagrams appear in his 1974 article for Scientific American, "The Search For Black Holes". In that article he states that "A hole rotating at moderate speed ... with a rotation parameter of .866 will be perfectly flat at the poles but will still be be rounded at the equator. Rotation does not affect the size of the equatorial circumference." This all seems to hang together with Taylor & Wheeler's Exploring Black Holes (1st ed.), section F-5, in which a circular polar cross-section of an extremal black hole is shown in BL coordinates, but it is noted that "When ... plotted in terms of the reduced circumference R/M instead of [BL coordinate] r/M, then the radius of the horizon is greater in the equatorial plane than along the axis of rotation, giving the horizon the approximate shape of a hamburger bun." I'd quite like to be able to plot this oblate event horizon in reduced circumference coordinates (although I understand that this isn't strictly possible in Euclidean space beyond the threshold of a = √/2). However, although I can find extensive discussions of the applicability of reduced circumference to the equatorial plane of the Kerr metric, I can't seem to find any discussion of the overall shape of the event horizon in these coordinates, beyond that throwaway line from Taylor & Wheeler. Can anyone help?