the Johns Hopkins University, L. Schläfli of Bern, W. I.
Stringham of the University of California, W. Killing of
Münster, T. Craig of the Johns Hopkins, R. Lipschitz of
Bonn. R. S. Heath and Killing investigated the kinematics
and mechanics of such a space. Regular solids in -dimensional space were studied by Stringham, Ellery W. Davis
of the University of Nebraska, R. Hoppe of Berlin, and
others. Stringham gave pictures of projections upon our space of regular solids in four dimensions, and Schlegel at
Hagen constructed models of such projections. These are
among the most curious of a series of models published by L. Brill in Darmstadt. It has been pointed out that if a
fourth dimension existed, certain motions could take place which we hold to be impossible. Thus Newcomb showed the
possibility of turning a closed material shell inside out by simple flexure without either stretching or tearing; Klein pointed
out that knots could not be tied; Veronese showed that a
body could be removed from a closed room without breaking the walls; C. S. Peirce proved that a body in four-fold space
either rotates about two axes at once, or cannot rotate without losing one of its dimensions.