the triangular prisms are joined to the truncated tetrahedra. The truncated tetrahedra correspond with the tesseract's vertices, and the triangular prisms correspond with the tesseract's edges. Images Projections In the truncated cuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows: The projection envelope is a non-uniform truncated cube, with longer edges between octagons and shorter edges in the 8 triangles. The irregular octagonal faces of the envelope correspond with the images of 6 of the 8 truncated cuboctahedral cells. The other two truncated cuboctahedral cells project to a truncated cuboctahedron inscribed in the projection envelope. The octagonal faces touch the irregular octagons of the envelope. In the spaces corresponding to a cube's edges lie 12 volumes in the shape of irregular triangular prisms. These are the images, one per pair, of 24 of the triangular prism cells. The remaining 8 triangular prisms project onto the triangular faces of the projection envelope. The remaining 8 spaces, corresponding to a cube's corners, are the images of the 16 truncated tetrahedra, a pair to each space. This layout of cells in projection is similar to that of the cantellated tesseract. Alternative names Cantitruncated tesseract (Norman W. Johnson) Cantitruncated 4-cube Cantitruncated 8-cell Cantitruncated octachoron Great (George Olshevsky) Grit (Jonathan Bowers: for great rhombated tesseract) 012-ambo tesseract (John Conway) Related uniform polytopes It is second in a series of cantitruncated hypercubes: References T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 H.S.M. Coxeter: Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1) Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and