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The Dalí cross, a net of a tesseract The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.. In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. [1]
The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first perspective projection, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube. Note that the other 5 faces of the cube are not seen here.
In the process of cantellation, a polytope's 2-faces are effectively shrunk.The rhombicuboctahedron can be called a cantellated cube, since if its six faces are shrunk in their respective planes, each vertex will separate into the three vertices of the rhombicuboctahedron's triangles, and each edge will separate into two of the opposite edges of the rhombicuboctahedrons twelve non-axial squares.
The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows: The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract.It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.
The tesseract is one of 6 convex regular 4-polytopes. In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope.They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not).
The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular cubic pyramid is 1/8. The regular 24-cell has cubic pyramids around every vertex. Placing 8 cubic pyramids on the cubic bounding cells of a tesseract is Gosset's construction [2] of the 24-cell. Thus the 24-cell is constructed from exactly 16 cubic ...