Hyperspace structures 

Inflated hypercubeProduced by Keith Beardmore 
The 2D analogue is the octagon. You start with a square and create a new vertex at the midpoint of each edge, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one circle. The resulting regular polygon  the octagon, has eight verticies and eight equal edges. For the 3D analogue, you start with a cube and create a new vertex at the midpoint of each edge and also at the centre of each face, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one sphere. The resulting semiregular polyhedron has 24 faces, 48 edges and 26 verticies. For the 4D analogue, you start with a hypercube and create a new vertex at the midpoint of each edge, at the centre of each face and also at the centre of each 3cube, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one hypersphere. The movie here demonstrates what happens if we view a projection the inflated hypercube into threedimensional space as it is rotated around. The movie shows what the 3d projection looks like as we rotate a model inflated hypercube slowly through a complete revolution, revolving about three orthogonal planes at different rates. Note: The projection into three dimensional space was achieved here by selecting a point in 4space and "casting shadows'' from that point onto a 3dimensional slice of the space. The fourth dimension is then made more apparent as extra depth in the image (objects further away in the 4th direction appear to shrink away). The edges of the shape are coloured according to their initial distance (in 4D) from the camera. 

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