Point… hypercube

The sequence point, segment, cube can be extended indefinitely. 

At each step, one starts from a figure, defined in an n-1 dimensional space, a new direction is fixed, perpendicular to all the previous ones (!), and the figure is dragged in this new direction, as when, to pass from 2 to 3 dimensions, the prism is built on a square to obtain the cube. After the cube we have the 4-dimensional hypercube, but we can go on and on, ad infinitum…

Starting from n=4 it becomes difficult to visualize the process because our imagination was formed by living in a world that has (apparently) only three spatial dimensions. But a geometry in a four-dimensional space has no contradiction and can be studied and explored. And it’s a fascinating exploration. It may also have a practical utility.

The numbers on the left indicate the number of dimensions. Click on a number to switch from the current dimension to the indicated one (the path passes through all intermediate dimensions). When the dimension is greater than 2 you can click & drag with the mouse on the left image to rotate the model.

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