Multi Dimensional Geometry The
geometry of symmetrical shapes familiar from 3-D, viewed
in their 4th or higher dimensional aspects.
Here is how we make a hypercube then we can watch it's
mysterious looking rotations
below: a square ascends the dimensional ladder - from
left: 2-d square, 3-d cube, 4-d tesseract and 5-d hypercube
a view of the same process seen from another angle
In Geometry another dimension is simply another direction
at right-angles to the previous -
since this creates a whole new type of space we can only
view the "shadows" of 4-D objects in 3-Space.
The shadow only r
epresents a vague outline of the true 4-D form. For example
note that in Higher Dimensional space Rotation is percieved
by us as
"turning inside out" as well as our normal concept
of rotationJust as an object in 3-space creates
a flat 2-D shadow that
changes shape when rotated, 4-dimensional objects can
only be visualized as 'shadows' that are 3-D forms
which change when the objects is rotated in 4-Space.
Below is a rotating Four Dimensional
Cube seen as it's shadow in 3-D space.
Notice how it fluidly turns inside out
from math.harvard.edu