MultiDimensional or HyperDimensional Geometry
The
geometry of symmetrical shapes familiar from 3D, viewed
in their 4th or higher dimensional aspects.
"the fourth dimension is a space with literally 4 spatial dimensions, or four mutually orthogonal directions of movement.
This space, known as 4dimensional Euclidean space, is the space used by mathematicians when studying geometric objects
such as 4dimensional polytopes. It is not to be confused with Minkowski space, where time is the fourth dimension;
the
latter space is not a metric space. The possibility of spaces with dimensions higher than three was first studied by
mathematicians
in the 19th century. In 1827 Möbius realized that a fourth dimension would allow a threedimensional form to be
rotated
onto
its mirrorimage,
and by 1853 Schläfli had discovered many polytopes in higher dimensions, although his work was
not published until after his death." (from Wiki)
At Right: A rotating Four Dimensional
Cube seen as it's shadow in 3D space.
Notice how italmost magically turns inside out!
from math.harvard.edu
6 simple 3 dimensional polygons and their 4 dimensional equivalents:
3 DIMENSIONS 


Tetrahedron

Hexahedron/Cube

Octahedron

4 DIMENSIONS 


5Cell or Pentachron or Hypertetrahedron

8Cell or Tesseract or Hypercube

4orthoplex or hexadecachoron




3 DIMENSIONS 


Cuboctahedron

Dodecahedron

Icosahedron

4 DIMENSIONS 


24Cell or Octaplex or Polyoctahedron

120Cell or Hyperdodecahedron

600Cell or Hexacosichoron

mysterious looking rotations of the tessseract
Here is how we make a hypercube starting with a 2 dimensional square
below: a square ascends the dimensional ladder  from
left: 2d square, 3d cube, 4d tesseract and 5d hypercube
a view of the same process seen from another angle
In Geometry another dimension is simply another direction
at rightangles to the previous 
since this creates a whole new type of space we can only
view the "shadows" of 4D objects in 3Space.
The shadow only r
epresents a vague outline of the true 4D 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 rotation Just as an object in 3space creates
a flat 2D shadow that
changes shape when rotated, 4dimensional objects can
only be visualized as 'shadows' that are 3D forms
which change when the objects is rotated in 4Space.
This is the outline of some of the cubes within the hypercube (8 total)
Here is the process for
constructing a hyperpyramid
A paper by Gabe Brisson and Cliff Reiter
describes a method for visualizing generalizations
of the Sierpinski Triangle in any dimension.
The generalization of the Sierpinski Triangle
to three dimensions is a Sierpinski Tetrahedron.
This animation shows the four dimensional version,
rotated so that its structure and symmetry is more apparent.
The structure of these fractals are based on a strokebased
construction.
Note the fourfold branching
and the many Sierpinski Triangles
appearing in planes throughout this example.
1]
Gabriel F. Brisson and Clifford A. Reiter, "Sierpinski
Fractals from Words in High Dimension",
Chaos, Solitons
& Fractals, 5 11 (1995) 21912200. [2] Clifford A. Reiter,
Fractals, Visualization
and J, Iverson Software, Inc., Toronto
(1995).
Visionary Artist Paul
Laffoley's Awesome Design of
an Unfolded Internally Mirrored Hypercube
House
Closeup view of a projection of the Hyperdodecahedron or 120cell
4 dimensional object with the characteristic shape of the human brain???
. The image below is the result of drawing a sphere in four dimensions, with a moderate
adjustment to
one of the dimensional parameters and displaying the result as a 3dimensional surface.
Very Interesting!
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