Multi-Dimensional or Hyper-Dimensional Geometry
The
geometry of symmetrical shapes familiar from 3-D, 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 4-dimensional Euclidean space, is the space used by mathematicians when studying geometric objects
such as 4-dimensional 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 three-dimensional form to be
rotated
onto
its mirror-image,
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 3-D 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 |
|
|
5-Cell or Pentachron or Hypertetrahedron
|
8-Cell or Tesseract or Hypercube
|
4-orthoplex or hexadecachoron
|
| |
|
|
| 3 DIMENSIONS |
|
|
Cuboctahedron
|
Dodecahedron
|
Icosahedron
|
| 4 DIMENSIONS |
|
|
24-Cell or Octaplex or Polyoctahedron
|
120-Cell or Hyperdodecahedron
|
600-Cell 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: 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 rotation Just 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.
This is the outline of some of the cubes within the hypercube (8 total)
Here is the process for
constructing a hyper-pyramid
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 stroke-based
construction.
Note the four-fold 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) 2191-2200. [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
Close-up view of a projection of the Hyperdodecahedron or 120-cell
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 3-dimensional surface.
Very Interesting!
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