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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 Dimensional Tetrahedron
3 Dimensional Tetrahedron
3 Dimensional Tetrahedron

5-Cell or Pentachron or Hypertetrahedron

8-Cell or Tesseract or Hypercube

4-orthoplex or hexadecachoron

3 Dimensional Tetrahedron
3 Dimensional Tetrahedron

24-Cell or Octaplex or Polyoctahedron

120-Cell or Hyperdodecahedron

600-Cell or Hexacosichoron

mysterious looking rotations of the tessseract
Hpercube Rotation / Involution

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)

tesseract cubes

Here is the process for constructing a hyper-pyramid



4-d sierpenski gasketA 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

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!
4 dimensional sphere .. looks like a brain!

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