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MichaelGibbs has written a Java Applet that shows 4-dimensionalpolytopes, and lets you rotate them.

He has put his Applet is on oneof his web pages. Here, on this page, is a version of hisApplet:

Draw4D Applet

Here is an excerpt from what Michael Gibbs says about his Appleton his webpage:


This applet does not run on some older Java Virtual Machines. In particular, if you have an iMac with system 8.5 that is using Apple's Macintosh Runtime for Java (MRJ) version 2.0, you should upgrade to the latest version of MRJ, which can be downloaded by going to URL http://asu.info.apple.com/. The applet runs fine on Macintosh Runtime for Java (MRJ) version 2.1.4. I have had no problem with the applet on any PCs.

The ... applet will let you rotate fifteen different polytopes.

There are six scroll bars which rotate the image about the six principal planes. The ones on the right and bottom are the rotations in normal 3-dimensional space.

  • Far right - rotates up vectors towards you
  • Near right - rotates image clockwise
  • Bottom - rotates left vectors towards you
  • Far left - rotates up vectors into the fourth dimension
  • Near left - rotates out vectors into the fourth dimension
  • Top - rotates left vectors into the fourth dimension

The fifteen polytopes the applet will rotate are:

  • Hypercube (4-D equivalent of a cube)
  • Simplex (4-D equivalent of a tetrahedron)
  • Cross Polytope (4-D equivalent of an octahedron)
  • 24-Cell
  • 120-Cell (4-D equivalent of an dodecahedron)
  • 600-Cell (4-D equivalent of an icosahedron)
  • Truncated Simplex
  • Truncated Hypercube
  • 3x3 Cube
  • 3x3 Cross
  • 6x6 Cube
  • 6x6 Cross
  • Icosa-Cylinder (Icosahedral prism)
  • Icosa-Cone (Icosahedral pyramid)
  • Hypercubic Grid (2x2x2x2)

The polytope may be viewed in any of 5 modes:

  • Monoscopic white lines
  • Stereoscopic white lines
  • Monoscopic colored lines
  • Stereoscopic colored lines
  • Stereoscopic red/blue

There are two stereo modes:

  • Cross eyed
  • Wall eyed

In cross-eyed mode, your left eye looks at the right image and your right eye looks at the left image to get the 3-D effect. In wall-eyed mode, your left eye looks at the left image and your right eye looks at the right image.

The colors shade through the spectrum according to the location in the fourth dimension. The colors go through red, orange, yellow, green, cyan, blue, magenta, and gray, with red being the most negative values and gray being the most positive values.

The red/blue mode gives a good stereoscopic effect through red and blue 3-D glasses. The other stereoscopic modes require you to cross (or uncross) your eyes to get an image. If the red lens is over your left eye and the blue lens over your right eye, use the wall-eyed setting. If they are blue left, red right, use the cross-eyed setting.

Polytope Properties

Polytope

Vertices

Edges

Faces

Volumes

Self-Dual

Central

Hypercube

16

32

24

8

NO

YES

Simplex

5

10

10

5

YES

NO

Cross

8

24

32

16

NO

YES

24-Cell

24

96

96

24

YES

YES

120-Cell

600

1200

720

120

NO

YES

600-Cell

120

720

1200

600

NO

YES

Truncated Simplex

10

30

30

10

YES

NO

Truncated Hypercube

32

96

88

24

NO

YES

3x3 Cube

9

18

15

6

NO

NO

3x3 Cross

6

15

18

9

NO

NO

6x6 Cube

36

72

48

12

NO

YES

6x6 Cross

12

48

72

36

NO

YES

Icosa-Cylinder

24

72

70

22

NO

YES

Icosa-Cone

13

42

50

21

NO

NO

Hypercubic Grid

81

216

216

96

NO

YES

The first six of these are the regular polytopes in four dimensions. A regular polytope has every edge the same length, every face the same shape, and every volume the same shape.

The dual of a polytope is gotten by replacing each vertex by a volume, edge by a face, face by an edge, and volume by a vertex. The hypercube and cross polytopes are duals of one another, as are the 120-Cell and 600-Cell. A polytope is self-dual if its dual is a polytope of the same type.

The column labelled Central indicates whether the polytope has central symmetry, that is, if it is unchanged by reflection through its center.

As you can see from the table, the 24-Cell is very special, being the only centrally symmetric self-dual polytope. The only other polytopes with this property are the polygons in two dimensions with even numbers of edges. The 24-Cell is also an orientable polytope, which means that it is possible to assign a direction to each of its edges such that every two dimensional face is bounded by a cycle. All polygons have this property, and in three dimensions, the octahedron has this property. None of the other polytopes shown here are orientable. The 24-Cell is unique in having no analogue in any other dimension.

Note that the Hypercubic Grid is not technically a polytope, but is useful in visualizing hypercubic lattices. The truncated simplex is a polytope, but not a regular polytope, since it has 5 tetrahedral volumes and 5 octahedral volumes. The truncated hypercube has 16 tetrahedral faces and 8 truncated cube faces.

The 3x3 Cube, 3x3 Cross, 6x6 Cube, and 6x6 Cross are specific instances of the general MxN Cube and MxN Cross. The hypercube is the 4x4 Cube, and the cross polytope is the 4x4 Cross. The MxN Cube and the MxN Cross are dual polytopes. In both cases, start with a M sided polygon in the X1-X2 plane and a N sided polygon in the X3-X4 plane.

The MxN Cube is the product of these two polygons having MN vertices. Each point is a vertex of both an M sided polygon and an N sided polygon. There are 2MN edges, 2 for each point, running to the next vertex on the M sided polygon and to the next vertex on the N sided polygon. There are MN+M+N faces. M of them are N sided polygons, N of them are M sided polygons, and MN of them are rectangles. The MxN Cube can be viewed as a ring of M N-prisms wrapped through a ring of N M-prisms, giving a total of M+N volumes. Projecting the X4 dimension down to the X1 dimension makes the MxN Cube look like a (prismatic) torus.

The MxN Cross is the sum of the two polygons, having M+N points, the M sided polygon with its X3 and X4 coordinates equal to zero and the N sided polygon with its X1 and X2 coordinates equal to zero. Every point on one polygon is connected to all points in the other polygon, giving MN+M+N edges, not necessarily all the same length. There are 2MN triangular faces formed by going from a point on one polygon, to two adjacent points on the other polygon. There are MN tetrahedral volumes, containing two adjacent points from each polygon.

Polytope

Vertices

Edges

Faces

Volumes

MxN Cube

MN

2MN

MN+M+N

M+N

MxN Cross

M+N

MN+M+N

2MN

MN

Tiling

Of these polytopes, the following will tile flat four dimensional space:

  • Hypercube
  • Cross Polytope
  • 24 Cell
  • 3x3 Cube
  • 3x4 Cube
  • 3x6 Cube
  • 4x6 Cube
  • 6x6 Cube


 


 

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