Controls Screen Shots
The resulting numerical W-value is displayed in the W-Display text box.
Dragging the W-slider will generally stop the animation (unless the animation has no W-shift).
The slider has 1000 increments but you will not be able to mouse-drag it with that precision due to the limited pixel-resolution of your monitor. However you can click on the slider bar above or below the pointer to make it move by one increment, causing the W-value to change by 0.002. Note that during animation, the W-value will generally change by a different increment, dictated by the W-Shift resolution.
When you resize the window, the contents of the applet are automatically resized. In a 2-image (Stereo) Viewing mode, only the vertical dimension is taken into account. This means that you have to make the window wide enough to show the two images, or adjust the Size Slider and/or Separation Slider so that the images fit. In a 1-image (Mono) Viewing mode, the size of the contents is determined by the smallest of the vertical and horizontal dimensions, so a square Graphics Panel is most efficient.
After you detach the applet, it is nice to "minimize" the browser to get it out of the way. Don't close the browser though, or the applet will go away.
Depending on your computer's processor speed, the processing time for an animation frame may be longer than the specified frame delay. Under these circumstances, the animation rate will vary, going faster when the calculation is easy, as it is with extreme W-values.
|Polytope Object||3D Cells||Faces||Edges||Vertices||Vertex Figure|
|Cross Polytope||16 tetrahedra||32||24||8||octahedron|
|FC 600-cell||1200 dipyramids||?||?||?||multiple|
|FC 120-cell||720 dipyramids||?||?||?||multiple|
|FC 24-cell||96 dipyramids||?||?||?||multiple|
|FC Hypercube||24 dipyramids||?||?||?||multiple|
|FC Cross Polytope||32 dipyramids||?||?||?||multiple|
|FC Simplex||10 dipyramids||?||?||?||multiple|
The FC (Face Center) polytopes are built by placing hyperplanes at all the face-centers of the corresponding regular polytopes. The hyperplanes are tangent to the hypersphere that contains all the face-centers. The FC polytope is the volume of hyperspace that is bounded by all these hyperplanes. This process is similar to the way a rhombic dodecahedron can be made by placing planes at all the edge-centers of a cube or octahedron. I imagine that these FC polytopes have their own names but I don't know what they are.
For the regular (non-FC) polytopes:
With extreme W-values, you will see a Cell when the Orientation is Cell-first. You will see a Vertex Figure when the Orientation is Vertex-first.
For the FC polytopes:
With extreme W-values, you will see a Cell when the Orientation is Face-first.
Obviously when moving a 3-dimensional polyhedron across a 2-D plane, the orientation can be Vertex-First, Edge-First, or Face-First. In 4 dimensions, you have the additional freedom to orient the polytope Cell-First. This is because the 3-space which contains a cell is a hyperplane which can be aligned to be perpendicular to the W-axis.
|A Dodecahedral Cell :|
For the FC Polytopes, the orientation choice indicates the orientation of the corresponding regular polytope. To view a cell of the FC polytope, chose the Face-First orientation and adjust the W-Shift to a value near -1.0 or 1.0.
The W-Range choices are:
For example, R1 is the 4-space distance from the center of the regular polytope to the center of an edge (an edge is 1-dimensional, hence "R1"). When a regular polytope is oriented Edge First, it will just begin to intersect our 3-dimensional space when it is W-shifted by that distance. So with the Edge First orientation it is appropriate to make the -1.0 and 1.0 limits of the W-Slider equal to R1.
R0 > R1 > R2 > R3
E.g. for a hypercube with an edge-length of 2.0:
|R0||sqrt(4) = 2|
|R1||sqrt(3) = 1.732|
|R2||sqrt(2) = 1.414|
|R3||sqrt(1) = 1.0|
The more complex the polytope, the less the spread between R0 and R3.
If you are performing arbitrary 4-space rotations (via animation or by shift-dragging), you should probably set the W-range to the largest range: R0.
For the FC Polytopes, the the wrange is chosen empirically so that the polytope just disappears as the W-slider is moved to -1.0 or 1.0.
This control assigns colors to the polytope cells. These become the colors of the faces of the polyhedron that is seen in 3 dimensions.
In the following choices, the W- prefix means that the cells are grouped by their W-position prior to assigning colors, and all cells with the same W-position are assigned the same color.
Rainbow means that the colors are assigned from a rainbow spectrum (red through violet).
Mirrored means that the color assignments are mirrored across the W=0 hyperplane.
The Color Scheme choices are: