Screen Saver

Hyper Screen Saver: Freeware for Windows

The Hyper screen saver displays a rotating 4-dimensional object (hypercube or 4-simplex) projected onto 3-space using a 4-D perspective transformation.

Notes

• The axis of rotation (a plane) changes randomly after a few cycles unless you press the H key (hold). Continuing to press the H key toggles hold mode on and off.
• Press C to change the plane of rotation at the completion of the current rotation (even in hold mode).
• Press the F key to speed up the 4-D rotation or S to slow it down.
• Press U, D, R, or L to change the position of the light source.
• Press 3 to toggle 3-D rotation on and off.
• Press O to toggle opaque and wire-frame rendering.
• Pressing other keys tells the screen saver to terminate.

The screen saver fundamentals were based on an example (minimal screen saver) published online by Lucian Wischik (www.wischik.com/scr/). The math of n-dimensional rotation, perspective transformation, and hidden facet removal is my own work.

In 2009, I converted the original Hyper screen saver from Borland C++ to Microsoft Visual C++ and cleaned up the code. In March 2010, I added the hyperbrick option. The Borland version is still here if you really want it (view page source), but I’ve been using only Microsoft C++ in recent years.

Remarks

When a rotating cube in ℝ³ is projected onto a 2-dimensional display surface using a 3-D perspective transformation, the facets that are farther away from the center of projection (CP) appear smaller than the facets that are closer. In a transparent wire-frame rendering, the back facets (squares) seem to shrink and pass through the front facets and then grow as they again become front facets. When shading and hidden-surface removal are done, a facet disappears from view whenever it faces away from the CP and reappears when it rotates so that it faces the CP again.

Something similar happens when a rotating hypercube in ℝ⁴ is projected down to ℝ³ using a 4-D perspective transformation. In a transparent rendering, the back facets (which are 3-dimensional cubes in this case) shrink and pass through the front facets before starting to grow again. With shading and hidden-facet removal, the facets appear and disappear (but the rendering is complicated by the need to perform another projection to map the 3-D image onto a 2-dimensional display surface).

The animated GIF at the upper right corner of this page shows the rotating hypercube. Clicking the image selects a different version with a different rotation plane (using Javascript).

Since I started experimenting with the hyperbrick shape (see below), I’ve decided I like it better than the simplex or the hypercube. I considered a monolith à la 2001: A Space Odyssey (1 × 4 × 9 × 16), but I think using the golden ratio makes a more æsthetically pleasing image.

To-Do List

• Finish testing on wide-screen monitors.
• Double rotation in ℝ⁴? Since ℝ⁴ is roomier than ℝ³, two independent simple rotations can occur simultaneously at different rates. However, on the screen this double rotation just looks like a 4-D rotation of the object combined with a 3-D rotation of the projected image.
• Rewrite the program, making explicit use of geometric algebra? Using GA from the start would have simplified the code and promoted better structure. The existing code is ugly.