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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.

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 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.

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17 Nov 2003 – 12 Mar 2010 Keith