### NAME

hypertorus - Draws a hypertorus that rotates in 4d

### SYNOPSIS

hypertorus [-display host:display.screen] [-install] [-visual visual] [-window] [-root] [-delay usecs] [-fps] [-wireframe] [-surface] [-transparent] [-solid] [-bands] [-spirals-{1,2,4,8,16}] [-twosided] [-colorwheel] [-perspective-3d] [-orthographic-3d] [-perspective-4d] [-orthographic-4d] [-speed-wx float] [-speed-wy float] [-speed-wz float] [-speed-xy float] [-speed-xz float] [-speed-yz float]

### DESCRIPTION

The hypertorus program shows the Clifford torus as it rotates in 4d. The Clifford torus is a torus lies on the "surface" of the hypersphere in 4d. The program projects the 4d torus to 3d using either a perspec- tive or an orthographic projection. Of the two alternatives, the per- spective projection looks much more appealing. In orthographic projec- tions the torus degenerates into a doubly covered cylinder for some angles. The projected 3d torus can then be projected to the screen either perspectively or orthographically. There are three display modes for the torus: mesh (wireframe), solid, or transparent. Further- more, the appearance of the torus can be as a solid object or as a set of see-through bands or see-through spirals. Finally, the colors with with the torus is drawn can be set to either two-sided or to a color wheel. In the first case, the torus is drawn with red on the outside and green on the inside. This mode enables you to see that the torus turns inside-out as it rotates in 4d. The second mode draws the torus with a fully saturated color wheel. This gives a very nice effect when combined with the see-through bands or see-through spirals mode. The rotation speed for each of the six planes around which the torus rotates can be chosen. This program is very much inspired by Thomas Banchoff's book "Beyond the Third Dimension: Geometry, Computer Graph- ics, and Higher Dimensions", Scientific American Library, 1990.

### OPTIONS

hypertorus accepts the following options: -window Draw on a newly-created window. This is the default. -root Draw on the root window. -install Install a private colormap for the window. -visual visual Specify which visual to use. Legal values are the name of a visual class, or the id number (decimal or hex) of a specific visual. -delay microseconds How much of a delay should be introduced between steps of the animation. Default 25000, or 1/40th second. The following three options are mutually exclusive. They determine how the torus is displayed. -wireframe Display the torus as a wireframe mesh. -surface Display the torus as a solid surface (default). -transparent Display the torus as a transparent surface. The following seven options are mutually exclusive. They determine the appearance of the torus. -solid Display the torus as a solid object. -bands Display the torus as see-through bands (default). -spirals-1, -spirals-2, -spirals-4, -spirals-8, -spirals-16 Display the torus as see-through spirals with the indicated number of spirals. The following two options are mutually exclusive. They determine how to color the torus. -twosided Display the torus with two colors: red on the outside and green on the inside. -colorwheel Display the torus with a fully saturated color wheel (default). If the torus is displayed as see-through bands each band will be displayed with a different color. Likewise, if the torus is displayed as see-through spirals each spiral will receive a different color. The following two options are mutually exclusive. They determine how the torus is projected from 3d to 2d (i.e., to the screen). -perspective-3d Project the torus from 3d to 2d using a perspective projection (default). -orthographic-3d Project the torus from 3d to 2d using an orthographic projec- tion. The following two options are mutually exclusive. They determine how the torus is projected from 4d to 3d. -perspective-4d Project the torus from 4d to 3d using a perspective projection (default). -orthographic-4d Project the torus from 4d to 3d using an orthographic projec- tion. The following six options determine the rotation speed of the torus around the six possible hyperplanes. The rotation speed is measured in degrees per frame. The speeds should be set to relatively small val- ues, e.g., less than 4 in magnitude. -speed-wx float Rotation speed around the wx plane (default: 1.1). -speed-wy float Rotation speed around the wy plane (default: 1.3). -speed-wz float Rotation speed around the wz plane (default: 1.5). -speed-xy float Rotation speed around the xy plane (default: 1.7). -speed-xz float Rotation speed around the xz plane (default: 1.9). -speed-yz float Rotation speed around the yz plane (default: 2.1). -fps Display the current frame rate, CPU load, and polygon count.

### INTERACTION

If you run this program in standalone mode you can rotate the hyper- torus by dragging the mouse while pressing the left mouse button. This rotates the hypertorus in 3D, i.e., around the wx, wy, and wz planes. If you press the shift key while dragging the mouse with the left but- ton pressed the hypertorus is rotated in 4D, i.e., around the xy, xz, and yz planes. To examine the hypertorus at your leisure, it is best to set all speeds to 0. Otherwise, the hypertorus will rotate while the left mouse button is not pressed.

### ENVIRONMENT

DISPLAY to get the default host and display number. XENVIRONMENT to get the name of a resource file that overrides the global resources stored in the RESOURCE_MANAGER property. X(1), xscreensaver(1)

### COPYRIGHT

Copyright (C) 2003-2005 by Carsten Steger. Permission to use, copy, modify, distribute, and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. No representations are made about the suitability of this software for any purpose. It is provided "as is" without express or implied war- ranty.

### AUTHOR

Carsten Steger <carsten@mirsanmir.org>, 28-sep-2005.