## 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 perspective or an orthographic projection. Of the two alternatives, the perspective projection looks much more appealing. In orthographic projections 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. Furthermore, 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 Graphics, 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 projection.

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 projection.

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 values, 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 hypertorus 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 button 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.