4-Dimensional Rotation

Source: 4-Dimensional Rotation, TheHeadlessIndian, 0:24, uploaded 2008-12-22, playlist index 391.

A white horse model appears against a black background. Over the 24-second clip, its body stretches, folds through itself and moves partly out of the frame. The horse becomes difficult to recognise as its visible surface passes through shapes that look like an inside-out version of the original model.

A horse seen through another space

An ordinary rotation in three-dimensional space changes the horse’s viewpoint and keeps its shape. This animation changes the shape on screen because the motion takes place through a space that the viewer cannot see directly. The torso swells into a broad surface, limbs bend into unlikely positions and the head reappears from a new direction. The clip gives the eye a sequence of two-dimensional images from which to infer a transformation with a fourth spatial coordinate.

The upload contains no spoken explanation beyond “Thanks for watching!” The description supplies the method in a sentence: Euclidean 3D space is projected onto a hypersphere, that non-Euclidean space is rotated in four dimensions and the result is shown again in ordinary 3D.

Stereographic projection and the 3-sphere

The linked explanation calls the hypersphere a 3-sphere. An ordinary sphere is a two-dimensional surface embedded in three-dimensional space, while a 3-sphere is a three-dimensional object embedded in four-dimensional space. In the construction used here, inverse stereographic projection moves points from everyday 3D space onto the 3-sphere. A 4D rotation acts there before stereographic projection sends the result back into flat 3D space.

That projection chain gives the horse its peculiar behaviour. The visible horse is a three-dimensional image of a rotation carried out in four-dimensional space. The mesh can expand, collapse and appear to pass through itself even though the underlying operation follows a defined rotation in the higher-dimensional space. The clip leaves the coordinate system, rotation plane and equations unstated, so its force comes from the image rather than from a worked derivation.

Limits of the demonstration

The video establishes a visual example of the stated projection method. It does not explain 4D rotations in general or identify the exact parameters used for this horse. The description gives the broad mathematical route, while the linked article supplies the longer explanation and code. The complete audio transcription contains only the closing sentence, so details about the mesh, software and animation settings remain outside the source evidence.

Further reading / references

• 4-Dimensional Rotations, the explanation linked in the video’s description.