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Rotating 4-Space


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Interaction Controls

Dragging the mouse will rotate three-space around an axis within the plane of the screen.

Dragging with the middle mouse button, or while the Option or Alt key is held down, rotates space about an axis perpendicular to the screen.

Dragging with right mouse button, or while the Command or Control key is held down, rotates space about an axis in the fourth dimension.

The readout at the bottom represents the rotation matrix for the coordinate axes (see below). Because the 4D to 2D projection is orthographic rather than in perspective, it is sometimes hard to tell what is in front and what is behind (far away things do not get smaller). Look at the colors at the overcrossings to get your orientation back if you get confused. All the line segments that are the same color are parpallel to the same coordinate axis.

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This demonstration is one of a series: you can return to the 2D version, or the 3D version next.

What this Demonstrates:

Dragging the mouse will cause the axes in two-space to rotate. The first column of the array at the bottom of the picture keeps track of the four new coordinates of the unit vector that points along the x axis. Similarly the other columns represent the new positions of the unit vectors along the y, z and w axes.

You can add the squares of the components in each column to see that the length of either coordinate vector remains the same, i.e. equal to one, throughout the rotation.

Also, for those familiar with the dot product, it is possible to check that the columns of this array, or matrix as it is known, are perpendicular throughout. Any matrix whose columns form mutually perpendicular vectors is called an orthogonal matrix.

Once we know the positions of the four axes, we can apply the same rotation to any four-dimensional figure at all. In this case, we rotate a hypercube (the figure in four-space that is analogous to the cube in three-space and the square in two-space) along with the axes. This is the basis of the rotations that show up in computer animation, and in other applications in computer graphics to both three-space and four-space.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.