In brief. Below is the DigiTorus, the Cartesian product of the letters d and p, projected onto three dimensions while freely rotating in four-dimensional space. By dragging your mouse over the figure, you can rotate it in three dimensions. By using the upper scroll bar at the bottom of the window, you can homotope the DigiTorus into a Mathematicians torus.
In detail. The rotation of the DigiTorus above is composed of three parts: (1) rotation in the plane containing the letter d, (2) rotation in the plane containing the letter p, and (3) "free" rotation -- a continous sequence of arbitrary rotations (if you must know, its a quadratic spline in the four-dimensional special orthogonal group with randomly chosen control points). The d on/off button toggles the rotation in the d-plane; the p on/off button toggles the rotation in the p-plane; the rotation on/off button toggles the free rotation.
The lower scroll bar beneath the DigiTorus controls the size of the letters d and p. Scrolling to the left of the middle position decreases the size of d, while keeping the size of the p constant; scrolling to the right of middle decreases the size of p, while keeping the size of d constant.
The upper scroll bar controls the homotopy (continuous deformation) between the DigiTorus and the Mathematicians torus. In the extreme left position, the DigiTorus is displayed in full; in the extreme right position, the Mathematicians torus is displayed in full. From the left to the middle scroll bar position, the "stems" of the d and p are simultaneously contracted; from the middle to right position, the stemless d and p are simultaneously deformed into circles.
By dragging on the DigiTorus with your mouse, you can rotate the projected DigiTorus/Torus in three dimensions (before it gets projected onto the screen).
The wire/solid/ribbon frame button toggles how the DigiTorus is represented. Note that, just like the Mathematicians torus, the DigiTorus has self-intersections.
Fun things to do with the DigiTorus. Turn off the free rotation (rotation off mode), turn on the rotation in the d-plane (d on mode), and turn off the rotation in the p-plane (p off mode). This applies a (constant speed) rotation to the d-plane, while leaving the p-plane fixed. You can see this by sliding the bottommost scroll bar to the extreme left and right: on the extreme left, the p will be stationary (the d has been shrunk to a point), while on the extreme right, the d will be rotating (the p has been shrunk to a point). When the scroll bar is in the middle position, can you see the stationary p?
While in the same d on mode as above, slide the bottommost scroll bar all the way to the right, so that you see the rotating d; and then slide the upper scroll bar all the way to the right position, so that the d is homotoped into a circle. Note that the circle is stable under the rotation: it is rotating about its center point. Now slide the bottommost scroll bar to the middle, so that you see the Mathematicans torus. Since the p-plane is already fixed (and not rotating), the rotation keeps the torus stable. Now turn on the rotation in the p-plane (p on mode); this (composite) rotation also keeps the torus stable.
Exercise (advanced). Identify the coset space SO(4)/Stab(S x S), where Stab(S x S) is the stablizer subgroup of the Mathematicians torus.