4D Translation. When a 4D object moves along the fourth dimension, its geometry is differently sliced based on the current cross-section hyperplane, producing a visual effect similar to morphing. When there is no intersection with the hyperplane, the projection does not generate a mesh, making the 4D object "disappear." If we imagine shooting a bullet with a hypothetical 4D gun, generating a sinusoidal movement along w, the perceived effect would be a bullet continuously entering and exiting the crosssection (i.e. appearing and disappearing).

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... In cross-section, a 4D object moving along w can be differently sliced based on its intersection with the current hyperplane, until all of its vertices exit the hyperplane, making the object itself invisible to the eye. In Fig. 2, for instance, an imaginary weapon shoots a bullet in the shape of a hyperdodecahedron (also known as a "120-cell"), enforcing upon such bullet a sinusoidal movement along w. The effect perceived by the viewer is that of a morphing geometry that periodically appears on and disappears. In frustum projection, an object moving further ...

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... explore different content and authoring patterns to facilitate the editing of 4D scenes. A subset of the design considerations outlined here will be then operationalized in the use case presented in Section 5. In Continuum, physics on 4D objects are treated as straightforward generalizations of 3D physics. In the example above, the same gun from Fig. 2 shoots a bullet which exits the current cross-section and hits a hidden hyperplane. As result of the collision, the hyperplane slowly rotates around the t, u, and v planes and becomes partially visible, while the bullet falls to the ground due to gravity, rolling away until its w coordinate again falls outside of the ...

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... to a normal radar display, the user's location is positioned at the center, with the radar plane defined in the x and z directions, and positioned at the same w coordinate as the player. 4D objects appear on the radar as 3D pins with an altitude proportional to their w offset with respect to the player, suggesting that objects showing on the radar plane are currently visible, while Figure 12: Camera control. Across Dimensions supports smoothly transitioning from cross-section to frustum projection. ...

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... only one additional variable, the orbit angle, which the user can control via two dedicated keys. Since moving the 4D camera and shifting the w vanishing point are hard to imagine, two new objects appear in the scene when frustum projection is activated, dynamically reflecting the the focus point position and 4D camera position, respectively (Fig. 12). These two in-game indicators are also shown on the 3D radar display, allowing the player to control them even if they are not currently on ...

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... the results produced by 4D rotation, we apply the concept of "ghost geometries" [LRY * 15,Kag16b] and visualize floating previews of the shapes that the 4D object could assume (Fig. 13). The player can also perform more complex actions through a set of weapons which he acquires throughout the game. One example is the weapon already introduced in Fig. 2 and Fig. 5, which shoots hyperdodecahedral bullets that fluctuate along the w axis. These bullets periodically become invisible as they exit the current cross-section, but their 4D movement can be continuously tracked using the 3D radar display. Each bullet has enough power to move simple 4D objects on contact, allowing one to remove obstacles ...