Editing Four-dimensional space (section)

===Projections===
A useful application of dimensional analogy in visualizing higher dimensions is in [[Graphical projection|projection]]. A projection is a way of representing an ''n''-dimensional object in {{math|''n'' − 1}} dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places, and things are represented in two dimensions by projecting the objects onto a flat surface. By doing this, the dimension orthogonal to the screen (''depth'') is removed and replaced with indirect information. The [[retina]] of the [[human eye|eye]] is also a two-dimensional [[Array data structure|array]] of [[Sensory receptor|receptor]]s but the [[brain]] can perceive the nature of three-dimensional objects by inference from indirect information (such as shading, [[foreshortening]], [[binocular vision]], etc.). [[Artist]]s often use [[perspective (graphical)|perspective]] to give an illusion of three-dimensional depth to two-dimensional pictures. The ''shadow'', cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections.

Similarly, objects in the fourth dimension can be mathematically projected to the familiar three dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.

The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer four-dimensional "depth" from these effects.

As an illustration of this principle, the following sequence of images compares various views of the three-dimensional [[cube]] with analogous projections of the four-dimensional tesseract into three-dimensional space.

{|class=" wiki table"
|-
!Cube
!Tesseract
!Description
|-
|[[File:Cube-face-first.png|160px]]
|[[File:Tesseract-perspective-cell-first.png|160px]]
|The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the '''cell-first perspective projection''', shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube.

Note that the other 5 faces of the cube are not seen here. They are ''obscured'' by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
|-
|[[File:Cube-edge-first.png|160px]]
|[[File:Tesseract-perspective-face-first.png|160px]]
|The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the '''face-first perspective projection''', shown on the right. Just as the edge-first projection of the cube consists of two [[trapezoid]]s, the face-first projection of the tesseract consists of two [[frustum]]s.
The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.
|-
|[[File:Cube-vertex-first.png|160px]]
|[[File:Tesseract-perspective-edge-first.png|160px]]
|On the left is the cube viewed corner-first. This is analogous to the '''edge-first perspective projection''' of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 [[kite (geometry)|deltoids]] surrounding a vertex, the tesseract's edge-first projection consists of 3 [[hexahedron|hexahedral]] volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.
|-
|[[File:Cube-edge-first.png|160px]]
|[[File:Tesseract-perspective-edge-first.png|160px]]
|A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has ''three'' hexahedral volumes surrounding an edge.
|-
|[[File:Cube-vertex-first.png|160px]]
|[[File:Tesseract-perspective-vertex-first.png|160px]]
|On the left is the cube viewed corner-first. The '''vertex-first perspective projection''' of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has ''four'' hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on the boundary of the projected volume, but at its center ''inside'', where all four cells meet.

Only three of the cube's six faces can be seen here, because the other three faces lie ''behind'' these three faces, on the opposite side of the cube. Similarly, only four of the tesseract's eight cells can be seen here; the remaining four lie ''behind'' these four in the fourth direction, on the far side of the tesseract.
|}