from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension. By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from the three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers. Cross-sections As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a cross-section of the three-dimensional object within this plane. For example, if a sphere passed through a sheet of paper, beings in the paper would see first a single point, then a circle gradually growing larger, until it reaches the diameter of the sphere, and then getting smaller again, until it shrank to a point and then disappearing. The 2D beings would not see a circle in the same way as three-dimensional beings do; rather, they only see a one-dimensional projection of the circle on their 1D "retina". Similarly, if a four-dimensional object passed through a three dimensional (hyper) surface, one could observe a three-dimensional cross-section of the four-dimensional object. For example, a hypersphere would appear first as a point, then as a growing sphere (until it reaches the "hyperdiameter" of the hypersphere), with the sphere then shrinking to a single point and then disappearing. This means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of Charles Howard Hinton. And, in the same way three-dimensional beings (such as humans with a 2D retina) can see all the sides and the insides of a 2D shape simultaneously, a 4D being could see all faces and the inside of a 3D shape at once with their 3D retina. Projections A useful application of dimensional analogy in visualizing higher dimensions is in projection. A projection is a way for representing an n-dimensional object in 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 eye is also a two-dimensional array of receptors but