dimensions. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces. Visual scope People have a spatial self-perception as beings in a three-dimensional space, but are visually restricted to one dimension less: the eye sees the world as a projection to two dimensions, on the surface of the retina. Assuming a four-dimensional being were able to see the world in projections to a hypersurface, also just one dimension less, i.e., to three dimensions, it would be able to see, e.g., all six faces of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as people can see all four sides and simultaneously the interior of a rectangle on a piece of paper. The being would be able to discern all points in a 3-dimensional subspace simultaneously, including the inner structure of solid 3-dimensional objects, things obscured from human viewpoints in three dimensions on two-dimensional projections. Brains receive images in two dimensions and use reasoning to help picture three-dimensional objects. Limitations Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the area of a circle () and the volume of a sphere (). One might guess that the volume of the 3-sphere in four-dimensional space is , or perhaps , but neither of these is right. The actual formula is . See also 4-polytope 4-manifold Exotic R4 Four-dimensionalism Fourth dimension in art Fourth dimension in literature List of four-dimensional games Eugene the Jeep Time in physics Spacetime References Further reading Andrew Forsyth (1930) Geometry of Four Dimensions, link from Internet Archive. Extract of page 68 E. H. Neville (1921) The Fourth Dimension, Cambridge University Press, link from University of Michigan Historical Math Collection. External links "Dimensions" videos, showing several different ways