We live in a three-dimensional space. All objects around us are three-dimensional. When we consider only areas or lengths, we are dealing with two or one dimension, respectively. Objects with more than three dimensions do not perceptibly exist in our world. However, in mathematics, such objects can certainly be conceived and calculated. Visualizing and representing these objects is difficult. The graphics here, like illustrations in a book, are two-dimensional. When three-dimensional objects are represented in this way, they are projected onto a plane, resulting in a loss of information. The loss of information is even greater when projecting multi-dimensional objects onto a plane.
The number of four-dimensional geometric objects is, of course, infinite. Only a few particularly important ones are described here. What is called a polygon in two dimensions and a polyhedron in three dimensions, shapes with exclusively straight boundaries, is generalized to any number of dimensions as a polytope. The best-known 4-polytope is the tesseract, or four-dimensional hypercube, the four-dimensional extension of the square and the cube. A shape that is not a polytope is the four-dimensional hypersphere, the extension of the circle and the sphere into an additional dimension. For these two shapes, the derivation of the formulas used to calculate them remains rather easy to follow. If visualizing four-dimensional shapes is possible at all, it is likely easiest with these two.
Even though four-dimensional shapes do not exist in our everyday world, they play a role in practical models and calculations. Many principles from two and three dimensions can be generalized to four or more dimensions, thereby revealing underlying relationships. General Relativity describes the three dimensions of space and the single dimension of time as an interconnected four-dimensional spacetime.