String theory provides an elegant way to unify quantum mechanics and the general theory of relativity. The theory postulates that the most fundamental physical objects are vibrating strings, each vibrational mode corresponding to a different elementary particle. In one version of string theory, these strings must exist in a ten-dimensional spacetime. We humans only experience four of these dimensions (three spatial dimensions and time), and so the remaining six are theorized to be “curled up” into extremely small, complex shapes on the order of the Planck scale. These shapes are known as Calabi-Yau manifolds — an example of which is depicted in this demonstration. For the mathematically inclined, the shape below is a representation of the quintic (fifth-degree) polynomial in four-dimensional complex projective space, known as a Fermat quintic threefold, given by

$$z_1^5+z_2^5+z_3^5+z_4^5+z_5^5=0.$$

(Note that you can change the degree of the manifold below with the \(n\) slider.) To compute the surface, we fix two of the four complex variables, and what remains can be projected from 4D to 3D. In this demonstration, you can alter the surface parameter — in a sense the “projection” from 4D — to see a variety of different aspects of this fundamental space.