If a 3D being could manipulate that square like a playing card on a table, it could flip it Now extrapolating up gives insight on 4D rotation-a 4D object must rotate about a (4 - 2)D = 2D pivot In this case, we can confirm the analogy checks out by working out the same math in 2D, in which objects always rotate about aĠD = (2 - 2)D pivot (a point). No matter how a 3D object rotates, it rotates about a 1D = (3 - 2)D pivot (a Suppose you want to gain a better understanding of how rotation works in 4 dimensions. Going through this thought process might be helpful in seeing how this program works. Regarding 4 dimensions can be understood, if not intuitively then logically, by creating an analogy that applies to 3 or fewer dimensions. Really feasible to perceive what this would look and feel like, but since the applicable math works out so nicely when extrapolating up aĭimension, a program like this one can give a glimpse into how geometries in higher dimensions would behave. In 4-dimensional space, a fourth coordinate axis exists that is orthogonal to the x, y, and z axes. This was written as a personal excerciseĪnd proof of concept, rather than to create an efficient and optimized rendering system. Spherical coordinate systems and the parametric equations of lines and planes in n-space. Was derived by myself with limited outside knowledge of linear algebra or graphics programming, other than the basic concepts of

The program utilizes no builtin matrix operations, quaternion transforms, etc. This OpenGL program in C++ is capable of rendering edges of the 'shadow' of arbitrary 4-dimensional polytopes

4d rendering