Is there such a thing as 4-D shapes?

Yes, there are 4-D shapes, also known as four-dimensional polytopes or "polychora." Just as 3-D shapes are defined in three spatial dimensions, 4-D shapes extend this concept into a fourth dimension. While we cannot directly visualize four dimensions, we can represent 4-D shapes mathematically or through projections into 3-D space, similar to how a 3-D object casts a 2-D shadow. Examples include the hypercube (or tesseract) and the 24-cell.