Even in low dimensional examples, basic object in differential geometry, such has tangent spaces and bundles, can only be perceived in at least 4-dimensions. This makes intuitive understanding of these objects difficult. This group created some tools for understanding these objects a little better.
In these “tangent space visualisers,” we explore the concept of tangent vectors and tangent spaces. Below we have two examples, the figure-8 space (left) and the “theta” space (right). Click on the images to download the applets. The arrows indicate the tangent vectors which span the tangent space at any given point, click on any point in the space to move the arrow.