Expressing the surface area and the volume by the distance p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q/2 and r = p − q/2), yields

As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earths magnetic field, where "poloidal" was used to denote "the direction toward the poles".In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topology
Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus.
The NGF gene provides instructions for making a protein called nerve growth factor beta (NGFβ) that is important in the development and survival of nerve cells (neurons), including sensory neurons. The NGFβ protein functions by attaching (binding) to its receptors, which are found on the surface of neurons. Binding of the NGFβ protein to its receptor transmits signals to the cell to grow and to mature and take on specialized functions (differentiate). This binding also blocks signals in the cell that initiate the process of self-destruction (apoptosis). Additionally, NGFβ signaling plays a role in pain sensation. Mutation of the NGF gene leads to the production of a protein that cannot bind to the receptor and does not transmit signals properly. Without the proper signaling, sensory neurons die and pain sensation is altered, resulting in the inability of people with HSAN5 to feel pain. Type 6 -
Hereditary sensory and autonomic neuropathy type 6 (HSAN6) is a severe autosomal recessive disorder characterized by neonatal hypotonia, respiratory and feeding difficulties, lack of psychomotor development, and autonomic abnormalities including labile cardiovascular function, lack of corneal reflexes leading to corneal scarring, areflexia, and absent axonal flare response after intradermal histamine injection.Genes related to Hereditary sensory and autonomic neuropathy Type 6:
Mutations in the DST (Dystosin) gene are associated HSAN6. In two separate studies of families with presentations of HSAN6, the DST gene was found to be mutated, in one case with a truncating mutation, and in the other a non-sense or missense mutation.