Radii and radius vectors are used to express spatial concepts like *extension* and *containment*. For example, consider a particle P characterized by some repetitive chain of events

where each orbital cycle is a bundle of quarks

that are described by their quark coefficients and internal energies . Recall that notes the enthalpy of the chemical quarks in P. And let a constant have a positive value, with units of a force. These quantities are used to describe the shape and extent of P as follows. The **chemical radius** is

The **magnetic radius** of P is

the **electric radius** is defined by

and the **polar radius** of P is given as

An ordered set of these radii defines the **radius vector** of particle P as

The three components of are conserved because they are defined from sums of quark coefficients, and quarks are conserved. So if some generic particles , and interact like then by the associative properties of addition, radii are related as

Also and . So particles and anti-particles have symmetrically opposed radius vectors

Going forward, we use these radii and radius-vectors to develop a description of particle *shape*. As a starting example; if P has the same radius vector for every cycle, then we call it a **rigid** particle.

## Radii and Range

Some particles are not very solid or exactly located. Then an analysis of shape might not provide a useful description. So instead we may use the following radii to describe their *containment* or *range*. The **inner radius** is defined as

And the **outer radius** of P is

We say that a particle is **free** when and . That is, when the inner radius is small and the outer radius is big. But if then we say that P is a **core** particle.

## Sensory Interpretation

Here is a sensory explanation of the radius vector. In these formulae means that contributions from sensations on the right side are cancelled by sensations felt on the left. The radius vector depends on their *net* magnitude. Coefficients of baryonic quarks do not appear in these definitions, only dynamic quarks. So the radius does not depend on thermal sensation, only somatic and visual sensations. Also the null value for energy is referred to down-quarks which are objectified from black sensations. So overall, the radius vector is interpreted as a description of size for visual sensations, relative to black sensations, net right from left.

Next we look at how to use these radius vectors to make a mathematical vector space for quark models.