in 3 dimensional geometry
In formulating a theory of existence, this is not an insignificant problem ... especially if one wishes to posit a universe populated by quantized entities, i.e. fundamental building blocks. If we put a "spot" on our fundamental blocks, they no longer possess fundamentality ... they have a "blemish". Where would the blemish be positioned relative to the axis of rotation? What shape would it be? Why? Because there are an infinite number of possible blemish types, the concept of such a rotation-verifying blemish is non-fundamental. Do you see the logical problem here? We can "decree" that the object is rotating but this sidesteps the fundamental rule requiring a geometric show of some sort to ...
Embodying an effect rather than the action
Suppose that we wish to embody the concept of rotation in a spherical field. We arbitrarily construct a model of the field with lines emanating from a central point and surround that point with concentric spherical shells. We can imagine seeing the individual lines pass by ... their position is arbitrary and don't actually ... logically indicate ... quantization. That is, if we set the lines 15o apart, there is really no logical justification for spacing them out like that ... and spacing them out is what is required to embody (to make visible) the concept of rotation. There are an infinite number of possible spacings so why is one preferred over another? Any reason would necessarily be ad hoc with no justification within the model's initial parameters.
I propose this solution.
That the concept of rotation be embodied by the "oblateness" of the field. Then, at zero rotation the oblateness is just zero and at unit velocity, the oblateness would be infinite. Oblateness (a flattening at the poles of rotation is non-arbitrary because it uniquely defines the axis and the rate of rotation. The equation governing the degree of oblateness would be similar to that which would govern a ball accelerated so that it approached the speed of light and was progressively flattened along the line of acceleration.
By doing this we geometrically embody rotation without an arbitrary "blemish" even though the action itself would not be detectable, i.e. another particle would respond to the oblateness of the quantized field that was supposed to be rotating.
A related problem is twisting the Euclidean field
Such a field cannot rotate because it has nothing to rotate relative to because it constitutes the "stage" upon which the activities of the universe are played out. It is an absolute reference frame. However, such a field could have twists in it as well as volumes of lesser or greater density adequately characterized by more or less lines per unit volume. The reference frame only requires that for every compression of the field, there is an equal an opposite expansion. It is then a constant reference overall. I have identified "charge" as such an expansion-compression pair. So, it is necessary to indicate how a twist in the field could be carried out without arbitrariness as was done for rotation.
An expansion of the reference field (a charge) was shown to be related mathematically to mass (the spherical field) in the ground state of the hydrogen atom (as a most probable value problem). One would then expect a twist to be related to rotation of the spherical field in some similar way.
To create a non-arbitrary twist effect I propose that as such a twist is made (with the necessary corresponding anti-twist), the field develops "folds" from drawing in the field closer to the axis of the twist. That is, when one twists the field, the field lines are stretched into spirals at the point of the twist normal to the axis of the twist and that the spiral's degree of stretch diminishes in as they approach parallels to that axis.
Such stretching lends itself to quantization in a way similar to a sheet on a table top which one twists with one's finger in the manner shown below.
As more table cloth is drawn to the center, it develops "folds" and ... there are "x" number of folds. We might imagine the folds as forming a sine wave around the central twisting point. Confining them to the sine form means that there is now a non-arbitrary way of associating wavelength to rotation, i.e the afforementioned oblateness of a rotating particle.
Thus, rotation of a particle as charactized by the oblateness of its spherical field will have a corresponding effect on the Euclidean field in a manner similar to that between charge and mass as given in The Electron-Proton Mass Ratio and Fine Structure Constant.
Now, I just have to work out the math .... hmmmmmmmmm...