# Non-trivial field limits' equivalence?

#### Loren Booda

What is the field F(r) with the least symmetry and which obeys

lim F(r) as r --> oo = lim F(r) as r --> 0

?

#### Hurkyl

Staff Emeritus
Gold Member
What sort of symmetry? In what way are you quantifying it?

#### Loren Booda

I thought that the putative field may have a minimal symmetry where r is of constant magnitude and isotropic in direction, but that there might be a fractal field of lesser symmetry that obeys the given conditions. Permit me a modification:

"What is the field F(r) with the least symmetry and which obeys

lim F(r) as r --> oo = lim F(r) as r --> 0

?"

#### Hurkyl

Staff Emeritus
Gold Member
What sort of symmetry? In what way are you quantifying the symmetry?

#### matt grime

Homework Helper
What do 'constant magnitude', and 'isotropic in direction', or 'fractal fields' mean (in this context, or any context for the last two; isoptropic means 'equal in all directions, so how can anything be isotropic in direction?' )?

#### Loren Booda

Hurkyl,

What sort of symmetry? In what way are you quantifying the symmetry?
Geometric symmetry. I am quantifying the symmetry as a (metric) space.

Matt Grime,

What do 'constant magnitude', and 'isotropic in direction', or 'fractal fields' mean (in this context, or any context for the last two; isoptropic means 'equal in all directions, so how can anything be isotropic in direction?' )?
"Constant magnitude" (which is incorrect; r is variable) and "isotropic in direction" (which is redundant) refer to the radius vector r correctly being "isotropic at every point." By "fractal field" I speculate that the embedded field as defined may not fill the original space at every point.

#### matt grime

Homework Helper
Yep, that still makes no sense.

How are you defining 'symmetry of a field' to be a vector space? What is a symmetry of a field? (I am asking for your definition, since I don't know that you're using symmetry to mean an automorphism Given your non-standard usage of terms I am assuming not.) How are they, whatever they are, metrized?

It appears r is a vector. You've not said in what vector space? What does it mean for a 'field' to be embedded in a vector space? What is the definition of 'fill' that you're using.

All these questions, and probably more, mean we have no idea what you're talking about.

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#### Loren Booda

Consider a one-dimensional space. Apply the constraint that the assigned value at any point equals the value at a distance from the point approaching infinity. What distribution, with greater complexity than where all values are constant and equal, obeys the constraint? Might a "class" of fractal distributions meet this condition?