- #1
CharlesRKiss
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Please excuse the length; a simple question but difficult for me to ask!
If all great circles of a sphere are the radial axes of ring tori, how many
dimensions are required for "every" [use of quotation marks explained below]
point on every tori to be unique? If this is an example of an n-tours; what is n?
Or what is the shape of shared points?
Shared points are allowed if the shape of shared points is a point, or
form a ring, or form an n-sphere -is compact and differentiable.
**To be redundant, in hopes to clarify the question, here is an example:
SAMPLE ANSWER: One Dimension, Spindle Torus
SAMPLE QUESTION: If in a single plane, I were to draw a circle of radius R, and a point P on the circle was used as a common point to every other possible circle of radius R in the same plane, then all circles have a common point, P, but some pairs of circles have 2 common points. How many dimensions are required for "every" point on every circle to be unique? What is this form called?
Well, in the sample answer not "every" point is unique; that's why I'm using the quotes, it's just easier for me to ask the question this way.
**In the sample answer an infinite number of common points were reduced to one common point, by rotating every circle 90 degrees off the plane, thus adding one dimesnsion.
As to the original question, perhaps there's a ring or sphere of
common points, I won't presume. That's why I'm asking the question ;)
The larger problem is these tori are also completely nesting, and being
completely nested in, other tori! They share a common radial
axis, called 'b', but 'b' is not radially symmetric to all tori unless a=b=c;
I made a little drawing of half the nested tori cross-section to show the nesting.
postimage.org/image/28qaj7nms/
The surfaces of negative curvature (the inner surfaces) are closer to each other than the surfaces of positive curvature (the outer surfaces). (a sub i)*(c sub i)=b^2, if a=b=c; this would be the inner most
torus, a ring.
If all great circles of a sphere are the radial axes of ring tori, how many
dimensions are required for "every" [use of quotation marks explained below]
point on every tori to be unique? If this is an example of an n-tours; what is n?
Or what is the shape of shared points?
Shared points are allowed if the shape of shared points is a point, or
form a ring, or form an n-sphere -is compact and differentiable.
**To be redundant, in hopes to clarify the question, here is an example:
SAMPLE ANSWER: One Dimension, Spindle Torus
SAMPLE QUESTION: If in a single plane, I were to draw a circle of radius R, and a point P on the circle was used as a common point to every other possible circle of radius R in the same plane, then all circles have a common point, P, but some pairs of circles have 2 common points. How many dimensions are required for "every" point on every circle to be unique? What is this form called?
Well, in the sample answer not "every" point is unique; that's why I'm using the quotes, it's just easier for me to ask the question this way.
**In the sample answer an infinite number of common points were reduced to one common point, by rotating every circle 90 degrees off the plane, thus adding one dimesnsion.
As to the original question, perhaps there's a ring or sphere of
common points, I won't presume. That's why I'm asking the question ;)
The larger problem is these tori are also completely nesting, and being
completely nested in, other tori! They share a common radial
axis, called 'b', but 'b' is not radially symmetric to all tori unless a=b=c;
I made a little drawing of half the nested tori cross-section to show the nesting.
postimage.org/image/28qaj7nms/
The surfaces of negative curvature (the inner surfaces) are closer to each other than the surfaces of positive curvature (the outer surfaces). (a sub i)*(c sub i)=b^2, if a=b=c; this would be the inner most
torus, a ring.