What is an n-Line?
In EPG (Encyclopedia of Poly Geometry) an n-Line is defined as a system consisting of 'n' random lines occurring in a plane, no three of which are concurrent.
There are no points involved. There is no order in these lines. Just 'n' random lines. Nothing more and nothing less.
Every line in an n-Line is exchangeable with one of the other lines.
Whatever is valid for a subset of these lines is also valid for another subset of equal amount of these lines.
An n-Line is a flexible framework that can be used to construct many objects upon.
In EPG these objects often will be prefixed with “nL-”.
Just like in a triangle and in a quadrilateral new (derived) points can be developed.
These new points will be called ‘points’ or ‘centers’.
They will be noted with codes nL-n-Px or nL-e-Px or nL-o-Px or nL-s-Px (x=serial number).
The prefix “nL” means that we deal with an n-Line.
“n” can be substituted by a natural number.
When the infix is “n” then the point occurs for all natural n.
When the infix is “e” then the point occurs for all even n.
When the infix is “o” then the point occurs for all odd n.
When the infix is “s” then the point occurs for only one specific n.
The suffix “Px” tells that we deal with a Point with serial number x.
Apart from Points also Lines, Circles, Conics, Cubics, other Curves, Transformations, Triangles can be constructed, then the suffixes resp. will be Lx, Cix, Cox, Cux, Cvx, Tfx, Trx, where x is the serial number for this category.
When the infix is “s” then the point occurs for only one specific n.
The suffix “Px” tells that we deal with a Point with serial number x.
Apart from Points also Lines, Circles, Conics, Cubics, other Curves, Transformations, Triangles can be constructed, then the suffixes resp. will be Lx, Cix, Cox, Cux, Cvx, Tfx, Trx, where x is the serial number for this category.
Examples:
nL-n-P1 = the point P1 existing in a 3-Line, 4-Line, 5-Line, 6-Line, etc.
5L-n-P1 = the point nL-n-P1 as occurring in a 5-Line.
5L-s-P1 = the point P1 which is only and specifically occurring in a 5-Line.
So there are different series of points in the nL-n-range, 3L-s-range, 4L-s-range, 5L-s-range, etc.
Points (Centers) in the nL-n-range are centers constructed in a recursive way to a higher level. Each higher level is one more reference line in the n-Line.
nL-n-P1 = the point P1 existing in a 3-Line, 4-Line, 5-Line, 6-Line, etc.
5L-n-P1 = the point nL-n-P1 as occurring in a 5-Line.
5L-s-P1 = the point P1 which is only and specifically occurring in a 5-Line.
So there are different series of points in the nL-n-range, 3L-s-range, 4L-s-range, 5L-s-range, etc.
Points (Centers) in the nL-n-range are centers constructed in a recursive way to a higher level. Each higher level is one more reference line in the n-Line.
Related to an n-Line several point, lines, circles, conics, cubics, transformations and triangles do exist, which can be obtained from the pulldown menu at the left of this page.
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