nL1: Notes on nLObjects
Many general objects in an nLine are described by Prof. Frank Morley in the period 18861930.
Morley’s discoveries were all made purely by algebraic approaches. In his time it even wasn’t possible to check his discoveries in drawings because of the complicated recursive character.
His documents are often hard to understand in detail. However involved documents were piece by piece “decoded” at the end of 2014 by Bernard Keizer, Eckart Schmidt and the author of this encyclopedia in a discussion at the Yahoo QuadriFigures Group (Ref34). See especially messages #826#917. Accordingly they are mentioned in EPG and completed with drawings using Cabri or Mathematica software.
For quick insight pictures of nLines often are represented by figures bounded by n linesegments.
Many general objects in an nLine are described by Prof. Frank Morley in the period 18861930.
Morley’s discoveries were all made purely by algebraic approaches. In his time it even wasn’t possible to check his discoveries in drawings because of the complicated recursive character.
His documents are often hard to understand in detail. However involved documents were piece by piece “decoded” at the end of 2014 by Bernard Keizer, Eckart Schmidt and the author of this encyclopedia in a discussion at the Yahoo QuadriFigures Group (Ref34). See especially messages #826#917. Accordingly they are mentioned in EPG and completed with drawings using Cabri or Mathematica software.
For quick insight pictures of nLines often are represented by figures bounded by n linesegments.
How many (n1)lines can be made up from an nLine?
Many of the recursive constructions are based upon the property that from an nLine exactly n different (n1)Lines can be made up. This can easily be deduced by omitting one line from the nLine. This will leave behind an (n1)Line. Since exactly n Lines can be omitted there will be n different (n1)Lines contained in an nLine. The (n1)Lines in an nLine will be called the Component (n1)Lines. The remaining line after choosing an (n1)Line in an nLine will be called the omitted line.
In descriptions we say “an nLine contains n (n1)Lines” or “an nLine has n Component (n1)Lines”. When we want to indicate different objects occurring in (n1)Lines we say that there are n versions of these (n1)Lobjects.
The n versions of an object often will be noted with a suffix consisting of un underscore and a number 1, …, n, indicating the number of the omitted line. For example a 5Line contains 5 4Lines and therefore has 5 4LMVPCentroids (4LnP8). They will be noted as 4LnP8_1, 4LnP8_2, 4LnP8_3, 4LnP8_4 and 4LnP8_5. The suffix number at the end is the number of the omitted line.
Many of the recursive constructions are based upon the property that from an nLine exactly n different (n1)Lines can be made up. This can easily be deduced by omitting one line from the nLine. This will leave behind an (n1)Line. Since exactly n Lines can be omitted there will be n different (n1)Lines contained in an nLine. The (n1)Lines in an nLine will be called the Component (n1)Lines. The remaining line after choosing an (n1)Line in an nLine will be called the omitted line.
In descriptions we say “an nLine contains n (n1)Lines” or “an nLine has n Component (n1)Lines”. When we want to indicate different objects occurring in (n1)Lines we say that there are n versions of these (n1)Lobjects.
The n versions of an object often will be noted with a suffix consisting of un underscore and a number 1, …, n, indicating the number of the omitted line. For example a 5Line contains 5 4Lines and therefore has 5 4LMVPCentroids (4LnP8). They will be noted as 4LnP8_1, 4LnP8_2, 4LnP8_3, 4LnP8_4 and 4LnP8_5. The suffix number at the end is the number of the omitted line.
Ratiopoint
A Ratiopoint R is a point collinear to two other given points X,Y and with the distances to these two other points in a given ratio.
This method is for example used for nLnP5, nLnP7, nLnpi.
Recursive Eulerline situated points in an nLine
There are different ways of construction of Eulerline points to a higher nLine level.
All methods are based upon the central property that from any nLine n versions of (n1)Lines can be constructed.
Triangle points are X(2)=Centroid, X3=Circumcenter, X(4)=Orthocenter, X(5)=Ninepoint Center.
All methods are based upon the central property that from any nLine n versions of (n1)Lines can be constructed.
Triangle points are X(2)=Centroid, X3=Circumcenter, X(4)=Orthocenter, X(5)=Ninepoint Center.
Morley’s points
3Lpoint

4Lpoint

5Lpoint

6Lpoint


X(2)

5LnP2

6LnP2

Etc.


X(3)

5LnP3

6LnP3

Etc.


X(4)

5LnP4

6LnP4

Etc.


X(5)

5LnP5

6LnP5

Etc.

MVP Points: Multi Vector Points
3Lpoint

4Lpoint

5Lpoint

6Lpoint


X(2)

4LnP8 = QLP12

5LnP8

6LnP8

Etc.

X(3)

4LnP9 = QLP6

5LnP9

6LnP9

Etc.

X(4)

4LnP10 = QLP2

5LnP10

6LnP10

Etc.

X(5)

5LnP11

6LnP11

Etc.
