Top Tweets for #KT_A_Geometric_Approach
#KT_A_Geometric_Approach
Testing the (8x8) Configuration's set with another #symmetric_Solution by G.Monge (circa 1780), produced 4 new Solutions, 1 new Bisection, and 8 new Trisections. This time, notice we got 3 Solutions with the last Trisection; and 2 ways to get another one
#KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CD03]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions.
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CD03]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions. https://t.co/aFr4irKUcr](https://pbs.twimg.com/media/FovEBhHXwAcRHIY.png)
#KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CL12]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions.
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CL12]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions. https://t.co/jdXpz4Dpis](https://pbs.twimg.com/media/Fop44sIXgAAsPLq.png)
#KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CL05]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions.
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
#Arrow_Diagram for Solution S:{T:{Jagusa 05-II-23} + [CL05]} (see #Family_Tree on Tweet Feb. 06, 2023).
There are 7 algorithms to define a Polytour as a function of a Solution (and vice-versa).
Notice the algorithm defines the #Numeric_Descriptions. https://t.co/jhuWukwb1u](https://pbs.twimg.com/media/FokybviWYAE8WWS.png)
#KT_A_Geometric_Approach
A new (9x9) symmetric Solution produced by modifying Pentasection P5:{T:{Jagusa 05-II-23} + [CD01]} (see Family Tree on Tweet Feb. 07, 2023).
1st we produce a new Trisection, then we connect the (2x02) open segments by means of "Virtual" Configurations.
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
A new (9x9) symmetric Solution produced by modifying Pentasection P5:{T:{Jagusa 05-II-23} + [CD01]} (see Family Tree on Tweet Feb. 07, 2023).
1st we produce a new Trisection, then we connect the (2x02) open segments by means of "Virtual" Configurations. https://t.co/YMTOqGrMdl](https://pbs.twimg.com/media/FoiTMXsWYAI5vrj.png)
#KT_A_Geometric_Approach
A new (9x9) symmetric Solution by modifying Tetrasection P4*:{B:{Jagusa 05-II-23} + [RV18]} + [CD01]} (see Family Tree on Tweet Feb. 07, 2023).
First we produce a new Trisection, then we connect the (2x02) open segments with "Virtual" Configurations.
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
A new (9x9) symmetric Solution by modifying Tetrasection P4*:{B:{Jagusa 05-II-23} + [RV18]} + [CD01]} (see Family Tree on Tweet Feb. 07, 2023).
First we produce a new Trisection, then we connect the (2x02) open segments with "Virtual" Configurations. https://t.co/GeHDmpEXOw](https://pbs.twimg.com/media/FofrCV4WIAE3ybJ.png)
#KT_A_Geometric_Approach
Our Tweet on Feb. 06, 2023 showed the #Family_Tree of (9x9) Trisection {Jagusa 05-II-23}. Today's one shows the new Solutions produced while modifying this Trisection with 9 "Real" Configurations.
#9x9_symmetric_Trisection
#New_9x9_symmetric_Solutions
#KT_A_Geometric_Approach
After transforming an asymmetric (9x9) Solution obtained with an heuristic by J-P Zanotti (https://t.co/SPB2jXkg4S) into a new #symmetric_Trisection, we obtained its #Family_Tree by testing which Configurations would produce new Solutions or Polytours.
#KT_A_Geometric_Approach
On the Family Tree for S:{Jagusa 230123}, we obtained a Trisection by applying [CL06]. Five new Solutions - depicted on the accompanying diagram, were produced with Configurations connecting the 3 circuits in 5 different ways
#Knights_Tour_New_Solutions
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
On the Family Tree for S:{Jagusa 230123}, we obtained a Trisection by applying [CL06]. Five new Solutions - depicted on the accompanying diagram, were produced with Configurations connecting the 3 circuits in 5 different ways
#Knights_Tour_New_Solutions https://t.co/28c25NUstM](https://pbs.twimg.com/media/FoF9sNeXgAA3on_.png)
#KT_A_Geometric_Approach
While developing the Family Tree for S:{Jagusa 230123}, we obtained {T*} by applying [CL02]. A new Solution - which is depicted on the accompanying diagram, could be produced by applying 4 "Virtual" Configurations.
#Knights_Tour_New_Solution_Method
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
While developing the Family Tree for S:{Jagusa 230123}, we obtained {T*} by applying [CL02]. A new Solution - which is depicted on the accompanying diagram, could be produced by applying 4 "Virtual" Configurations.
#Knights_Tour_New_Solution_Method https://t.co/ffhBxMGQ41](https://pbs.twimg.com/media/FoDeia6WIAExT2Y.png)
#KT_A_Geometric_Approach
While developing the Family Tree for S:{Jagusa 230123}, we obtained 9 new Solutions, 13 new Trisections and 1 new Bisection. All of them by applying only "Real" Configurations.
#Knights_Tour_New_Solution_Method
#Tour_Du_Cavalier
#KT_A_Geometric_Approach
While developing the Family Tree for P4:{Jagusa 230123}, we obtained {P4*} by applying [CD07]. And a new Solution - which is depicted on the accompanying diagram, could be produced by applying 5 "Virtual" Configurations.
#Knights_Tour_New_Solution_Method
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
While developing the Family Tree for P4:{Jagusa 230123}, we obtained {P4*} by applying [CD07]. And a new Solution - which is depicted on the accompanying diagram, could be produced by applying 5 "Virtual" Configurations.
#Knights_Tour_New_Solution_Method https://t.co/YLfqDxOFyn](https://pbs.twimg.com/media/Fn2ZUy6WIAEoqHs.png)
#KT_A_Geometric_Approach
The method also allows you to modify other known Polytours such as this (10x10) Hexasection with open segments.
The use of "Virtual" Configurations allowed us to circumvent some of its connecting spots limitations.
#Knights_Tour_New_Solution_Method
#KT_A_Geometric_Approach
Our Tweet on Jan.27, 2023 showed the Family Tree for P4:{JAGUSA 230123}.
On this one, we show the Family Tree for
B:{P4:{Jagusa230123} + [CL05]}, and we could go on and on, as this method is recursive ...
#New_Knights_Tour_Solution_Method
#10x10_Board
![Jarigusa's tweet photo. #KT_A_Geometric_Approach
Our Tweet on Jan.27, 2023 showed the Family Tree for P4:{JAGUSA 230123}.
On this one, we show the Family Tree for
B:{P4:{Jagusa230123} + [CL05]}, and we could go on and on, as this method is recursive ...
#New_Knights_Tour_Solution_Method
#10x10_Board https://t.co/8CEqznunl5](https://pbs.twimg.com/media/FnhzA26WYAEeoit.png)
#KT_A_Geometric_Approach
The method also allows you to modify known Polytours (such as this 10x10 Tetrasection).
By systhematically testing the geometry of every "Configuration", we can generate its Family Tree.
#Symmetric_Polytour_Family_Tree
#Knights_Tour_New_Solution_Method
#KT_A_Geometric_Approach
Sometimes, there's no easy way to connect one of the closed chains of a Polytour, to obtain a new #symmetric_Solution.
However, we can always count on "Virtual Configurations" to do the task, as shown on this example (see our Tweet on Jan. 19, 2023).
#KT_A_Geometric_Approach
Our Tweet on Jan. 22, 2023 showed the Geometry match between the Configuration's Set for (8x8) boards and the tested Solution {JAGUSA (200123)}.
This one shows the Family Tree of #Solutions_and_Polytours as the outcome of that test ...
#Knights_Tour
#KT_A_Geometric_Approach
To obtain new #symmetric_Solutions just modify known Solutions or Polytours (Bisections, Trisections, or else). You have to change the connections at geometric spots we call "Configurations" which are fixed for every board size. There are 6 types of them.
#KT_A_Geometric_Approach
Another example on how to transform a non-symmetric (8x8) Solution by De Mairan (see our Tweet on Jan. 19, 2023) this time by applying "Virtual Configurations".
#TourDuCavalier #SymmetricSolution #KnightsTour
#KT_A_Geometric_Approach
A known Bisection including 4 "Centric-Configurations" can generate 15 additional Bisections by applying the "Free Passes" method described by #Philippe_Jolivald in 1882. Here's an example (see our Tweet on Jan. 15, 2023).
#Cavalier
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