Circular Boards Again

 These new versions of the previous four tours keep strictly within the circle marked by the black dots. There are no cells whose corners go outside the circle. 

Radius 5

Radius root-50

Radius root-65

Radius root-85

As before the fourth shows 180 degree rotation, the others 90 degree.

Smaller Circular Boards

Here are four tours on the smallest circular boards, with at least 12 fixed corners on the circle. These show boards with radii 5, root-50, root-65, root-85 corresponding to the smallest double-pattern fixed-distance leapers. The first three tours show 90 degree rotational symmetry but the fourth only has 180 degree rotation, due to the quarter-board containing an even number of cells (64). 

Radius 5

Radius root-50

Radius root-65

Radius root-85

There are 17 other two-pattern fixed-distance leapers to consider before the first three-pattern leaper of this type (root 325) is encountered, as used in the large tour previously shown. I doubt if I will get round to constructing examples of all of these! 

There is a certain arbitrariness concerning which cells, whose corners go beyond the circle, should be included or not.

Circular Board

 This is a knight's tour with 90 degree rotational symmetry on a circular board of 964 cells. The 24 black dots on the circle are at exact distance root-325 (approx 18.03) from the centre point. The four corner cells that make possible the birotary symmetry by ensuring the quarter-board has an odd number of cells (241) extend slightly beyond this circle to root-338 (approx 18.38).

The black dots are at points with coordinates {1, 18}, {6, 17} and {10,15} relative to the centre, while the corner points are at {13, 13}. 

Antelope Pettern

Here is another to add to the list. I hope I've got it right. It was difficult to be sure a move wasn't missed.  

I suppose this is a sort of "painting by numbers".

Leaper Move Patterns

 I was inspired to produce these patterns as a contribution to the #Maydala season on Twitter for mathematics and art. They use rainbow-sequence colouring in place of the numbers of moves from the centre to other cells. 

Zebra {2,3}

Giraffe {1,4}

Knight {1,2}

Wazir {0,1}

 I suppose the Giraffe pattern could be cut off at the 9x9 size, if we stop when the first 8-move cells are reached.

Dawsonian Tours on Odd Boards

 It occurred to me that I haven't seen any examples of Dawsonian tours on odd square boards, so I composed a set. Of course closed circuits using all the square numbers are not possible. Other formations have to be considered. Here simple up and down steps.

A 5x5 knight solution is not possible but there is a simple wazir tour with the squares in this formation with a uniquely determined path, and also emperor solutions using knight and wazir moves.

 Diagrams: