It was only when I received examples of work from Michael Creighton of Liverpool at the end of April 2013 that I decided to publish this section on rotary tours. Ten examples by Mr Creighton are now included. He has a particular interest in using board shapes that fit together to tile a larger area. He has also produced some versions in which the cells are coloured, and also more elaborate artworks based on tours. These can be seen on his Flickr page: https://www.flickr.com/photos/95665358@N08
Here we collect tours that show rotary symmetry, invariant to 180° rotation, but altered by 90° rotation or reflection. This type of symmetry is also termed 'binary oblique' or 'diametral' symmetry. The assortment of tours shown here is not the result of any particularly systematic study, just an accumulation of various special cases. Many more examples of this type of symmetry will be found in the separate studies of Square and Oblong boards. For examples on boards of less than 16 cells see the section on Smallest Knight-Tourable Boards.
Rotary symmetry is of two types:
Eulerian where the path circles the centre point without passing through it. This is the symmetry that is most often seen in knight's tours, since it is the only one possible on the orthodox chessboard. Leonhard Euler was the first to construct multiple examples of it. The containing rectangle for an Eulerian tour can be of any of the three types: even by even (EE), even by odd (EO), or odd by odd (OO).
Bergholtian where the path passes through the centre, twice. Bergholtian symmetry is closely related to open paths with centrosymmetry. This type was first studied by Ernest Bergholt. The frame of the board for Bergholtian symmetry must be of EO type (one side even the other odd) so that the centre is at the mid-edge where two cells meet.
16 cells. The 4×4 board admits no complete tour, but moving two corner cells will produce two centro-symmetric boards with equal numbers of dark and light cells when chequered. One of these has four Eulerian symmetric closed tours (shown below). The other has no closed or symmetric tours but has 16 asymmetric open tours. (See the Asymmetry section). I also show the four symmetric tours of a holey board chanced upon while constructing the 20-cell tours shown further below.
18 cells There are two closed rotary tours on the 4×5 board with two corner cells omitted, one Bergholtian and one Eulerian.
Some impromptu Eulerian examples including three with holes.
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20 cells. The four 20-cell tours here (Jelliss 1996) on a board with birotary symmetry have only rotary symmetry. The first two have four 1-1 moves around the centre in what on a square board would be diagonal symmetry, and the other two have the 1-1 and the 0-2 moves in apparent lateral symmetry. However they are not examples of 'mixed quaternary symmetry' because the board does not have diagonal or lateral axes. The other diagram shows a 20-cell board (4×5 with two corners moved) with a unique Bergholtian tour (Jelliss 2009). It also has 16 asymmetric tours.
The 4×6 board with corners removed has four symmetric tours (Jelliss 2016) showing a type of 'pseudo-mixed' symmetry. Here the main symmetry is biaxial with the 1-1 moves deviating in the first two and the 0-2 moves in the other two.
22 cells. The first tour here is given by Godron and Vatriquant in L'Echiquier Jan 1929 and has Eulerian rotary symmetry. I find four others of this type and one having Bergholtian symmetry.
24 cells. This cross-shaped board seems to have been neglected.
I found the following 16 symmetric tours (Jan 2017) while compiling these pages.
The letters indicate the central angles. The pairs with the same lettering differ only by the positioning of the 1-1 links.
No tours are possible on the centreless 3×9 board due to colour mismatch, but by omitting two corner cells the three Eulerian tours shown become possible (Jelliss 2003).
The first tour here was constructed by Ernest Bergholt (1917) specifically as an example of Bergholtian symmetry on a non-rectangular board. The other two are my own (Jelliss 2009). These three are the only symmetric closed tours possible on this board. Three symmetric Bergholtian closed tours (Jelliss 2009) on a board-shape used for an asymmetric tour in L'Echiquier Nov 1928.
The other shaped Eulerian tour, above right, is by Godron and Vatriquant from L'Echiquier Jan 1929.
The following symmetric board, used for an asymmetric tour in L'Echiquier July 1928, is derived from the 4×6 rectangle with two corner cells moved. It has a large number of tours (total unknown). The following is an enumeration (Jelliss 2015) of the symmetric tours on this board. Total 56.
1. Tours with diagonal acute angle at end, total 24.
2. Tours with lateral acute angle at end, total 14.
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3. Tours with right angle at end total 18.
26 cells. Eulerian tour by Godron and Vatriquant from L'Echiquier Jan 1929.
28 cells. These four Eulerian tours (on a sort of 'tanned hide' shape) appeared in L'Echiquier Dec 1928 where they are attributed to four composers: Tolmatchoff, Marques, Godron and Vatriquant.
30 cells. Of course no tours are possible on the 4×8 with opposite corner cells removed, since there are 16 cells of one colour and 14 of the other. Here are an Eulerian and a Bergholtian tour by S. Vatriquant, L'Echiquier May 1929 on a shaped board.
32 cells. Rotary tours on the 6×6 without corners by Euler (1759) and Jelliss (1999). Naidu (1922) gives a symmetric closed tour on the 6×6 board with four holes.
This 32-cell board, can be regarded as a 'modal transformation' (a sort of 45 degree rotation) of the white or black cells on an 8×8 chequered chessboard (as was pointed out by T. R. Dawson). It has just four distinct knight tours with Bergholtian symmetry (Jelliss 2009). They correspond exactly to the four {1,3}-leaper (camel) tours on the chessboard.
Five tours of a 32-cell board (Jelliss 2018) formed by moving corners of a 4×8 board to the mid-edge. These tours relate to the five magic tours of squares and diamonds type that have this board shape as their background pattern.
The 3×11, apart from 3×3, is the smallest odd-by-odd centreless board that admits tours.
36 cells. This tour is from L'Echiquier Aug 1928. Tours are impossible on the centreless 3×13 board due to colour mismatch, but this can be overcome by omitting two more cells.
38 cells. H-shaped tour from Warnsdorf 1823 (rotated 90 degrees).
40 cells. A rotary tour of an octonary board (Jelliss 2017) formed by omitting 6 cells from each corner of the 8×8. Birotary is impossible on this board since 40/4 = 10 is even.
42 cells. A shaped board with Bergholtian tour.
44 cells. Two more rotary examples from L'Echiquier Apr 1929. The first by Vatriquant and Post, the second by Godron and Tolmatchoff. Also a tour on centreless 5×9 board (Jelliss 2003).
On the centreless 3×15 board there are 14 tours, of which I show four examples.
48 cells. Quaternary symmetry is possible on the centreless 7×7 board but those shown here (Jelliss 2003) have the simpler (or is it more complex?) binary symmetry. See also 120-cells.
The next two tours on rather amorphous boards (Jelliss 29 May 2013), one Bergholtian and one Eulerian, use no acute angles. See also 52 cells.
52 cells. Tour (Jelliss 30 May 2013) shows 'pseudo-mixed quaternary symmetry' and no acute angles. Only the 1-1 moves deviate from oblique quaternary, but their diagonal axes are not axes of symmetry of the board.
There are just these three geometrically distinct rotative tours (Jelliss 2015) on this 8×8 board with a cross-shaped hole.
56 cells. Rotary tour by Michael Creighton (2013) on expanded oblong (4×12 + 8).
58 cells. Two boards with Bergholtian tours (Jelliss 2014).
64 cells. Here are two tours with 180 degree rotational symmetry (Jelliss 2014) on an alternative 64-cell board of the truncated Greek Cross-shape used to designate medical services:
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Two rotary tours on expanded oblongs (4×12 + 16) by Michael Creighton. Besides fitting together with copies of themselves, the first fits with the 56-cell tour shown earlier and the second with the 96-cell board shown later.
Board with rotary symmetry, above right, the tour (Cognet L'Echiquier Oct 1930) being of Bergholtian type.
68 cells. A rotary tour by Michael Creighton (2013). This fits with the 92-cell tour shown later.
72-cells: A board with octonary symmetry, depicting a lake with a platform in the centre, used for an asymmetric tour by the Rajah of Mysore, but here showing one with rotary symmetry (Jelliss 2009). This is a board within an even area, 12×12, on which bitotary symmetry is impossible, since 72 = 4×18 and 18 is even. For another 72-cell ring tour see the 120-cell entry.
80 cells. The Greek Cross tours below show rotary symmetry. The upper two are by Ernest Bergholt from Queen 22 Jan 1916 and from his First Memorandum 24 Feb 1916. A handwritten note by Bergholt alongside the first diagram on the copy of his article in Murray's collection asserts: "I have since constructed several still more remarkable for their elegance" but no others are given there. He may have meant tours with mixed quaternary symmetry, of which Murray composed many, or may refer to his 84-cell tour with birotary symmetry. The lower tour here is by H. J. R. Murray from his 1942 manuscript "selected from some 60 tours on this board in mixed quaternary symmetry composed by the author".
Binary tour, above right, on the centreless 9×9 board (Jelliss 2003): See also birotary section.
88 cells. These two tours with oblique binary symmetry by Bergholt are in his notes on symmetry sent to Murray in 1917. They bear the titles 'Terra Ignota' and 'Terra Cognita'. The latter is a tour showing mixed quaternary symmetry, with h:j:k values 3:6:13, the h-moves being a4-c3-b5 and d1-f2 and their reflections in the vertical and horizontal medians, the obvious octonary pattern j-moves being typically b2-d1, d2-e4-f5. The third tour (Jelliss undated) is more obviously binary.
92 cells Two tours by Michael Creighton (2013). They relate to the 68 cell tour shown earlier.
96 cells. An X-cross by Michael Creighton (2013).
100-cells. A board 10×10 with the four corners moved to the middle of two sides, with closed rotary tour from L'Echiquier Nov 1929 by Monsieur Post.
108 cells. A 'hashtag' board with a rotary tour by Michael Creighton (2013).
112 cells. A board (144 – 32) used for an asymmetric tour #12 in Harikrishna 1871. Rotary symmetry (Jelliss 2009) is shown on the same octonary board. It does not admit quaternary symmetry. This tour is not in mixed quaternary, as shown for example by the moves of type a4-c5.
116 cells. Another in my series of centreless odd-by-odd board tours, on the 9×13. (Jelliss 2003).
136 and 144 cells. These tours by Michael Creighton (2013), are on boards designed to fit together to form a continuous strip or tile an area.
152 cells. This X-shaped tour by Michael Creighton (2013) fits horizontally with the first 192 cell tour below, and vertically with the 248 cell tour further below.
160 cells. Edward Falkener (1892) gave a collection of 18 tours showing approximate quaternary or octonary symmetry on this four-handed chess board. However only #3 has true binary symmetry.
A. Falkener 1892 #3, and a version of his #8.
B. after Falkener #6 and #17.
The tours shown here are closely based on his designs and achieve proper binary symmetry.
C. after Falkener #5 and #18.
162 cells. A rotary border design for the Knight's Tour Notes title page (24×9 – 18×3).
188 cells. Another tour by Michael Creighton (2013).
192 cells. Three tours by Michael Creighton (2013). The first 192 cell tour fits with the 152 cell tour above. The other two 192 cell tours complement each other.
208 cells. Another example by Michael Creighton (2013).
232 and 248 cells. Another pair of tours on complementary boards by Michael Creighton (2013).
248 cells. Pseudotour by Michael Creighton (10 Feb 2017) the inner part of a much larger pattern which has three further outer bands of "symmetric ripples towards infinity" as he calls them.
The centre is Euler's 12-cell cross. The surrounding rings use 80, 156 cells (then 252, 332, 412 cells). His drawing uses alternate colours and leaves out the border lines. See also 120 cells.
252 cells: This 36×12 – 30×6 tour was designed as a border for the Bibliography section.
336 cells. A 20×20 board with an 8×8 central hole by Michael Creighton (2013). He has also produced tours on even larger shaped boards.
A coloured version of the 248 cell tour. Click on the image to see a larger version. Photo from Michael Creighton.
