** Knight's Tour Art **

The following Knight's Tour reflects simplicity, beauty, and semi-symmetry
in which the two left quadrants show the same path of the knight, while the
two right quadrants mirror the path. Please contact me if you have created
interesting knight tours. I will pick the best ones and add them to this
section of the webpage.

(click on the image above to see a static view)

Another interesting tour gem, a mini-knight tour of only 16 moves, can be
used to create the most amazing geometric pattern called a 4th dimensional
hypercube. The path of the knight could be: d8, b7, a5, b3, d2, f3, g5,
f7, d6, c4, e5, c6, d4, e6, c5, e4. By placing a dot in the center of each
square that the knight moves to, then connecting each dot with a line that
represents a legal knight move, the end result is a perfect 4th dimensional
hypercube. Notice that there are 16 different cubes, all of which have the
same exact dimensions, inside the hypercube.

(click on the image above to see all 16 cubes)
Rudi Ashdown noticed that my mini-knight tour of 16 moves provides only half the total moves necessary to make a complete hypercube. He has graciously provided the following completed "knight tour hypercube".

To see how a 10x10 Knight's Tour was artistically used to relieve writer's block, and to provide structure for a literary masterpiece, check out Georges Perec's 10x10 Knight's Tour.

Below you will find an interesting pattern that can be repeated over and over again to
make larger and larger open knight tours. The inspiration to create this tour came from a request I received in email from Viv at the British Library. She asked: "Do you know of anywhere that shows the solution to a 9 x 9 knight's tour board starting in a corner square?" I could not think of a nice 9x9 tour so I created one for her.

I used Mathematica to generate all 5x5 tours that started with the first move at the bottom left corner of the board, a total of 304 tours. I selected a 5x5 tour pattern that was symmetrical and expanded it to a 9x9 tour by making a rotational pattern of knight moves around the board. I later realized that I could continue the process of making larger open knight tours. In fact, the process can continue "To Infinity & Beyond!" This phrase was made popular by Buzz Lightyear, the animated space ranger, in the movie *Toy Story* by Disney-Pixar in 1995.

Notice the blue letter 'S' in the center of the tour. The 'S' represents the first letter of 'Springertour,' which is the German word for 'Knight's tour.' I might incorporate this letter 'S' design as my logo for Springer Geometry. Claude Bragdon illustrates the same 'S' pattern design labeled 'Magic Lines From Euler's Squares' in his book: *Projective Ornament* published in 1915 by The Manas Press in Rochester, New York.

(click on the image above to see another infinite open Knight's Tour with blue lines)
The same first 80 moves of the infinite open Knight's Tour with blue lines were also created by Johannes (Hans) Secelle and Albrecht Heeffer on March 09, 2002 to demonstrate an infinite positive integer sequence. Their sequence of moves are posted on "Sloane's Database of Integer Sequences," Part 55 of the On-Line Encyclopedia of Integer Sequences under the A068614 section.

Below you will find another interesting pattern that can be repeated over and over again to
make larger and larger closed knight tours. The pattern comes from my
Closed Knight's Tour Solution Key. Ceramic tiles, wood, or plastic can be cut
into these basic shapes to make the secret closed knight's tour pattern.
The cut pieces could then be put on floors, walls, or table tops. For
simplicity, create the pattern for an 8x8 square board. Do not put numbers
on the top of the pieces. If you want to make it into a puzzle, cut out the
pieces and put the corresponding numbers on the back of the pieces in a very
light shade or impression.

When I practice creating knight tours on paper with checkerboard squares, I
can quickly complete an 8x8 square closed knight's tour in 15 seconds by
marking off each knight move on the squares with the following symbols: /,
-, \, |. I first use '/' for all diamond shape patterns leaning in the same
direction as '/' in all four quadrants. I then use '-' for square shape
patterns in all four quadrants, then '\' for the opposite diamond patterns,
and finally '|' for the opposite square patterns.

After doing hundreds, possibly thousands of knight tours, using those four
simple marks (which by the way are much easier to jot down with a pencil
than numbers - especially if the tour is 10,816 squares), I realized that
they made a neat pattern on the chessboard which could be reproduced with
tiles. I just follow the path pattern that I use for the Closed Knight's
Tour Solution Key and other tours that are displayed on the Knight Tours web
page.

I might consider making a cool poster out of it or some of the other tours
I've created. There is a definite symmetrical pattern. Just think, by
creating that one quadrant, it can be used as computer wallpaper, web page
background, watermarks, or quilts by tiling it. Experiment by changing the
line color and thickness, and the tile color. By combining these four
shapes in various ways, alphabets, number systems, and ciphers can be easily
created. Who knows, maybe there could be a fictional story written that
centers its plot around the patterns in the Secret Closed Knight's Tour.

See www.borderschess.org/KTtess for
information about how the following Astersphaira (Star Sphere) was conceived.

Check out other renderings of my Star Sphere by Paul Bourke at
local.wasp.uwa.edu.au/~pbourke/geometry/starsphere.

Look at the following excellent 3-d artwork by Jason Martineau. His
Zonohedron encompasses all of the patterns (square, diamond, and rhombus)
made by the moves of the knight on the chess board.

The following closed knight's tour comes from www.borderschess.org/KTtess where it is used to show how knight tours can contain tessellational patterns. This same tour also contains a pattern that if animated could be called a pinwheel, windmill, water mill, fan or whatever you might think it resembles. Look closely at other knight tours to see what familiar patterns you can find.

A gentleman by the name of Biniam Gebremichael has made a very nice 64x64 color gradient Knight's Tour.

(click on the image above to enlarge the graphic)
In the above image on the left, the Knight starts at square (0,1) - top left corner, second column, and first row. Using a Warnsdorf heuristic algorithm, the Knight moves around the board visiting all squares. A gradient color pattern fills the board with the first few squares being black, then dark green, then finally light green for the last few moves. The last square visited is (63,1) colored red (enlarge the image for a better view). Different designs can be obtained by varying the colors, the generation formulas, or the starting square of the Knight, as in the above image on the right. Look at the PHP source code ktour.php to see how the images are created.

I've finally decided to let computers solve additional knight tours. There are many algorithms already published that can solve knight tours such as the Warnsdorf algorithm previously mentioned. The algorithm I like best is from Dr. Colin Rose at www.borderschess.org/KTcode. Michael Taktikos, a german mathematician, and I have modified the code to include an end list so that both the first few moves and last few moves can be chosen to solve specific types of tours (especially closed tours).

For now, please take a look at some unique 3x10 closed knight tours and (3x4, 3x7), 3x9, and 5x5 symmetrical open knight tours generated by my PC using Dr. Colin Rose's Mathematica code. Mathematica has also solved all 9,862 unique closed knight tours of a 6x6 square board. Here are the first 10 of 168!/(63!*(168-63)!), just slightly less than 1 octillion (1x10^{48}), 8x8 sequentially backtracked knight tours. Here are some nice 10x10 90-degree rotational quaternary symmetric tours.

Juro Bystricky from Canada was trying to calculate all 8x8 Knight Tours on his computer. He wrote a program that printed out all tours in a text format. After a few months, precisely 1973 hours, he had to stop calculating new Knight Tour solutions. There were simply too many viruses that required applying service packs, and he finally had to reboot the machine, which terminated the program. The program itself ran as an idle task on a 2.4G Pentium 4 machine. It calculated over 18 Million solutions that were logged into a 4GB size text file. Since he found no tool that can actually handle text files of this size, Juro split the file into chunks of 100,000 solutions each. He then zipped the file chunks, compressing them into about a 400 MB file that would easily fit onto a single CD.

### Knight Tours and Music

There are all sorts of correlations to music from knight tours. In fact, if you think about it, the chessboard is a grid that when written in algebraic notation, directly resembles musical notes. a1 to a8 would be the first column on the chessboard counting from bottom left to upper left of the board. The basic notes in music are a, b, c, d, e, f, g. The last column on the right is h, which in German music, h represents sharps (I think - you can research this to make sure and correct me if I am wrong).

Therefore, you can actually use a Knights Tour to play interesting tunes. I will be adding a section to my Web page for orchestrated music via knight moves. For instance, I could have multiple knights on the chessboard where every square equates to the actual notes, then have multiple tours running at the same time to make chordal type music. a1 would be the lowest octave for note "a," a2 would be the next octave higher for note "a," a3 ..., etc.

In western music, we normally have 12 tones for one octave which consists of 12 half steps. For instance, if we take a simple major scale of c: we would have c - d - e - f - g - a - b - c. Notice that the steps between each note are c - (1) - d - (1) - e - (1/2) - f - (1) - g - (1) - a - (1) - b - (1/2) - c, a total of 12 half steps, or 12 tones. For a minor scale, we would have a - (1) - b - (1/2) - c - (1) - d - (1) - e - (1/2) - f - (1) - g - (1) - a.

If you move a single knight across the board from a5 to c4 to e3, you have played a simple arpeggio for A-minor. An arpeggio is like plucking the individual strings of a chord on a guitar, or playing the notes one at a time on the piano. In fact, chords are built from every third note of the scale. A basic A-minor chord would consist of the following three notes played at the same time: A - C - E.
By adding multiple knights, say three knights, they can be spaced on a musical chessboard to play chords. Thus, put one knight on any "a" square, the second knight on any "c" square, and the third knight on any "e" square. If they all start their tours from those first squares, the first chord played will be A-minor. Adding more knights extends the range of the chords into 7ths, 9ths, 11ths, or 13ths.

www.BordersChess.org/KTart.htm * modified 2006.12.14*