Symmetrical Knight's Tours
The following Knight's Tours are examples of millions of similar Knight's Tours. The first two graphics show four 180 Degree Rotation Symmetric Knight's Tours. The third graphic shows four Knight's Tours made up of 2 (32 moves) circuits in each tour.
The following graphic shows the making of a 180 Degree Rotation Symmetric Knight's Tour. Reconnecting the four Knight's moves (red lines) from graphic A to different squares in graphic B makes two new Knight's Tour circuits. Likewise, graphic B makes two new Knight's Tours in graphic C. Moving just two Knight's moves (red lines) from graphic C makes the single 64 move Knight's Tour shown in graphic D.
The next couple of images show additional examples of making a 180 Degree Rotation Symmetric Knight's Tour. Move the Knight's moves (red lines) from graphic A to new squares to make graphic B.
Here are some additional images showing how to make the above 180 Degree Rotation Symmetric single circuit Knight's Tour also referred to as radially symmetric. CMG Lee made a colorful redrawing of this same tour on wikimedia.org.
I made one mini-knight's tour and replicated it three more times coloring the moves blue, red, green, and orange. I kept the same orientation of the blue knight's moves and placed them on a chessboard. I then rotated the red moves 90 degrees left, the green moves 90 degrees right, and the orange moves 180 degrees and placed them on the same board. Afterwards, I connected all four mini-knight's tours with four black lines (knight moves) to make a single closed knight's tour.
I made one mini-knight's tour and replicated it three more times coloring the moves blue, red, green, and orange. I kept the same orientation of the blue knight's moves and placed them on a chessboard. I then rotated the red moves 90 degrees left, the green moves 90 degrees right, and the orange moves 180 degrees and placed them on the same board. Afterwards, I connected all four mini-knight's tours with four black lines (knight moves) to make a single closed knight's tour.
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The following graphic shows a beautiful symmetrical image of four Knight's Tour circuits overlapping each other. The composite image is 180 degree rotation symmetric as well as quaternary rotation and reflection symmetric.
I decided to repeat the previous figure as graphic A below. Graphic B can be made by moving eight Knight's moves (red lines) in graphic A to new squares. Graphic C is a combination of two Knight's Tour circuits that makes a 180-degree rotation symmetric image. Graphic C has some similarities to graphic A. Graphic D has the same type of symmetries as Graphic A. With a little bit of imagination, one might be able to see a Maltese Cross in graphic D.
Graphic D from the above image is reproduced in the image below as graphic A and B. Graphic B reveals a Maltese Cross from the Knight's Hospitallers from the island of Malta just south of Sicily, Italy. "Tour cavaliere croce maltese" is Italian for Knight's Tour Maltese Cross. Originally, the outermost eight pointed vertices represented the eight Beatitudes from the Biblical Scriptures. The Beatitudes was Jesus Christ's first sermon, 'Sermon on the Mount' from Matthew 5:3-11. I've reassigned the Beatitudes to the eight points of the star at the middle of the cross that represents the foundation or heart of the Knight's Hospitallers. The eight outermost points also symbolize the duties (obligations or aspirations) of the Knights.
- to live in truth
- to have faith
- to repent one's sins
- to give proof of humility
- to love justice
- to be merciful
- to be sincere and wholehearted
- to endure persecution
By slightly modifying graphic B from above, I made the following Great Commission Cross. If you copy the following image of the cross and use it on your web pages or in your publications, please reference my name as the designer.
The following is a plain copy of the above cross. Remember to reference my name if you publish or post this cross on your websites.
Graphic A below is a copy of graphic D: Quaternary Rotation Symmetric from above. Notice how simple reflection and rotation art can be obtained from the Knight's moves within graphic A. I placed four Knight moves (green lines) at the center of graphic C to complete the geometric image.
