** Knight Tour Tessellations **

(click on image to see a 32x32 Knight's Tour)
In the above Closed Knight's Tour animation, notice the basic
patterns being made by each consecutive 16 moves of the knight. By
putting each pattern on a separate chess board,

closing the open pattern to make a geometric shape, and filling in the
pattern with the same color as the line path's color, there now exists
four pieces that can be connected into one piece,

in such a way that it can be tessellated. Tessellation means the complete covering or tiling of a plane
or space with a non-overlapping shape.

(Click image to see another Knight Tour tessellation.

Be the first to recreate the original closed Knight's Tour from that tessellation and WIN $500.00)

[Contest is over as of May 29, 2003. Here is the Solution along with the best two entries.]

Michiel de Bondt used the above tessellational patterns to make a very nice web-page background for http://www.math.ru.nl/lo_shu_tot_sudoku/. As you can see, Arno van den Essen used the same tessellational background for the cover of his book: *Magische Vierkanten - Lo Shu Tot Sudoku*.

Michiel and Arno are both mathematicians in the Netherlands at the Radboud Universiteit Nijmegen Institute for Mathematics, Astrophysics and Particle Physics. Arno states the following about his Dutch book:

On pages 100-103 I describe your work on tessellations and mention on page 101
that also the tessellations on the cover are created by you.

On page 227 I refer to your internet site www.borderschess.org/KnightTour.htm.
The translation of the Dutch text is: "A beautiful site of Dan Thomasson on knight tours, containing very nice and artistic applications."

Of course, on page 11, I thank you (and some other people) for their help!

Knight Tour tessellations can be used to create beautiful symmetrical patterns. A few of the many
possibilities are shown below.

Astersphaira (Star Sphere)
represents the surrounding stars of the Universe.
It is a 3-d tiling of 12 stars. Each of the five limbs
of the stars were created by the diamond patterns
made from the knight's move on the chess board. If a
boarder-line was drawn around the outer perimeter of each star, 12 pentagon patterns
would appear, thus making a Dodecahedron
(12 sided figure). Historically the Dodecahedron symbolizes the Universe.

At the same time, when looking at the Astersphaira, one may notice
20 triangular shapes made with 3 diamond patterns each. This construction would then be
called an Icosahedron. Both the Dodecahedron and the Icosahedron are harmoniously intertwined.
The Icosahedron has 20 faces and 12 vertices while the Dodecahedron has 12 faces and 20
vertices, an exact reciprocal of each other.

Here are a couple other regular and irregular tessellational patterns made from
the same knight's tour diamond shapes.

What do you get when you cross a Dodecahedron with an Icosahedron? Answer.

(click on the image above to see a tiled sphere)

Did you notice when looking at the tiled sphere that both the Icosidodecahedron and the Hexastersphaira graphics are actually identical. By replacing the perimeter edges in the Icosidodecahedron with star limbs, the Hexastersphaira takes its form.

Polarsphaira (Polar Sphere) represents the polarity of the earth and earth itself. It is a 3-d tiling of six compass shapes. Each compass contains four rhombus patterns (representing North, South, East, and West) made from the knight's move on the chess board. If a boarder-line was drawn around the outer perimeter of each compass, 6 square patterns would appear, thus making a Cube or Hexahedron (6 sided figure). Historically the Hexahedron symbolizes the Earth.

At the same time, when looking at the Polarsphaira, one may notice 8 triangular shapes made with 3 rhombus patterns each. This construction would then be called an Octahedron. Both the Hexahedron and the Octahedron are harmoniously intertwined, as is the Dodecahedron and Icosahedron mentioned earlier. The Hexahedron has 6 faces and 8 vertices while the Octahedron has 8 faces and 6 vertices, an exact reciprocal of each other.

Four of the five platonic solids are represented by the knight's move on the chess board. The perimeters of these platonic solids can all be seen in "Creation Designed". The only other pattern made by the knight is a square pattern which makes a cube. The fifth platonic solid, not yet mentioned, is the Tetrahedron which is contained within the cube. The Tetrahedron represents the concept of system. It is the underlying basic structure of the earth, sun, stars, and life itself. In reality, approximately 90% of all the earth's crust is made from silicate minerals (a group of rock forming minerals) which are based on the fundamental structural unit: a Tetrahedron.

The paper model shown above combines the Astersphaira (Universe) and Polarsphaira (Earth). I made an ornament of sorts where the Polarsphaira resides inside the Astersphaira. Inside of the Polarsphaira resides the final shape of the knight's tour, a square which makes a cube containing the hidden Tetrahedron. It is a quarter inch block of wood that will have six different jeweled tiles covering each of the faces. The tiles represent the cyrstals or precious gemstones of the earth that have a cubic or hexagonal crystaline structure (diamond, ruby, emerald, sapphire, topaz, amethyst). They act as the inner light or energy source for the world.

I could use the individual spheres for geometric analysis. Combined into a model, they can make lamps, mobiles, and different types of ornaments. The cube in the center of the ornament can still be the various jeweled tiles but with a small lightbulb in the center to radiate the various colors of red, green, blue, purple, yellow, and clear. Rotating the individual spheres in opposite directions while lighting the cube would make an interesting effect.

The stand and the combined spheres were extremely easy to make. I used 67 lb white cover paper, a drink sterring straw, coated copper wire, thick piano wire, and Elmer's glue. I got the quarter inch wood cube and quarter inch jeweled tiles from Michaels, a local craft store. When you make the spheres out of paper, you can actually collapse the spheres inward to make other star or spur shapes. Collapse the spheres until the edges of the limbs touch each other. Pretty cool patterns appear similar to the beautiful candle lantern shown on either side of the paper model. The candle lantern is a one-of-a-kind handcrafted piece made in a small mountain town in the heart of Mexico. The lantern was distributed to a local gift shop via Norwich, Vermont under the label: Viva! the Evolution.

Isn't it interesting about the dichotomy of creation. The Astersphaira (Star Sphere) contains the Dodecahedron and Icosahedron. The Polarsphaira (Polar Sphere) contains the Hexahedron and Octahedron. Notice that those shapes can be easily created from spheres when using the 3-d tiling of the knight move patterns. I like using the model as a teaching aid to show the relationships of the platonic solids. The cube at the center of the Polarsphaira contains the Tetrahedron but it is not visible to the eye.
The ornament can have several different stories associated with it such as the following:

The Christian Trinity of the Father (Astersphaira), the Son (Polarsphaira), and the Holy Ghost (Tetrahedron)

The Biblical book of Genesis 1:1 - In the beginning, God created the heavens (Astersphaira) and the earth (Polarsphaira)

The Hindu mythological male Purusha (Icosahedron) and female Prakriti (Dodecahedron) giving birth to Purusha Junior

The Chinese Yen and Yang, nature's two channels of electric-flows which bring about changes in the Universe

The Chemistry or Physics "polaron", a conducting electron in an ionic crystal together with the induced polarization of the surrounding lattice

I made up the words Astersphaira, Polarsphaira, and Hexastersphaira using mostly the Greek language to give homage to Greek geometry. Astersphaira is from the Greek Aster (Star) + sphaira (sphere). Polarsphaira is from the new Latin Polaris (Polar) + sphaira (sphere). Hexastersphaira is from the Greek Hex (Six) + aster (star) + sphaira (sphere).

With 3-d glasses (red-blue, or red-green), you can enjoy looking at 3-d models of the platonic solids that change shape automatically in a Java applet found at dogfeathers.com web site. I felt like I could touch the shapes coming out of my monitor. Check out some 4-d hyperspace polytopes in the following links. A polytope is a 4th dimension polygon.

Hyperspace Star Polytope Slicer

Stellations of the Dodecahedron

The following tessellational patterns encompass all the basic shapes
(square, diamond, and rhombus) made by the knight's tour moves. They show two different
geometrical cubic staircase patterns.

### Four Color Chain-Linked Tessellation

While playing around with knight moves on the chessboard from the home page, I ended up with an interesting symmetrical pattern that could be turned into a very nice four color chain-linked tessellation. Enjoy the following animation.

After making the above animation, I connected the four symmetrical patterns from the 8x8 square to make a 7x7 square tessellation piece. I then divided each individual pattern within the square into three smaller shapes representing the 2x1 "L" shaped move of the Knight. I changed the colors of the red squares within the four symmetrical patterns of the Knight's Tour to include blue, green, yellow, and white. As you will see, the white patterns become the background, and there are no white colored links. All the white shapes should be transparent, thus only showing the red, green, blue, and yellow links. The center square of the 7x7 tessellation piece is actually a gap, as can be seen in the animation.

Four copies of the 7x7 square shown above on the right, can be connected together in such a way as to make a single larger tessellation piece. The center white square of the four combined pieces is also a gap. Therefore, the pattern is not a true regular tessellation, but a complex tessellation where small gaps do exist. None-the-less, the tiling of the tessellation piece makes for an interesting interlocking chain of red, green, blue, and yellow square links.

Now we can replicate the larger tessellation piece, and tile the pieces into a work of art that I call the **Four Color Chain-Linked Tessellation (derived from Knight's Tour)**:

(Click image to see a Four Color Chain-Linked Regular Tessellation)

Here are a couple other examples of the chain-linked tessellation. When you create your own similar tessellations, experiment with different colors for the links and the background. You may even use actual scenic pictures for the background. If the pictures look fuzzy, or the edges of the square links look jagged or rough, expand the browser window to full screen, then press F11 (Internet Explorer). To get Internet Explorer back to the original view, press F11 again. Using 1024x768 or greater screen resolution size may also help.

Black and White Chain-Linked Tessellation

Multi-Colored Chain-Linked Tessellation

Patriotic Chain-Linked Tessellation 1

Patriotic Chain-Linked Tessellation 2

While creating these square chain-linked tessellations, I decided to go ahead and make a simple interlocking ring tessellation. The curves are a bit rough, so I'll need to redo the rings when I get a better graphics program. Perhaps someone can suggest a good, yet simple, geometric graphics program I can use.

Here are a couple other crude drawings of circle tessellations.

Circle Tessellation

Multi-Circle Tessellation

Here is a tile called "Illusion" that I created. It can be used as a web-page background image such as http://www.borderschess.org/illusion.htm, or for other artistic creations.

Depending on one's perception of the tiling, different objects may appear with various 2-D or 3-D orientations. The flow of the objects appear to move in opposite directions. Experiment with different color combinations when making your own tessellation pieces using my "Illusion" design.

ILLUSION

© 2005, Dan Thomasson

Just for fun, I animated only two frames of the "Illusion" tile to create a new tessellation called "Still Motion."
See http://www.borderschess.org/Still-Motion.htm.

STILL MOTION

© 2005, Dan Thomasson

The further your eyes are from the screen, the more the lines in the "Still Motion" tessellation appear to oscillate when looking at the web-page background. Actually, the only thing that changes is the medium grey and white colors alternating within the tile pattern. If you copy my tessellation designs, or any other graphics from my web-site for publication, or posting on your web-site, please reference my name and copyright date.

As for the unique geometry created by knight moves that can make beautiful 2-dimensional, 3-dimensional, and multi-dimensional tessellations, I call this geometry: Springer Geometry (patented as Geometric Craft and Educational Kit - Patent #7029364, April 18, 2006). Check out the book *POLYHEDRA 2 Part 2* (under *Symmetry: Culture and Science* - Latest Issues), where you will find a 17 page article I wrote about Springer Geometry. The article was published in Budapest, Hungary, in the journal of Symmetrion called *Symmetry: Culture and Science* Volume 13, Numbers 3-4, pages 401-417, 2002 (ISSN #0865-4824, ISBN #963 214 761 8). It is published by the International Symmetry Foundation.

See the following abstract for the article:

*Polygonal shapes made by chess pieces on the chessboard, especially those made by the knight, are the subjects of this text. The article is named "Springer Geometry," to give homage to the great German chess players and mathematicians throughout the ages. 'Springer' is the German translation for the English word: 'knight.' I dedicate this article to Grandmaster Karsten Müller, a top German chess player and doctor of mathematics who graciously took the time to translate my Knight's Tour Web site, http://www.borderschess.org/KnightTour.htm, into German. While focusing on chess-piece polygons, this article covers the following three topics: 1) Polygons as Counters, 2) Tessellations, and 3) Ornament-type Symmetrical Spheres.*

Many thanks goes to Paul Bourke for his excellent and professional renderings of my Springer Geometry concepts. Check out the following letter from Paul Bourke to see how he is using my geometry in Astrophysics.

Much of my work on Knight Tours is referenced on hundreds of other Web sites, and used by schools, universities, and technical institutes to help their students learn programming logic, create artwork such as tessellations from Knight Tours, and to learn basic concepts about geometry (platonic solids) and other areas of mathematics. See the following small sampling of who is using my Knight's Tour information:

Philipps-Universität Marburg, Fachbereich Mathematik und Informatik: Ubungen zu, Parallelität in funktionalen Sprachen Nr. 12, Abgabe: 5. November in der Vorlesung

University of Edinburgh, Science and Engineering, School of Informatics:
MSc - Fundamentals of Artificial Intelligence, First Assessed Practical

University of Maryland, Computer Science, College of Engineering: UMBC CMSC 201 Project 4 Fall 2006

www.BordersChess.org/KTtess.htm * modified 2007.4.30*