Knight's Tour "Springer" Geometry
As for the unique geometry created by the Knight's moves that can make beautiful 2-dimension, 3-dimension, and multi-dimension 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 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.
Check out the book POLYHEDRA 2 Part 2 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.
Symmetry: Culture and Science, - Springer Geometry
See the following abstract for my article about Springer Geometry in the math journal: Symmetry: Culture and Science, Budapest, Hungary.
Many thanks go 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's Tours is referenced on hundreds of other Websites, and used by schools, Universities, and Technical Institutes to help their students learn programming logic, create artwork such as tessellations from Knight's Tours, and to learn basic concepts about geometry (platonic solids) and other areas of mathematics.
- 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 go 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's Tours is referenced on hundreds of other Websites, and used by schools, Universities, and Technical Institutes to help their students learn programming logic, create artwork such as tessellations from Knight's Tours, and to learn basic concepts about geometry (platonic solids) and other areas of mathematics.
The following Gallery of pictures are computer graphic renderings by Paul Bourke in June 2002 of my Springer Geometry including Astersphaira, Hexastersphaira, Polarsphaira and other 3d Knight's Tour Sphere images. View several of these images on my Knight's Tour Tessellations webpage.
Select the images below to see a full view of each rendering. Using your cellphone, select the first image then slide your finger to the left across the middle of the screen to scroll through all the images.
Select the images below to see a full view of each rendering. Using your cellphone, select the first image then slide your finger to the left across the middle of the screen to scroll through all the images.
