The K-Dron, A New Geometrical Shape – Its Nature, Properties, and Consequences
Year: 1999 Authors: Janusz Kapusta
Core claim
The K-Dron is a pure geometric form with distinctive symmetry and mathematical properties that link cube geometry, wave equations, and practical applications.
Topics
geometric form, symmetry, wave equation, map projection
Domains
geometry, symmetry, partial differential equations, polyhedra, sculpture, architecture, visual pattern design, theatrical presentation
Methods
geometric analysis, surface equation, stacking arrangement, pattern generation
Media
three-dimensional form, flat surface patterns, building material, exhibition display
Paper text
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BRIDGES Mathematical Connections in Art, Music, and Science
The K-Dron, A New Geometrical Shape – Its Nature, Properties, and Consequences
Janusz Kapusta 1060 Ocean Ave., Apt. D5 Brooklyn, NY 11226 E-mail: kapusta@earthlink.net
In 1985 in New York I discovered a new geometrical shape, which I called K-Dron, and in 1987, I received a patent for some of its applications. A three-dimensional shape emerged suddenly from drawings about the nature of infinity that I made 6 years earlier during my study of the History of Philosophy in Warsaw, Poland.
The K-Dron is both a remarkably simple and complex structure, but once discovered, it cannot be improved upon – just as a cube, it is a pure form. Its basic form has a square base, 11 multifaceted sides and a diamond-shaped face with a 45-degree angle of inclination. Viewed from above, it is a square within a square. The surface structure is both symmetrical and asymmetrical, concave and convex. You can also consider it as a bisection of a cube; but it is much more complicated. It took me 8 years to discover a natural placement of K-Drons in the universe.
In his 1991 book Connection: The Geometric Bridge Between Art and Science (McGraw-Hill) Jay Kappraff describes the K-Dron. Concerning “the striking optical properties that the K-Dron possesses” Kappraff asked the question, “Why do K-Dron structures exhibit such a strong relationship between form and function?” The answer to this question he saw in the close relation to the symmetry of a cube (orthoschemes).
Professor Stanislaw Kwapien (Texas A&M University, Department of Mathematics) formulated a simple equation for the surface of the K-Dron. He also noticed that the K-Dron surface represents a special solution to the one-dimensional wave equation (vibrating string) in a space-time coordinate system. The fact that the K-Dron surface can be related to both the symmetry of a cube and the solution to the wave equation gives a deeper meaning.
The K-Dron possess many other interesting properties. The interior of 8 stacked K-Drons maps out a rhombic dodecahedron. K-Drons can also be shown to be juxtapositions of pyramids seen from both the outside and inside reminiscent of Escher, and 8 K-Drons (4 white and 4 black) can generate 38,416 flat surface patterns.
A book, K-Dron, Patented Infinity, a complete study and description of K-Dron’s incipience, nature and applications, was published by Science and Educational Publishing House in 1995. In the same year the K-Dron was used as a building material for the Alias Studio in Hollywood, CA. K-Dron shapes allowed for a new map projection of the Earth and Celestial sphere.
In October 1999 a K-Dron exhibition will take place in the Museum of Modern Art in Lodz, Poland, and a theatrical spectacle for children about K-Dron will be presented in the Actor and Puppet Theater in Katowice, Poland.