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Spidron and Spidron system
A Spidron is a planar polygon of finite area but infinite sides, which is centrally symmetric and has vertices that can be placed at the points of two logarithmic spirals.
It can be created by joining the sides of two types of triangles that decrease according to a geometric series (or, when viewed from the opposite direction, increasing) or by breaking down a regular hexagon.
Because of its recognizable, emblematic shape, the author gave this shape the name Spidron, which he trademarked. He named the whole system after that.
In the classic case, the angles of the two types of triangles are respectively 60°, 60°, 60° and 30°, 120°, 30°, countless other Spidron versions can be produced from different pairs of triangles. All solutions form a system and can be assembled into extraordinary planar, spatial, even dynamic, movable structures, flat and space-filling elements in different compositions.
In the first approach, it is worth discussing the so-called "classic Spidron", during which we introduce concepts that we will be able to apply according to these definitions.
The Spidron was an independent work of Dániel Erdély, which he completed in the 1979-1980 school year as a solution to a homework given by Ernő Rubik. He had tried plane experiments before at the Printing Vocational Training Institute, but he created the spatially movable, accordion-like version for the format class of the College of Applied Arts (today's MOME), to which the legendary teacher simply said: "I've never seen anything like this before". This unsolicited comment provided the initial push for further development and work.
Concept collection:
Semi-spidron
This is the shape from which - by joining the longest edges - a Spidron is obtained from two pieces. The area of an equilateral triangle raised to its longest edge is the same as the area of a semi-spidron, and can be divided into it.
Spidron Ring
A ring-shaped zone containing two types of adjacent triangles of six semispidrons that can be inscribed in a regular planar hexagon. There are endless pieces of this that can be embedded in each other. Its area is always 3 times or 1/3 of its adjacent ring.
Interestingly, any ring can be reversed. (This means that with the exception of any ring, or if the Spidron arm is already folded in this way, by rotating it by 180° in relation to the base plane and by 60° in relation to the axis set in the center of the Spidron, the ring can be returned to its original place, thus by "controlling" the rings numbers can practically be coded in a binary number system.
Spidron nest
A planar or spatial ensemble of several, even infinite, nested spidron rings.
Hornflake
A mirror-symmetric assembly of two semispidrons when they are joined along the longest edges.
Comment
With equal right (S) and left (Z) spidrons - using both - the plane can only be covered without gaps and overlaps if hornflakes are inserted between them, while with pure S or pure Z Spidrones the plane can also be covered.
The Spidron versions created from decagons, on the other hand, are the so-called They provide an extremely elegant solution to Penrose tiling, which was also recognized by the Nobel Prize-winning physicist-mathematician. It was developed by the excellent geometer and artist Marc Pelletier, who passed away a few years ago. He was also one of the key members of the international team that, under the coordination of Dániel Erdély, worked almost non-stop from 2003 to 2009 on the production of the various Spidron versions and uses.
Additional members of the working group:
Amina Buhler-Allen
Craig S. Kaplan
János Erdős
Gergő Kiss
Cristiana Grigorescu
Paul Galiunas
Rinus Roelofs
Lajos Szilassi and many others