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Spidron
System |
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by Daniel
Erdely ©1980 - 2002
Technical and art collaborators |
H- 1015 Budapest, Batthyany st. 31. 1/12.
PACS 02.40.D, PACS 61.50.A
Keywords:
tiling, plane-filling, space-filling, polyhedron
"Crystallogenesis" in Hungarian
A planar, plane-filling object (which I call Spidron) is presented which, to the best of my knowledge, has not been described previously. Upon three-dimensional deformation subject to specific constrains, the planar system generates a regular, dynamic geometrical construction with remarkable features.
Spidron
From twenty-four uniform Spidrons, a closed polyhedron has also been created (which I call spidrohedron) which is a regular shell structure. This solid figure, together with its spatial mirror-form, are able to fill the space so that each Spidrohedron is surrounded by eight counter-rotating, that is mirrored, Spidrohedrons.
The experiments have led to the idea of creating a system of space-filling which, instead of defining the solid figures that fill in the space, classifies the possible structures according to the characteristics of the surfaces that border the solid figures.
The paper also attempts to suggest a classification of new spatial forms, which are based on the vertices of regular polyhedrons, however, their vertices are not connected with edges but curved surfaces.
The Spidron, which is presented here, is a centrally symmetric planar, geometric form. It can be created out of numerous simple planar forms according to the system of rules described below. In addition to the Spidron, there is a related form, which is not centrally, but axially, symmetric and looks like a horn. On the basis of its appearance, it can be called, e.g. Hornflake.
Hornflake
The Spidron and the hornflake originate from a basic form, the Semi-Spidron, which is built of a series of gradually smaller triangles, forming a spiral curve. When the appropriate edges of the triangles are joined, the Semi-Spidron can be transformed into a cone surface. The figures above illustrate the construction and main parameters of the Semi-Spidron. The spiral form is created by alternately adjoining two types of triangles in an infinite series, consisting of equilateral triangles (60° , 60° , 60° – a, a, a,) and isosceles triangles (30° , 120° , 30° – a, a/Ö3, a/Ö 3). Each equilateral triangle is joined with an isosceles triangle of 1/3 the area, and this isosceles triangle is joined with another equilateral triangle having an area equal to the area of the isosceles triangle. Consequently, the area of the whole planar form is the sum of two geometrical series: if the area of the first equilateral triangle = 1, then 1+1/3+1/3+1/9+1/9+1/27+1/27+...+1/3n-1+1/3n-1+1/3n+1/3n, where n approaches Y . The entire sum approaches 2. The area of each equilateral triangle equals the sum of all the triangles with an area smaller than the given equilateral triangle.
Series of isosceles triangles and equilateral traingles
The constructed form that I call Spidron consists of two Semi-Spidrons, at a rotation of 180° . The Spidron is spiral-like, its convex angles always 150° (with the exception of the two angles in the middle, which are 120° ) and its concave angles 210° . Thus the sum of these two types of angles is 360° , which provides a wide range of possible junctions and enables the plane to be filled. Consequently, the Spidron System is able to fill in the plane without any gaps.
Planefilling Spidron System
A unique feature of this system is that we can add new elements to it and we can cut elements out of it without disturbing the continuity of the plane. The result of these interventions is that the plane is transformed into space, according to various regularities.
Hornflake System is not able to fill the plane itsef.
The hornflake consists of two mirrored Semispidrons. The hornflake system is not able to fill in the plane itself, but is able to fill in various surfaces when it is transformed in space.
Below is an account of further investigations regarding the Spidron System.
Spidron relief (click for animations)
I folded every second edge, reaching to the centre of the created hexagon in the given Spidron System, as a spine and folded every first edge as a groove. The resulting relief-like surface, under the impact of an external deforming force, does not show simple linear displacements, such as those produced with an accordion; instead, the edges in between the vertices and the centres of the original hexagonal system move in a vortex within each hexagon. If the relative position of any of the non-neighbouring vertices has been altered, this entails a similar alteration in the relative positions of all similarly situated pairs of vertices. In this deforming process, the cluster of all the vertices creates a three-dimensional system of points with variable density while still maintaining the regular ordering. This has come about in the system in such a way that the length of the edges and, consequently, between the neighbouring vertices, has remained constant. Due to the organisation of spines and grooves, changes in compactness always pertain to each second edge, i.e., every second vortex becomes closer or farther from one another. Thus, the grooves and spines become steeper and steeper until they reach a limit where the plane surfaces bordering the deepest groove/the highest spine, fold up vertically, i.e., become perpendicular to the original plane. Beyond this point, no further deformation is possible.
Spidrohedron
From twenty-four uniform Spidrons, one can create a closed spatial, shell-like form (Spidrohedron=SH), which can be defined as a transitional form between a regular hexahedron and a regular octahedron: the vertices of the spatial formation become concentrated in eight points in space, which correspond to the vertices of the hexahedron; at the same time, if we draw an octahedron around the same hexahedron, its vertices mark those points in space, from which four edges originate in four directions, at right angles with one another.
This spatial form has been created in practice as well. In order to be able to construct the solid figure, I did not describe the surfaces that are too tiny. The regular shell structure constitutes a beautiful, closed, spatial form. This solid form, together with its spatial mirror-form, are able to fill the space so that each SH is surrounded by eight counter-rotating, that is mirrored, SH’s. This is natural, since the complement of a spatial, vortex-like form, i.e., its negative in space, can only be a vortex of opposing direction. This may lead to the idea of creating a system of space-filling which, instead of defining the solid figures that fill in the space, classifies the possible structures according to the characteristics of the surfaces that border the solid figures. In the given case, each SH is bordered by four pairs of Spidron-reliefs which, in pairs, rotate in opposing directions.
Spacefilling with Hornhedrons and Spidrohedrons
From twenty-four uniform hornflakes a polyhedron can be created; such uniform polyhedrons joined, are able to fill the space. This is due to the fact that the hornflake itself is axially symmetrical.
One might begin to wonder whether a similar structure can be found in nature as well.
Further observations regarding the characteristics of the Spidron*:
