Spidron System
1. General Concept

 

A spidron is a plane figure consisting of an alternating sequence of equilateral and isosceles (30°, 30°, 120°) triangles. Within the figure, one side of a regular triangle coincides with one of the sides of an isosceles triangle, while another side coincides with the hypotenuse of another, smaller isosceles triangle. The sequence can be repeated any number of times in the direction of the smaller and smaller triangles, and the entire figure is centrally projected through the mid-point of the base of the largest isosceles triangle.

The semi-spidron is half of the spidron, consisting of increasingly densely packed and smaller regular and isosceles triangles. The sequence can also be continued in the opposite direction, with ever larger triangles, ad infinitum. It seems to me that this composition, or to use the term that is more customary in mathematics, this "triangle complex", or the half of it I call a semi-spidron is the foundation for all the other shapes with their interesting spatial and almost "spatially flexing" properties which according to the comments and the feedback I get promise novel developments in the fields of geometry, physics and other branches of science.

expanation

A spidron belt is a plane figure consisting of a regular hexagon, six regular triangles raised outward on its sides and the larger hexagon connecting the outside vertices of those triangles. The formation may be deformed along its external or its internal edges.

The spidron arm is the spatial version of the semi-spidron, whose sides are either all at convex or all at concave angles to each other. Those spidron arms can have either direction of spin determined by the interconnection of the triangles starting from the centre they can be clockwise or counter clockwise. The direction of "spin" is the result of the series of triangles rotated by 30° relative to each other. The semi-spidrons that, after deformation in space, have both concave and convex angles are not called spidron arms.

A spidron nest is a plane formation made of nested regular hexagons* and six-pointed stars that can be deformed into a spatial formation consisting of 6 semi-spidrons, but we can also view it as a nested sequence of similar plane spidron rings of spidron belts.

         

* It is possible to build a nest from four identical spidron arms as well, but that differs from the one made from six arms in that it can only be deformed to a more limited extent and it can not be laid out in the plane .
Six spidron nests with four spidron arms each can be assembled to form a body similar to a cube which has 3 left-handed and 3 right-handed spidron nests of this kind and as such it is also a space-filler.