貢中元的部落格

Unlimited number of periodic decagonal tiling (2D-crystal)

The fat and thin rhombus tiles with different acute angles of 36 and 72 degrees  can combine to form six different decagonal tiles, of different inner structures, Each decagons composed of ten rhombuses , five thin rhombus and five fat rhombus , where a, b, c, d, e, f, are marked for six different decagons respectively in Figure 1.  For the fat rhombus in the decagons of type-a, type-d, type-e and type-f are highlighted in red, purple red, yellow, and light blue, respectively. These six decagons which can be regarded (considered) as basic unit, instead of two types of rhombuses, to fabricate (infinity) tiles in this paper. And be noted that only type-a decagonal tile is five-fold symmetric and the rest decagons are only mirror symmetric.

所有十邊形彭羅斯瓷磚 

The five-fold symmetrical tile that can be self-coupled to build up a five-fold tiling,periodic and four-way symmetry tiles (crystals) to infinity

Some smaller unit cells that form different decagonal two-dimensional crystals

Four unit cells that intercepted from a translational decagonal style crystal and form four different crystals.

Tessellation of unlimited equilateral rhombuses

 If the acute angle of a rhombus is 90/N degrees, and N is an integer, then there can be N equilateral rhombuses whose acute angles are integer multiples of 90/N respectively. A standard drawing method with numerous variables, in which N equilateral rhombus can be tessellated into an infinite number of groups (kinds) of combinations of infinite pattern. There are four general rules of drawing, which are described hereafter.

new

tellation of three triangles

asymmetry  Penrose binary  tiling  不對稱彭羅斯平鋪 

1 one may start from any conventionally defined five-fold symmtry (around a specific center, center of blue star,)or on-five fold symmerric  penrose tile , either pentagonal or circular shape.as shown in fig 1

All  aperoidic and periodicPenrose (Kung)  tiling  (# one set全集之2)