Composing Different Tessellation from the Same Elements

Year: 2001 Authors: Imameddin Amiraslanov

Core claim

The same fish elements and their mirrors are uniquely versatile for composing many distinct tessellations and symmetry patterns.

Topics

tessellation, quasiperiodicity, symmetry, spirals, crystallography education

Domains

tiling theory, plane symmetry groups, quasicrystals, radial symmetry, pattern design, Escher-like imagery, educational toys, applied objects

Methods

geometric composition, mirror transformations, pattern enumeration

Media

fish-shaped paper forms, 2D pattern illustrations

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

BRIDGES Mathematical Connections in Art, Music, and Science

Composing Different Tessellation from the Same Elements

Imameddin Amiraslanov Institute of Inorganic and Physical Chemistry Academy of Science, Husein Javid Avenu 31, Baku, Azerbaijan imam@gate.sinica.edu.tw

Obviously, creating tessellations of Escher-like, is not an easy task. As a rule the solutions found (pattern elements) are design only can be composed although there are few examples where the elements found allow for composing two or three different patterns. In this respect, the fish elements shown below (pic. 1(a,b)) are unique.

Pic.1

a*

b

b*

These two thin and thick fish forms together with their mirror view (pic. 1 $a^{\ast }$, $b^{\ast }$ ) can compose a lot of patterns, such as:

• 10 fold quasiperiodical $(a,b)$ pic. 3
• 5 fold quasiperiodical $(a,b,a^{\ast },b^{\ast })$ pic. 4
• 10 fold radial symmetry $(a,a^{\ast })$ pic. 5
• 5 fold radial symmetry $(b,b^{\ast })$ pic. 6
• spiral 1 $(b,b^{\ast })$ pic. 7
• spiral 2 $(b,b^{\ast })$ pic. 8
• spiral 3 $(b,b^{\ast })$ pic. 9
• many different $2D$ periodical patterns pic. 10, 11, …

Pic.2

$a^{\ast }{a}^{\ast }$

bb

$b^{\ast }{b}^{\ast }$

These pictures may with be used with success in explaining the geometrical basis of quasicrystals, spiral growing, phenomenon of polymorfism, the plane groups of symmetry in teaching of crystallography and relevant subjects. Also these two fish forms can serve as a basis for preparation of a number of applied objects and educational toys.

Imameddin Amiraslanov

Pic.3. 1—fold quasicrystal fishes

pic.4. 5-fold quasiperiodical fishes

Composing Different Tessellation from the Same Elements 315

Pic.5. 10-fold radial symmetry

Pic.6. 5-fold radial symmetry

Imameddin Amiraslanov

Pic.7. Spiral 1

Pic.8. Spiral 2

Composing Different Tessellation from the Same Elements 317

Pic.9. Spiral 3

Pic.10-13. Some of 2D periodical fishes