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DIY and Craftsmanship:3D printing & CAD

## Complex patterns generated by synthesizing multiple waves and printing by helical 3D printing

For more than a year, I have investigated patterns generated by synthesizing three types of waves by helical 3d printing. Because waves are synthesized by linear models by this method, I thought no unexpected patterns are generated. However, very recently I found that unexpected complex patterns are generated by using very fine deformation waves.

Linear synthesis works in limited ranges in physical world. When it goes out of range, 3D printing, which is a physical phenomenon, generates non-linearity. どの波でも振幅がおおきくなると，隣接するべきフィラメントのあいだにすきまができて，かたちをなさなくなる． 変調による波の場合は，振幅がおおきくなると逆に突起ができることがあることがわかり，クリスマス・オーナメントなどでつかってみた． しかし，これらの場合には，予想できない複雑なパターンがあらわれることはなかった．

I happened to synthesize crossing wave by vibrato (i.e., vertical waves of filaments) and very small and high-frequency waves, very complex patterns were generated. Several examples are shown by Repeater graphics.

These patterns are generated by two vibrato waves and a fine vertical deformation wave. Because the vibrato waves are difficult to see, the following figures shows eight-times larger patterns.

The lower left figure shows only the vertical deformation wave. Although there is only a vertical wave, there seems to be horizontal waves. 波数がとてもおおいので螺旋が 1 回上昇するあいだに 20 波経由するからだ． In the lower right figure, the amplitude of this wave is eight-times larger.

Finally, printed examples are shown by photos. The above graphics show only the surface shapes, but these photos show brilliance caused by transmission, reflection, and refraction of light.

P.S. (2018-4-4): If you look the above figure that draws only crossing waves, you can see moiré patterns. They are similar to the patterns with vertical waves. It probably means that the complex patterns shown above are essentially moiré patterns. Usually moiré patterns are less complex; however, sine-waved filaments generates very complex patterns.

Keywords: Moiré