'Concentric orthogonal field progressions' in the 'Root square' and connection to angles within regular Polyhedron

August 4, 2018

Below is the basic geometry structure I call the 'Root square'. Here it is arbitrary truncated at 32. Each parallel x and y line is plotted at square-root(n)

Some interesting results. I suspect there are a bunch more angles found in Platonic solids hiding in these square root progressions.

Below: A cursory observation connecting progressions above for different value n to the Dihedral angle of the regular Polyhedron


Polyhedron Dihedral angle f(n)
Tetrahedron 70.53° 90 - f(8) = 70.53
Cube 90°
Octahedron 109.47° 90 + f(8) = 109.47
Dodecahedron 116.57° 90 + f(4) = 116.57
Icosahedron 138.189685° ?
4-simplex 75.5225° f(15) = 75.5225
Hypercube 90°
Cross-polytope 120°
24-cell 120°
120-cell 144°
600-cell 164.478° 240 - f(15) = 164.478


Launch in visualizer

Above: visualization of Inverse wave pattern

Below: visualization of standard wave pattern

Launch in visualizer


Self simliar Moire patterns in square root concentric progression

August 6, 2018

Scale

Move Scale slider above to see concentric moiré patterns that seem to echo the parent progression of alternating black and white circles of sqRt(n) radius. The browser anti-aliasing is turned off with css for these images to prevent the browser's own smoothing algorithms from blurring out the Moiré effect. These images do not get swapped out when scaling. We find any pattern formed with this progression produces self echoing concentric moiré patterns. The effect is much more pronounced viewing in Firefox due to different rendering algorithms I assume.

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