{"id":8266,"date":"2026-09-06T08:51:06","date_gmt":"2026-09-06T15:51:06","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8266"},"modified":"2026-09-06T14:58:22","modified_gmt":"2026-09-06T21:58:22","slug":"the-beauty-of-roots","status":"publish","type":"post","link":"http:\/\/theoryofeverything.org\/theToE\/2026\/09\/06\/the-beauty-of-roots\/","title":{"rendered":"The Beauty of Roots"},"content":{"rendered":"\n

For a nice introduction to roots and their fractal nature, see this post<\/a> from 2011 on John Baez’s Azimuth blog or his 2023 UCR page here<\/a>. Some of my code here is sourced from a great repository of interesting math visualizations from many others including the author of the pywonderland GitHub<\/a> of Zhao Liang.<\/p>\n\n\n\n

The SVG images below can be clicked to zoom in on another tab, but here<\/a> is a Mathematica Notebook with Tooltip info on the selected roots.<\/p>\n\n\n\n

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All roots of polynomials up to 224<\/sup> (~16.7 million) with +\/-1 coefficients.<\/figcaption><\/figure>\n\n\n\n

The roots are complex valued (Real x-axis and Imaginary y-axis) with normalized Abs@binValue defining the color from 0 to 1. The ColorRules are as follows:<\/p>\n\n\n

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The analysis below provides more detail on the Roots-of-Unity Circle (RUC) at the center of the light blue area. These are interesting in that there are 4 quadrants each with 12 roots that all have 1\/2< Abs@binVal <1 at the center of holes. <\/p>\n\n\n\n

There are 6 roots that sit in holes surrounded by zeros (i.e. 2\u03c0\/{12,10,8,6,5,4}) and 6 roots in holes with values <1\/2 (i.e. 2\u03c0\/{24, 14, 7, 11\/2, 29\/6 (75\u00b0), 437\/90 (77\u00b0)}). The only other binVal roots >1\/2 are on the 1 pixel wide x-axis with 5936 of 8000 (~75%) from the x-y grid of 8000×8000 (64 million) bins which sum over each of the 224<\/sup> (~16 million) roots of polynomials up to with -\/+1 coefficients. The rest of the x-axis roots are 2020 bins equal to 0 and only 44 bins between 0 and 1\/2).<\/p>\n\n\n

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The upper-right quadrant of the SVG image has the RUC roots ringed in colors assigned by their order in each quadrant. To the right is an annotated band at the center of the black ring with the detail (below) with the angle at which the root is located. Click the image below to see this SVG and zoom further.<\/p>\n\n\n

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Zoom detail of the upper-right quadrant RUC root values – click SVG to create another tab and zoom in your browser.<\/figcaption><\/figure>\n<\/div>\n\n\n

Below are 100 detail areas of “special” algorithmically searched roots with a 400×400 ArrayPlot and an 11×11 zoom of that Abs@binVal matrix with its ArrayPlot. This includes the 48 RUC roots and 12 x-axis roots with binVal>1\/2, and 40 off-axis roots with binVal<1\/2. This off-axis list includes one set of 4 quadrant roots that is a proper hole (in the 4 yellow rings located just above\/below the bottom\/top apex in the full image or 10-14th from the bottom in the detail list below). The other 9×4 roots are what I call “pseudo-holes” that have no enclosed high central binVal.<\/p>\n\n\n\n

FYI – The color scheme on these detail zoom MatrixPlots are default to highlight numeric differences, so they don’t match the ColorRules in the large 8000×8000 MatrixPlot.<\/p>\n\n\n\n

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Detail areas of special roots with a 1000×1000 ArrayPlot and an 11×11 zoom of the Abs root values and its ArrayPlot <\/figcaption><\/figure>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

For a nice introduction to roots and their fractal nature, see this post from 2011 on John Baez’s Azimuth blog or his 2023 UCR page here. Some of my code here is sourced from a great repository of interesting math visualizations from many others including the author of the pywonderland GitHub of Zhao Liang. The … Continue reading The Beauty of Roots<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[11,14],"class_list":["post-8266","post","type-post","status-publish","format-standard","hentry","category-physics","tag-2d","tag-math"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t