I'd like to make something like this programatically.
Does anyone have a suggestion? Could you approach this programmatically with Illustrator, Inkscape, HTML5, or Three.JS?
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You should use Scalable Vector Graphics (SVG) for this type of graphic. You can code SVG directrly in a text editor or use some vector graphics software. If you want to control the colors, borders etc. you have to make them variables. You can also assign id and class to different elements within the SVG and control them with CSS ... it's just the way to go.