How did I make this?! Radiating triangle sphere shape

With some work, I got this.

Will describe later this evening (if I have more time) how I got this result.

The spirograph part can be created with a simple script. 

Select a circle (the count can be changed in the first line):

var count = 25;
var doc = app.activeDocument;
var source = doc.selection[0];
var r = source.width / 2;
var first = doc.pathItems.ellipse(source.top - r / 2, source.left, r, r);
first.stroked = true;
first.filled = false;
for (var i = 0; i < count; i++) {
    var duple = first.duplicate();
    duple.rotate(360 / count * (i + 1), undefined, undefined, undefined, undefined, 
        Transformation.RIGHT);
}

I can't find a formula for the concentric circles. 

You need a circle and a curve. Then duplicate/rotate around the center and then mirror and duplicate.

Then use Live paint to fill this: