Well, they all have the same boundary points. A boundary point is a limit of a point sequence lying inside the shape. But if we define a different concept of "outline point" as a limit of a path lying inside the shape, then I think the three shapes won't have the same outline.

The construction shown on the page makes me think of this as if the three fluids are infinitely mixed, if that makes any sense.

The article indeed conveys no meaning at all to those not acquainted with the mathematical jargon it relies on.

Many scientific Wikipedia articles have improved in this regard over the last few years, but this one (along with many others in the field of mathematics) remains of little interest to non-mathematicians unready to synthesize and internalize the vast quantity of information in the articles of relevant linked terms.

I don’t see this changing any time soon without a lot of concerted effort.

(For the record, I’m someone who did not grok the significance of the article’s subject in the slightest.)

It is a way to divide an area, let's say a square, into three "countries". Each country is connected: it constitutes a contiguous region without enclaves or exclaves. The countries, as usual, are also disjoint: no point is shared between two or more countries (points exactly on the border are not thought to belong to any country). Now, the border between the three countries has a very peculiar property: every point of the border separates all three countries!

That is, a bit more rigorously, no matter what border point you choose, you can always find points belonging to all three countries arbitrarily close to it. In non-pathological real-world borders this can only hold for a finite number of points (say, for instance, the point near Basel where the borders between France, Germany, and Switzerland meet).

I also had absolutely no idea what was going on, so I tried to emulate the physical generation of Wada basins. Post here: http://blog.jordan.matelsky.com/wada/

This article describes a way to split a square into three non-overlapping regions that all have the same border. It is very counterintuitive that this is possible. If you imagine a line on a square, you can define two parts of the square that share this line as a border – and how could there possible be three?

Hope that helps a little bit.

Out of curiosity, what wasn't clear aside from openness?

BTW, for those interested, openness is the property assigned to a set of points not containing its boundary. For instance, an open interval (a,b) doesn't contain boundary points a or b. An open unit square is a 1x1 square without it's edges included