Lakes of Wada | Hacker News Reader

Lakes of Wada

en.wikipedia.org

66 pointsramgorur15 comments

So imagine you have three fluids that don't mix which are colored red, green, blue. You can make drops of these fluids on a piece of paper, and manipulate their shapes e.g. with a pipette.

This article is saying that mathematically, it's possible to make three single, continuous but weirdly shaped drops of these fluids, that together fill a square, in a way such that if you consider the three drop outlines as seen from above (e.g. by a camera), they all have the same outline.

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.

This entry needs a simple english page. I read the first paragraph and didn't understand anything.

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.)

OTOH the vast vast majority of people who will be reading this will be mathematicians who are familiar with the jargon. IMO it's perfectly fine for wikipedia to optimize for the primary audience, instead of optimizing for the rare curious person who has no relevant background.

There’s too much math to fit into one brain. It’s infeasible to expect every topic explained to a common denominator. Though, I definitely invite anyone interested to grok through all the math they can and cannot understand. Math, I hope, is more than just the academic practitioners.

I can attest to the enjoyment value in reading through math way beyond and outside my understanding. It’s amazingly beautiful, humbling, and can be surprisingly useful.

During my master thesis, i went into the Wikipedia pages for mathematical theorems i mastered and i felt i was five years old.

The language used on the mathematics pages on Wikipedia almost makes me believe it is some sort of intentional gate-keeping scheme.

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.

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/

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).

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

>In mathematics, the lakes of Wada (和田の湖 Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point.

"disjoint connected open sets" - What is an open set? What does disjoint connected mean?

"plane or open unit square" - I know what a plane is, but the context of an "open unit square" which doesn't make sense to me shakes my confidence in that.

"on the boundary of one of the lakes" - What did we just define as a lake?

It is one counterexample in Topology for the Jordan Curve Theorem.

Wada basins exist for any number of open sets. As you could probably guess from the article, the Newton method applied to x^n - 1 gives a Wada basin of n sets. These are fun counterexamples to the claim "given three nontrivial disjoint sets on a plane, their boundaries are not mutally equal."

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