⋰|\  
        ⋰ A|B\  
      ⋰____|__\  
    ⋰      |   \  
  ⋰        |    \  
 ’——————————+—————`  

I coauthored the paper "Trailing the Dovetail Shuffle to its Lair" which solved the GSR model for riffle shuffling in closed form, leading to the recommendation to shuffle a 52 card deck seven times.

Your picture is a great geometric way to think about riffle shuffling. Your n-simplex represents a sorted deck of n cards. Choosing a point uniformly at random in this simplex makes the needed choices all at once to predetermine an arbitrary number of riffle shuffles. Think of k riffle shuffles as a single 2^k shuffle (as if you had 2^k hands); we can now study an "a-shuffle". Scale the n-simplex by a factor of a. Where does our random point end up? Either view all of space as tiled by your n-cubes, or reduce all coordinates mod 1. Either way the random point ends up in some n-simplex described by a different (or the same) order (mod 1) of the coordinates. That's your shuffle.

For a single shuffle of two cards, doubling the 2-simplex ("up-triangle") covers three up-triangles and one down-triangle, so the odds of reversing the cards is 1 in 4. This makes sense if you imagine two "stowaway" cards face up, placed at random in a face-down deck. Shuffling the deck, to reverse these cards they can't both be in the same packet, and being in different packets only reverses them half the time.

Increasing "a" for two cards, one sees the up and down-triangle counts converge to a 1:1 ratio. The error "fringe" looks like a numerical integration error.

So from a mathematical perspective, the issue is that your "line" (path) actually has zero thickness, and so there are gaps between the spiral windings. If you sample a point from any finite-length path then you have probability zero of hitting any of the points in the gaps between segments. (Maybe you don't care about this if the gaps are sufficiently small, but from a mathematical perspective it's not strictly valid.)

And you can't just say "the path is infinitely long and spirals infinitely many times" because then you have no way of uniformly sampling a random real number between zero and infinity (it can be proven that no such probability distribution exists).

But I think the basic idea is sound and you make it work by formulating it slightly differently:

1. Divide up the triangle into infinite concentric triangles

2. Choose a random triangle with probability proportional to that triangle's length

3. Choose a random point on that triangle's perimeter

The interesting part is step 2. If you parameterize the triangles by a scale factor s, ranging from 0 to 1, then the perimeter is proportional to s. So I believe you can choose s appropriately by choosing x uniformly at random from 0..1 and letting s = sqrt(x). See https://en.wikipedia.org/wiki/Triangular_distribution

Then step 3 is mathematically straightforward: if the lengths of the triangle's sides are a,b,c, you choose a random number between 0 and a+b+c, and "wrap" that length around the triangle's perimeter. Then you have to do the correct coordinate transformation to find the appropriate point on the scaled triangle you chose.

So I think it would work, but it probably wouldn't be any simpler to implement than the "accept-flip" method described in the article.

Not square, rectangle. The article speaks of a parallelogram but you can the distribution will be the same for the rectangle.

That's essentially the accept-flip method of the article, isn't it?

Wouldn't that subsequent transform distort the sample distribution? As you make the angle at one vertex more obtuse, the density of the point mapping increases more there relative to the points near the other, more acute vertices.

You mean inside the resulting rectangle?

That's what the article suggests.

What's a "whole triangle" ? Google leads me nowhere.

A parallelogram is just a rectangle with extra steps. Maybe they could have mentioned what those steps are but it seems fairly straightforward

I didn't want to be that critical right out of the gate, but yeah if an "experienced consultant" thinks that this is worth writing about at this level of detail, then that makes me trust them less, not more.

Go away kitten, your comment appears to be low-quality.

"Generate in a parallelogram" is pretty obvious, unless you are truly clueless. Every post of johndcook that I find posted to HN is interesting, and definitely of high quality.

Generating random points uniformly in a circle without rejection is at least mildly more interesting, as you need to scale one factor by a square root:

    const r = sqrt(random()) * radius;
    const theta = random() * TAU;

Obligatory xkcd: https://xkcd.com/1053/