Hironaka’s Curve

Start with a simple figure made up of a couple of parallel line segments.

seed figure

Next, make a copy of the figure, shrink it, and arrange copies in an M-shape, flipping the middle copy vertically, and finally superimposing the copies on the original. (The M-shape comment will make sense in a minute.) The arrowheads aren’t part of the figure, just here to try to make the construction easier to follow.

starting figure to part of first iteration

Superimposing the start figure with the scaled and rotated copies results in the first approximation to Hironaka’s curve.

first iteration

Repeat the process of scaling, translating, and flipping.

first iteration to part of second

Superimposing all the pieces gives us the second approximation to Hironaka’s curve. The M-shape comment should make a little more sense now.

second iteration, showing M shape

From the second iteration to the third:

second iteration to part of third

Do this enough times, and you’re approximating Hironaka’s curve, sometimes called Hironaka’s M-curve.

fourth iteration fifth iteration

The individual pieces look like this, which are interesting enough in a Cantor set kind of way:

components making up figure

The figure can be generalized for different numbers of copies made at each step, in this case five:

five-copy variant

Or seven:

seven-copy variant

Or nine:

nine-copy variant

It can also be generalized to 3-D, though not very legibly:

3-D variant

If you’re interested, the Mathematica code for the first figures:

Hironaka[0] = {Line[{{0, 0}, {0, 1}}], Line[{{1, 0}, {1, 1}}]};

Hironaka[n_] := 
 Hironaka[n] = 
  {Translate[Scale[Hironaka[n - 1], {1/3,  1/2}, {0, 0}], {0/3, 1/2}], 
   Translate[Scale[Hironaka[n - 1], {1/3, -1/2}, {0, 0}], {1/3, 2/2}], 
   Translate[Scale[Hironaka[n - 1], {1/3,  1/2}, {0, 0}], {2/3, 1/2}]}

  Table[{ColorData["SouthwestColors"][i/5], Hironaka[i]}, {i, 5, 0, -1}]}]

The others follow similarly.

See Gerald A. Edgar, Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990, pp. 203–204.

See also Curtis McMullen’s http://www.math.harvard.edu/~ctm/gallery/ and http://www.math.harvard.edu/~ctm/gallery/cx/Mcurve.gif.

Thanks to Roger Bagula for the generalization idea.

Figures created with Mathematica 10.

Robert Dickau, 2016.

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