H tree

In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering.

Construction

An H tree can be constructed by starting with a line segment of arbitrary length, drawing two shorter segments at right angles to the first through its endpoints, and continuing in the same vein, reducing (dividing) the length of the line segments drawn at each stage by

2

{\displaystyle {\sqrt {2}}}

. A variant of this construction could also be defined in which the length at each iteration is multiplied by a ratio less than

1

/

2

{\displaystyle 1/{\sqrt {2}}}

, but for this variant the resulting shape covers only part of its bounding rectangle, with a fractal boundary.

An alternative process that generates the same fractal set is to begin with a rectangle with sides in the ratio

1

:

2

{\displaystyle 1:{\sqrt {2}}}

, and repeatedly bisect it into two smaller silver rectangles, at each stage connecting the two centroids of the two smaller rectangles by a line segment. A similar process can be performed with rectangles of any other shape, but the

1

:

2

{\displaystyle 1:{\sqrt {2}}}

rectangle leads to the line segment size decreasing uniformly by a

2

{\displaystyle {\sqrt {2}}}

factor at each step while for other rectangles the length will decrease by different factors at odd and even levels of the recursive construction.

Properties

The H tree is a self-similar fractal; its Hausdorff dimension is equal to 2.

The points of the H tree come arbitrarily close to every point in a rectangle (the same as the starting rectangle in the constructing by centroids of subdivided rectangles).