Gosper curve - Citizendia

The Gosper curve, named after Bill Gosper, also known as the flowsnake, is a space-filling curve. Ralph William Gosper Jr, (born 1943 known as Bill Gosper, is an American mathematician and programmer from Pennsauken Township New Jersey. Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit It is a fractal object similar in its construction to the dragon curve and the Hilbert curve. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" A dragon curve is the generic name for any member of a family of self similar Fractal curves which can be approximated by recursive methods such as A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous Fractal Space-filling curve first described by the German mathematician


A fourth-stage Gosper curve	The line from the red to the green point shows a single step of the Gosper curve construction.

A Logo program to draw the Gosper curve using turtle graphics:

to rg :st :ln
make "st :st - 1
make "ln :ln / 2. Logo is a Computer programming language used for Functional programming. Turtle graphics is a term in Computer graphics for a method of programming Vector graphics using a relative cursor (the " Turtle " 6457
if :st > 0 [rg :st :ln rt 60 gl :st :ln  rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st :ln rt 60]
if :st = 0 [fd :ln rt 60 fd :ln rt 120 fd :ln lt 60 fd :ln lt 120 fd :ln fd :ln lt 60 fd :ln rt 60]
end

to gl :st :ln
make "st :st - 1
make "ln :ln / 2. 6457
if :st > 0 [lt 60 rg :st :ln rt 60 gl :st :ln gl :st :ln rt 120 gl :st :ln rt 60 rg :st :ln lt 120 rg :st :ln lt 60 gl :st :ln]
if :st = 0 [lt 60 fd :ln rt 60 fd :ln fd :ln rt 120 fd :ln rt 60 fd :ln lt 120 fd :ln lt 60 fd :ln]
end

The program can be invoked, for example, with rg 4 300, or alternatively gl 4 300.

The constant 2. 6457 in the program code is an approximation of √7.

Properties

The space filled by the curve is called the Gosper island. The first few iterations of it are shown below:

The Gosper Island can tile the plane. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps In fact, seven copies of the Gosper island can be joined together to form a shape that is similar, but scaled up by a factor of √7 in all dimensions. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other Repeating this process indefinitely produces a tessellation of the plane. The curve itself can likewise be extended to an infinite curve filling the whole plane.

External links

A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" A Fractal is a geometric object whose Hausdorff dimension (&delta strictly exceeds its Topological dimension.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Properties

See also

External links