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| |___ | |___ | |___ | |___ | |___ | |___ | |___ _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _| _ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | | _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _| _ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | | _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _| _ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | | _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ _|_| _|_| _|_| _|_| _|_| _|_| _|_| _|_| _| _ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | |___ | | | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| | | | | | | | | | _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ | | | | | | | | | / \ / \ / \ / \ / \ / \ __/ \__/ \__/ \__/ \__/ \__/ \ / \ / \ / \ / \ / \ \__/ \__/ \__/ \__/ \__/ \__/ / \ / \ / \ / \ / \ / \ __/ \__/ \__/ \__/ \__/ \__/ \ / \ / \ / \ / \ / \ \__/ \__/ \__/ \__/ \__/ \__/ / \ / \ / \ / \ / \ / \ __/ \__/ \__/ \__/ \__/ \__/ \ / \ / \ / \ / \ / \ \__/ \__/ \__/ \__/ \__/ \__/ / \ / \ / \ / \ / \ / \ __/ \__/ \__/ \__/ \__/ \__/ \ / \ / \ / \ / \ / \ The Flowsnake ------------- asciied by Michael Naylor Order 1 Flowsnake ________________ \ \ \ \ \ \ \________ \ / / / ________/ Order 2 Flowsnake ________ \ \ ________ \____ \ \ \ / / \____ \____/ / ____ / \ \ \ ____/ ________ \ \ \ / \ \ \ / / ____ \____ \ \/ \ \ \ / / \ \ \____/ / ____ \ / \ / / \/ ________ \/ / \ \ / \____ \____/ / ____/ Order 3 Flowsnake ____ ____ \__ \ \__ \__/ / __ __/ ____ \ \ \ ____ / __ \__ \ \/ / __ \__ \ ____ \ \ \__/ / __ \/ / __/ / __ ____ \__ \ \/ ____ \/ / __/ / __ \ \ \ \__ \__/ / __ \__ \__/ / __ \ \ \ \/ __/ ____ \ \ \__/ ____ \ \ \ \/ / __ / __ \__ \ \/ ____ \__ \ \/ / __ \/ / \ \ \__/ / __ \__ \__/ / __ \ \ \__/ \/ ____ \/ / __/ ____ \ \ \ \/ ____ \__ \__/ / __ \__ \ \/ / __ \__ \ __/ ____ \ \ \__/ / __ \/ / __/ / __ / __ \__ \ \/ ____ \/ / __/ / __ \/ / \/ / __/ / __ \__ \__/ / __ \/ / __/ __/ / __ \ \ \__/ ____ \ \ \__/ / __ / __ \ \ \ \/ ____ \__ \ \/ ____ \/ / \ \ \ \/ / __ \__ \__/ / __ \__ \__/ \/ / __ \/ / __/ ____ \ \ \__/ \ \ \__/ / __ \__ \ \/ \/ \ \ \__/ / __ \/ ____ \/ / \__ \__/ __/ Flowsnake coloring sheet (makes a great present for Mom!) _ / `-_ / `-_ / `-_ / `-_ / `-_ / `-_ / ____ `-_ / ____ \__ \ `-_ color this / \__ \__/ / __ `-_ region a dark / __/ ____ \ \ \ ____ `-_ color / / __ \__ \ \/ / __ \__ \ / / ____ \ \ \__/ / __ \/ / __/ / __ / ` / ____ \__ \ \/ ____ \/ / __/ / __ \ \ \/ ` / \__ \__/ / __ \__ \__/ / __ \ \ \ \/ ` / __/ ____ \ \ \__/ ____ \ \ \ \/ / __ ` , / / __ \__ \ \/ ____ \__ \ \/ / __ \/ / / \ \ \__/ / __ \__ \__/ / __ \ \ \__/ , / \/ ____ \/ / __/ ____ \ \ \ \/ ____ / \__ \__/ / __ \__ \ \/ / __ \__ \ , / __/ ____ \ \ \__/ / __ \/ / __/ / __ / / __ \__ \ \/ ____ \/ / __/ / __ \/ / , / \/ / __/ / __ \__ \__/ / __ \/ / __/ / __/ / __ \ \ \__/ ____ \ \ \__/ / __ , / / __ \ \ \ \/ ____ \__ \ \/ ____ \/ / `-_ \ \ \ \/ / __ \__ \__/ / __ \__ \__/ , `-_ \/ / __ \/ / __/ ____ \ \ \__/ `-_ \ \ \__/ / __ \__ \ \/ , Color this `-_ \/ \ \ \__/ / __ region a light `-_ \/ ____ \/ / , color, or `-_ \__ \__/ leave it blank `-_ __/ , `-_ / `-_ / , `-_/ ` , ` cut out around ` , the outside ` ` ` Flowsnake technical information The Flowsnake was invented by William Gosper. It is a space-filling fractal curve. "Space-filling" means that if this process were carried out infinitely, this curve would pass through every point in the region it occupies. It is a fractal in that if any "segment" is enlarged, it will be identical to the entire curve. This curve has a fractal dimension of log(3)/log(sqrt(7)) = ~1.129. Detail of construction: The Flowsnake can be thought of as being composed of seven segments. At each iteration, each of these seven segments is replaced with the original shape. (The new segments should have length equal to 1/sqrt(7) times the original length; in ascii, of course, we'll make do with an approximation). In the following illustrations, the original vertices are labeled to make it easier for you to follow what happens at each stage. Order 1 Flowsnake 4_______5______6 \ \ \ \ \ \ \3______2 \7 / / / 0______1/ Order 2 Flowsnake ________ \ \ ________ \____ \6 \ \ / / \____ \5___/ / ____ / \ \ \ 4___/ ________ \ \ \7 / \ \ \ / / ____ \____ \2 \/ \ \ \ / / \ \ \3___/ / ____ \ / \ / / \/ ________ \/ / \ \ / \____ \1___/ / 0___/ Order 3 Flowsnake ____ ____ \__ \ \__ \__/ / __ __/ ____ \ \ \6 ____ / __ \__ \ \/ / __ \__ \ ____ \ \ \__/ / __ \/ / __/ / __ ____ \__ \ \/ ____ \/ / __/ / __ \ \ \7 \__ \__/ / __ \__ \__/ / __ \ \ \ \/ __/ ____ \ \ \5_/ ____ \ \ \ \/ / __ / __ \__ \ \/ ____ \__ \ \/ / __ \/ / \ \ \__/ / __ \__ \__/ / __ \ \ \__/ \/ ____ \/ / __/ ____ \ \ \2\/ ____ \__ \__/ / __ \__ \ \/ / __ \__ \ 4_/ ____ \ \ \__/ / __ \/ / __/ / __ / __ \__ \ \/ ____ \/ / __/ / __ \/ / \/ / __/ / __ \__ \__/ / __ \/ / __/ __/ / __ \ \ \3_/ ____ \ \ \__/ / __ / __ \ \ \ \/ ____ \__ \ \/ ____ \/ / \ \ \ \/ / __ \__ \__/ / __ \__ \__/ \/ / __ \/ / __/ ____ \ \ \1_/ \ \ \__/ / __ \__ \ \/ \/ \ \ \__/ / __ \/ ____ \/ / \__ \__/ 0__/ Truchet : | | | | | | | | | | | | | | | | | `. 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`Y88ba, ,ad88P' `Y88ba, ,ad88P' `88888ba ad88888' `88888ba ad88888' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `"Y888b,d888P"' `"Y888b,d888P"' ``":([" __,:,__ "]):"'' `Y88ba, ,ad88P`Y88ba, ,ad88P' Normand `88888ba ad88888' `88888ba ad88888' Veilleux `Y88888b, ,d88888P' `Y88888b, ,d88888P' `"Y888b,d888P"' `"Y888b,d888P"' ``":"'' ``":"'' __,:,__ ,ad88P`Y88ba, __,:,__ ad88888' `88888ba __,:,__ ,ad88P`Y88bd88888P' `Y88888bd88P'Y88ba, ad88888' `888888P"' `"Y888888' `88888ba ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d888P"' `"Y888b, ,d888P"' `"Y888b, :([" "]): :([" "]): `Y88ba, ,ad88P' `Y88ba, ,ad88P' `88888ba ad88888' `88888ba ad88888' ,Y88888b, ,d88888P, ,Y88888b, ,d88888P, ad88888888b,d88888888ba ad88888888b,d88888888ba ,d88888P' ``":"'' `Y88888b, ,d88888P' ``":"'' `Y88888b, ,d888P"' `"Y888b,d888P"' `"Y888b, :([" Normand "]):([" Veilleux "]): `Y88ba, ,ad88P`Y88ba, ,ad88P' `88888ba __,:,__ ad88888' `88888ba __,:,__ ad88888' `Y88888bd88P`Y88bd88888P' `Y88888bd88P`Y88bd88888P' `"Y888888' `888888P"' `"Y888888' `888888P"' ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d888P"' `"Y888b, ,d888P"' `"Y888b, :([" "]): :([" "]): `Y88ba, ,ad88P' `Y88ba, ,ad88P' `88888ba ad88888' `88888ba ad88888' `Y88888b, ,d88888P, ,Y88888b, ,d88888P' `"Y888b,d88888888ba ad88888888b,d888P"' ``":"'' `Y88888b, ,d88888P' ``":"'' `"Y888b,d888P"' ``":"'' ,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b, :"'' ,ad88P`Y88ba, ``":"'' ,ad88P`Y88ba, ``":"'' ,ad88P`Y88ba, ``": ad88888' `88888ba ad88888' `88888ba ad88888' `88888ba ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d888P"' `"Y888b,d888P"' `"Y888b,d888P"' `"Y888b, :([" "]):([" "]):([" "]): `Y88ba, ,ad88P`Y88ba, ,ad88P`Y88ba, ,ad88P' `88888ba ad88888' `88888ba ad88888' `88888ba ad88888' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' :,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,: `Y88ba, ``":"'' ,ad88P`Y88ba, ``":"'' ,ad88P`Y88ba, ``":"'' ,ad88P' `88888ba ad88888' `88888ba ad88888' `88888ba ad88888' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `"Y888b,d888P"' `"Y888b,d888P"' `"Y888b,d888P"' "]):([" "]):([" "]):([" ,ad88P`Y88ba, ,ad88P`Y88ba, ,ad88P`Y88ba, ad88888' `88888ba ad88888' `88888ba ad88888' `88888ba ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b, :"'' ,ad88P`Y88ba, ``":"'' ,ad88P`Y88ba, ``":"'' ,ad88P`Y88ba, ``": ad88888' `88888ba ad88888' `88888ba ad88888' `88888ba ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d888P"' `"Y888b,d888P"' `"Y888b,d888P"' `"Y888b, :([" Normand "]):([" "]):([" Veilleux "]): `Y88ba, ,ad88P`Y88ba, ,ad88P`Y88ba, ,ad88P' `88888ba ad88888' `88888ba ad88888' `88888ba ad88888' `Y88888b, ,d88888P' `Y88888b, ,d88888P' `Y88888b, ,d88888P' :,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,:,__`"Y888b,d888P"'__,: `Y88ba, ``":"'' ,ad88P`Y88ba, ``":"'' ,ad88P'Y88ba, ``":"'' ,ad88P' _ _ _ _ _ _ (_.' ) .' ) .' )( `. 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