View source for Minkowski sausage

{{short description|Fractal first proposed by Hermann Minkowski}}
{{Distinguish|Minkowski cover}}
{{multiple image
 | align = <!--right-->
 | direction = vertical
 | width = 275
 | header = <!--Initiators and generators-->
 | image1 = Quadratic Koch curve type2 iterations.png
 | caption1 = First iterations of the quadratic type 2 [[Koch curve]], the Minkowski sausage{{efn|name=Type 2|Quadratic Koch curve type 2}}
 | image2 = Quadratic Koch curve type1 iterations.png
 | caption2 = First iterations of the quadratic type 1 Koch curve{{efn|name=Type 1|Quadratic Koch curve type 1}}
 | image3 = Koch quadratic L7 curves 0-2.svg
 | caption3 = Alternative generator with dimension of {{sfrac|ln 18|ln 6}}&nbsp;≈&nbsp;1.61{{efn|name=not|Neither type 1 nor 2}}
}}
[[Image:Quadratic Koch.svg|thumb|Higher iteration of type 2{{efn|name=Type 2}}]]
[[File:6452553 Vicsek Fractal Antenna.png|thumb|right|Example of a [[fractal antenna]]: a space-filling curve called a "Minkowski Island"<ref name=r4>{{cite journal|last1=Cohen|first1=Nathan|title=Fractal antennas Part 1|journal=[[Communication Quarterly]]|date=Summer 1995|pages=7–23|url=https://archive.org/stream/fea_Fractal_Antennas_Part-1/Fractal_Antennas_Part-1_djvu.txt}}</ref> or "Minkowski fractal"<ref>Ghosh, Basudeb; Sinha, Sachendra N.; and Kartikeyan, M. V. (2014). ''Fractal Apertures in Waveguides, Conducting Screens and Cavities: Analysis and Design'', p. 88. Volume 187 of ''Springer Series in Optical Sciences''. {{ISBN|9783319065359}}.</ref>{{efn|name=Type 1}}]]
{{multiple image
 | align = <!--right-->
 | direction = horizontal
 | width = 110
 | image1 = 32SegmentSmall.jpg
 | caption1 = Generator
 | image2 = 8 scale fractal.png
 | caption2 = island{{efn|name=not}}
}}

The '''Minkowski sausage'''<ref>{{cite book|last=Lauwerier |first=Hans |title=Fractals: Endlessly Repeated Geometrical Figures |url=https://archive.org/details/fractalsendlessl00lauw |url-access=registration |publisher=Princeton University Press |date=1991 |isbn=0-691-02445-6 |page=[https://archive.org/details/fractalsendlessl00lauw/page/37 37] |quote=The so-called Minkowski sausage. Mandelbrot gave it this name to honor the friend and colleague of Einstein who died so untimely (1864-1909). |translator-last1=Gill-Hoffstädt |translator-first1=Sophia }}<!--Original 1987 in Dutch.--></ref> or '''Minkowski curve''' is a [[fractal]] first proposed by and named for [[Hermann Minkowski]] as well as its casual resemblance to a [[sausage]] or sausage links. The initiator is a [[line segment]] and the generator is a [[polygonal chain|broken line]] of eight parts one fourth the length.<ref name="Addison">Addison, Paul (1997). ''Fractals and Chaos: An illustrated course'', p. 19. CRC Press. {{ISBN|0849384435}}.</ref>

The Sausage has a [[Hausdorff dimension]] of <math>\left ( \ln8/\ln4\ \right ) = 1.5 = 3/2</math>.{{efn|name=Type 2}} It is therefore often chosen when studying the physical properties of non-integer fractal objects. It is strictly [[self-similar]].<ref name="Addison"/> It never intersects itself. It is [[continuous function|continuous]] everywhere, but [[differentiable function|differentiable]] nowhere. It is not [[rectifiable curve|rectifiable]]. It has a [[Lebesgue measure]] of 0. The type 1 curve has a dimension of {{sfrac|ln 5|ln 3}}&nbsp;≈&nbsp;1.46.{{efn|name=Type 1}}

Multiple Minkowski Sausages may be arranged in a four sided polygon or [[square]] to create a quadratic [[Koch island]] or '''Minkowski island/[snow]flake''':
{{multiple image
 | perrow = 2
 | align = center
 | direction = <!--vertical-->
 | header = Islands
 | image1 = Karperienflakeani2.gif
 | caption1 = Island formed by a different generator<ref name="Archive">Weisstein, Eric W. (1999). "[https://archive.lib.msu.edu/crcmath/math/math/m/m263.htm Minkowski Sausage]", ''archive.lib.msu.edu''. Accessed: 21 September 2019.</ref><ref name="Pamfilos">Pamfilos, Paris. "[http://users.math.uoc.gr/~pamfilos/eGallery/problems/Minkowski.html Minkowski Sausage]", ''user.math.uoc.gr/~pamfilos/''. Accessed: 21 September 2019.</ref><ref name="MathWorld">{{MathWorld |id=MinkowskiSausage  |title=Minkowski Sausage |access-date=22 September 2019}}</ref> with a dimension of ≈1.36521<ref>[[Benoit Mandelbrot|Mandelbrot, B. B.]] (1983). ''The Fractal Geometry of Nature'', p. 48. New York: W. H. Freeman. {{ISBN|9780716711865}}. Cited in Weisstein ''MathWorld''.</ref> or 3/2<ref name="Archive"/>{{efn|name=Type 1}}
 | image2 = Minkowski island 1-3.svg
 | caption2 = Island formed by using the Sausage as the generator{{efn|name=Type 2}}{{efn|This has been called the "zig-zag quadratic Koch snowflake".<ref>Schmidt, Jack (2011). "[https://www.ms.uky.edu/~jack/2011-01-MA111/2011-04-22-Worksheet.pdf The Koch snowflake worksheet II]", p. 3, ''UK MA111 Spring 2011, ms.uky.edu''. Accessed: 22 September 2019.</ref>}}
| image3 = Anticross-stitch curve 0-4.svg
| caption3 = Anti-island (anti[[cross-stitch curve]]), iterations 0-4{{efn|name=Type 1}}
| image4 = Minkowski anti-island 1-3.svg
| caption4 = Anti-island: the generator's symmetry results in the island [[flipped image|mirrored]]{{efn|name=Type 2}}
| image5 = Minkowski island (angled quadratic Koch type 1) 0-4.svg
| caption5 = Same island as the first formed from a different generator <!--(resembling: /\/ )-->,<ref name="Pamfilos"/> which forms 2 [[right triangle]]s with side lengths in ratio: 1:2:√5<ref name="MathWorld"/>{{efn|name=Type 1}}
| image6 = Koch quadratic flake L7 0-3.svg
| caption6 = Quadratic island formed using curves with a different generator{{efn|name=not}}
}}

==See also==
*[[Self-avoiding walk]]
*[[Vicsek fractal]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==External links==
*{{WolframDemonstrations |title=Square Koch Fractal Curves |urlname=SquareKochFractalCurves |access-date=23 September 2019}}

{{Fractals}}

[[Category:De Rham curves]]
[[Category:L-systems]]
[[Category:Hermann Minkowski]]