KOCH SNOWFLAKE Fractals

L A B KOCH SNOWFLAKE 11 Fractals

louds, biological growth, and coastlines are examples of real-life phenomena that Cseem too complex to be described using typical mathematical functions or relationships. By developing fractal geometry, Benoit Mandelbrot (1924–) found a way to describe real-life phenomena using a new mathematical construct. Observations Mandelbrot first used the word fractal in 1975 to describe any geometric object whose detail is not lost as it is magnified. Being “endlessly magnifiable” means that you can zoom in on any portion of the fractal and it will be identical to the original fractal. Purpose In this lab, you will analyze some fractals. You will use Derive to “zoom in” on different parts of a fractal. References For more information about fractals see The Fractal Geometry of Nature by Benoit Mandelbrot. © Houghton Mifflin Company. All rights reserved.

LAB 11 KOCH SNOWFLAKE 1 2 Derive LabManual forCalculus Data Data Data O Stage 2 Stage 0 tions continuethisprocessindefinitely.Thefirstfourstagesareshownbelow. unit longisaddedinthecenterofeachsidefirstiteration.Successiveitera of eachsidetheoriginal.Inseconditeration,atrianglewithsidesone-ninth In thefirstiteration,atrianglewithsidesone-thirdunitlongisaddedincenter this The snowflake beginswithanequilateraltrianglewhosesidesareoneunitlong. mathematician HelgevonKoch(187 ne ofthe“classic”fractalsisKochsnowflake,namedafterSwedish Derive Derive file arestoredinthetextcalledLAB11.TXT. file correspondingtothislabiscalledLAB11.MTH.The instructionsfor 0 –1924). TheconstructionoftheKoch Stage 3 Stage 1 -

© Houghton Mifflin Company. All rights reserved. © Houghton Mifflin Company. All rights reserved. Exercises Exercises Exercises Instructor Date Name 2. 1. ______perimeter? Explainyourreasoning. sible foraclosedregionintheplanetohavefinite area andaninfinite Perimeter andArea. in thislab’sDatatocompletethefollowingtable. Complete theTable. ______Stage n 5 4 3 2 1 0 Ӈ ______Sides Perimeter 12 3 Use theiterationforKochsnowflakeasdescribed Using theresultsofExercise1,doyouthinkit’spos 4 3 Area Ί Ί 3 4 3 3 Class ______LAB 11 K OCH S NOWFLAKE - 3 4 Derive LabManual forCalculus 3. Stage process indefinitely.Thefirstfourstagesareshownbelow. unit longisaddedinthecenterofeachside.Successiveiterationscontinuethis each sideofthesquare.Inseconditeration,atrianglewithsidesone-ninth first iteration,atrianglewithsidesone-thirdunitlongisaddedinthecenterof time, theconstructionbeginswithasquarewhosesidesareoneunitlong.In Start withaSquare. Stage 2 Stage 0 n 5 4 3 2 1 0 Ӈ Sides Perimeter 16 4 1 4 3 6 A variationoftheKochsnowflakeisshownbelow.This 1 Area ϩ 1 Ί 9 3 Stage 3 Stage 1

© Houghton Mifflin Company. All rights reserved. © Houghton Mifflin Company. All rights reserved. 4. 5. are shownbelow. Successive iterationscontinuethisprocessindefinitely.Thefirstfourstages triangle withsidesone-ninthunitlongisaddedinthecenterofeachside. added inthecenterofeachsidepentagon.Inseconditeration,a one unitlong.Inthefirstiteration,atrianglewithsidesone-thirdlongis below. Thistime,theconstructionbeginswithapentagonwhosesidesare Start withaPentagon. Koch snowflakedescribed inExercise3?Explainyourreasoning. perimeter willbeequalto, greaterthan,orsmallerthantheperimeterof perimeter oftheKochsnowflake describedinExercise1?Doyouthinkits Exercise 4willhaveaperimeterequalto,greaterthan, orsmallerthanthe Comparing Perimeters. the KochsnowflakedescribedinExercise3?Explain yourreasoning. you thinkitsareawillbeequalto,greaterthan,orsmaller thantheareaof smaller thantheareaofKochsnowflakedescribed inExercise1?Do flake describedinthisexercisewillhaveanareaequalto,greaterthan,or How manysideswillthe Stage 2 Stage 0 Another variationoftheKochsnowflakeisshown n Do youthinktheKochsnowflakedescribedin th iterationhave?DoyouthinktheKochsnow- Stage 3 Stage 1 LAB 11 K OCH S NOWFLAKE 5 6 Derive LabManual forCalculus 6. A KochCurveUsingDerive. xmax, ymin, reproduces eachgraphbelow?Ifso,listtheviewingwindowvalues viewing windowyouused.Canfindmorethanonethat graphs ofaKochcurve.Use and ymax you usedforeachgraph. Derive Derive to reproduceeachgraphandrecordthe was usedtogeneratethefollowing xmin,