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  • efg's Fractals And Chaos -- von Koch Curve Lab Report
    base is So the area of the equilateral triangle two 30 60 90 triangles is In each step k we add the area of n k little equilateral triangles with sides s k The area at step k can be written Convince yourself that n 1 3 n 2 3 4 n 3 3 4 4 and n k 3 4 k 1 The sides of the little triangles are scaled down by a factor of 3 in each step i e s k 1 3 n a The area recurrence formula can now be written The total area can be computed by evaluating the geometric series formed by the above recurrence formula The total area after an infinite number of steps is For a more detailed explanation of the area computation see Peitgen92B p 167 or Eric Weisstein s World of Math In summary the von Koch snowflake has an infinite perimeter but a finite area The von Koch curve has a self similarity Hausdorff dimension D log 4 log 3 1 2619 A line is 1D and a square is 2D See Peitgen92A or Devaney89 The modified Koch curve Falconer90 pp 120 121 uses a rectangle in each iteration instead of a triangle The height of the rectangle is the same as the height of the triangle in Step 1 above Feel free to experiment with other rules Also see Experiments in Computing Laboratories for Introductory Computer Science in Turbo Pascal by Kenneth Abernethy and J Thomas Allen Jr PWS Kent Publishing Boston 1993 Chapter 20 Recursion in Fractal Geometry The Koch Snowflake Curve pp 342 348 Generalized Snowflake Curves pp 349 350 Materials and Equipment Software Requirements Windows 98 Delphi 4 5 to recompile KochCurve EXE Hardware Requirements VGA display with 640 by 480 screen in high true color display mode Procedure Double click on the KochCurve EXE icon to start the program Select various values in the spinboxes at the left and immediately observe a new von Koch curve on the screen Select the Print button to print the currently displayed von Koch curve perhaps in greater detail since many printers have many more dots than a display screen does To write a file to disk select the bitmap size and the press the Write to File button A save dialog will let you write this file anywhere you want Discussion Outline for now ScreenKochCurve unit and form DrawVonKoch method ButtonPrintVonKochClick ButtonVonKochFileClick ShellExecute to link to web site VonKochCurveLibrary unit separates computations from user interface TVonKochCurve Class with method Draw draws on any canvas screen printer or bitmap NextSegments Local routine to Draw which is called recursively Since NextSegments works with any line segment the points making up the polygon are rotated before this routine is ever called MapWorldToPixel unit TRealPoint RealPoint TRealRect RealRect TDigitalPantograph for mapping real x y to integer i j corrects direction of y dimension Conclusions Tell your friends the von Koch snowflake has an infinite perimeter but a finite area What

    Original URL path: http://www.efg2.com/Lab/FractalsAndChaos/vonKochCurve.htm (2016-02-14)
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  • efg's Fractals And Chaos -- Lyapunov Exponents Lab Report
    I just computed all base 2 logarithms in a brute force way Remember We should forget about small efficiencies say about 97 of the time premature optimization is the root of all evil Donald Knuth Click on any of the color boxes above to change the color using the standard color dialog The various regions of the ab parameter map are either stable or chaotic Different coloring schemes can be used to empahsize each region Feel free to experiment with various color schemes Markus89 often used a shades of gray gradient or a black to yellow gradient for coloring order and a black to red gradient for color chaos In some images in Markus90 a black to gray gradient was used to color order and a solid black was used to color chaos Another color scheme used in Markus90 was the black to yellow as described in Markus89 for order but a back to blue color mapping for chaos these are the current color scheme defaults Computation of the rainbow colors are explained in more detail in the Spectra Lab Report Press the Save button to store the image in a pf24bit BMP or a JPG Press the Print button and select portrait or landscape orientation and any other desired printer properties The following table summarizes the header and footer information printed with the above image Lyapunov Image efg1112 Sequence ab image a 3 8360 to 3 8405 360 pixels vs b 3 845 to 3 851 480 pixels Iterations point 600 warmup 4000 maximum 428 seconds creation time Coloring Order Rainbow Chaos Gradient clBlack to clBlue Discussion Outline for now ScreenLyapunov unit and form LyapunovUpdateCallback ResetStartPixel Procedure TimerDisplayTimer Procedure with elapsed and estimated completion time Keystroke validation EditFloatKeyPress EditIntegerPress EditSequencePress Range validation EditLimitsChange EditPixelsChange EditIterationsChange Start Pause Reset Procedures BitBtnStartClick BitBtnPauseClick BitBtnResetClick csOwnerDrawFixed style ComboBox with hidden parameters UpdateColorScheme storing IEEE singles as pixels in pf32bit bitmap displaying floating point value with ImageLyapunovRawMouseMove handling MouseMove events when TImage is stretched saving BMP or JPG file need to provide way to specify JPEG quality ShellExecute to link to web site LyapunovLibrary unit separates computations from user interface LyapunovExponent Function TLyapunovLambdaMap Class with methods Compute SetColorScheme ColorLambdaBitmap SpectraLibrary unit Rainbow Function WavelengthToRGB Function overloaded ColorToRGBTriple Function Lyapunov Benchmark 640 by 480 bitmap 600 warmup 4000 iterations Dewdney s Name Hours Minutes Seconds Sun SparcStation 2 C February 1992 33 MHz 486 Turbo Pascal July 1992 450 MHz Pentium II Delphi 5 Win 98 April 2000 650 MHz Pentium III Delphi 5 Win 2000 April 2000 LyapunovSpace 3 41 06 8 50 59 0 12 39 0 7 55 Jellyfish 3 38 33 8 49 20 0 12 40 0 7 52 Swallow 3 39 24 8 49 46 0 12 39 0 7 53 ZirconZity 3 32 16 9 04 25 0 12 53 0 7 59 pf32bit Enigma One Win 98 Pentium reports FFFFFFFF for the values in a scanline for a new pf32bit bitmap while two others report a

    Original URL path: http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm (2016-02-14)
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  • efg's Fractals And Chaos -- Sierpinski Triangle Lab Report
    is infinite A Sierpinski triangle has a self similarity Hausdorff dimension D 1 585 A line is 1D and a square is 2D Also see Experiments in Computing Laboratories for Introductory Computer Science in Turbo Pascal by Kenneth Abernethy and J Thomas Allen Jr PWS Kent Publishing Boston 1993 Chapter 20 Recursion in Fractal Geometry Constructing Sierpinski s Carpet pp 351 352 Materials and Equipment Software Requirements Windows 98 Delphi 4 5 to recompile SierpinksiTriangle EXE Hardware Requirements VGA display with 640 by 480 screen in high true color display mode Procedure Double click on the SierpinskiTriangle EXE icon to start the program Select various values in the spinboxes at the left and immediately observe a new Sierpinski Gasket on the screen Click on the colored boxes to change colors via a color dialog Select the Print button to print the currently displayed Sierpinksi triangle perhaps in greater detail since many printers have many more dots than a display screen does To write a file to disk select the bitmap size and the press the Write to File button A save dialog will let you write this file anywhere you want Discussion Outline for now ScreenSierpinskiTriangle unit and form DrawSierpinski method ButtonPrintSierpinskiClick ButtonSierpinksiFileClick ShapeTColorMouseDown to change color of TShape ShellExecute to link to web site SierpinskiTriangleLibrary unit separates computations from user interface TSierpinskiTriangle Class with method Draw draws on any canvas screen printer or bitmap SierpinskiTriangle Local routine to Draw which is called recursively Since Sierpinksi works with any triangle the points making up the triangle are rotated before this routine is ever called MapWorldToPixel unit TRealPoint RealPoint TRealRect RealRect TDigitalPantograph for mapping real x y to integer i j corrects direction of y dimension A Sierpinski Triangle can be formed in a variety of other ways In the Chaos

    Original URL path: http://www.efg2.com/Lab/FractalsAndChaos/SierpinskiTriangle.htm (2016-02-14)
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  • efg's Fractals and Chaos Page
    Delphi Sierpinski Triangle Gasket Sierpinski triangle Sierpinski gasket fractals self similarity Hausdorff dimension digital pantograph world to pixel mapping recursion von Koch Snowflake von Koch curve von Koch snowflake fractals self similarity Hausdorff dimension digital pantograph world to pixel mapping recursion Chaos Program Description Files Keywords Iterated Function System to create four ferns Also see Mathematical Recreations Scientific American How to transform flights of fancy into fractal flora or fauna

    Original URL path: http://www.efg2.com/Lab/FractalsAndChaos/index.html (2016-02-14)
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  • efg's Pepsi Challenge
    your kids to balance their soft drink cans Hint The cans are not full but they re not empty either Thanks to Jim Sisul for showing me this trick Updated

    Original URL path: http://www.efg2.com/Lab/ScienceAndEngineering/Pepsi.htm (2016-02-14)
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  • efg's Fireworks, Half a Dozen Pi’s, and the Fourth of July
    a configuration the gaps between knuckles are approx 3 2 3 degrees To estimate angles over 8 degrees while keeping your arm stretched out put your thumb tip and little fingertip as far apart as you can The angle between them is near 19 degrees This works because most people with long arms also have big hands I think the army said 21 not 19 but that one s easy to check right angles are easy to find and 5 X 19 is much closer to 90 than 5 X 21 is Maybe one can t stretch one s thumb and little finger so far apart as one gets older The 19 year old version of myself is no longer available Method 2 Since estimating angles is difficult make a simple astrolabe to measure the angles You will need the following six inch plastic protractor with hole plastic drinking straw piece of string or heavy thread about two feet long large paper clip or binder clip for a weight some tape To make the astrolabe follow these simple steps see Figure 2 below Tape the drinking straw to the straight edge part of the protractor Make a loop with the string that passes through the hole in the protractor Attach the two ends of the string together and to the paper clip weight Figure 2 Simple Astrolabe To use your astrolabe hold it from the top with your thumb and index finger Rotate the astrolabe up or down around the hole Now locate something by looking through the straw Read the angle on the protractor marked by the string To measure the angle of a firework measure the top and bottom angle with your astrolabe Then take the difference of the two angles A team of two works best One team member observes through the straw while the other reads the angle Use your astrolabe to determine the size of fireworks or to measure the position of the moon in the sky Why does the formula work How accurate is the formula Figure 1 shows two right triangles that have the same dimensions In a right triangle the tangent of an angle in radians is defined to be the ratio of the length of the opposite side divided by the length of the adjacent side See Figure 3 below Figure 3 Tangent of an angle in a right triangle If we use the definition of a tangent with one of the right triangles from Figure 1 the following formula can be written We divide by 2 in two places since this formula only applies to one of the two right triangles The full angle and full size are shared by both triangles For angles 30 degrees there is less than a 10 error in the following approximation For angles 10 degrees there is less than a 1 error in this approximation Assuming we re at a safe distance from the fireworks this approximation is appropriate Given this approximation for small

    Original URL path: http://www.efg2.com/Lab/ScienceAndEngineering/Fireworks/index.html (2016-02-14)
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  • efg's Computer Lab: Spectra Lab Report
    s approximations to R G and B as a function of wavelength Select the Intensity or Y inYIQ coordinates to display the corresponding intensity of a pixel of a given wavelength Print the TChart graph if desired See graph above Select the Info about Atomic Spectra for Hydrogen Tabsheet for information about the emission and absorption spectra of hydrogen Discussion The WaveLengthToRGB function is based on Dan Bruton s work www physics sfasu edu astro color html and is in the file SpectraLibrary PAS which is part of the download set PROCEDURE WavelengthToRGB CONST Wavelength Nanometers VAR R G B BYTE CONST Gamma 0 80 IntensityMax 255 VAR Blue DOUBLE factor DOUBLE Green DOUBLE Red DOUBLE FUNCTION Adjust CONST Color Factor DOUBLE INTEGER BEGIN IF Color 0 0 THEN RESULT 0 Don t want 0 x 1 for x 0 ELSE RESULT ROUND IntensityMax Power Color Factor Gamma END Adjust BEGIN CASE TRUNC Wavelength OF 380 439 BEGIN Red Wavelength 440 440 380 Green 0 0 Blue 1 0 END 440 489 BEGIN Red 0 0 Green Wavelength 440 490 440 Blue 1 0 END 490 509 BEGIN Red 0 0 Green 1 0 Blue Wavelength 510 510 490 END 510 579 BEGIN Red Wavelength 510 580 510 Green 1 0 Blue 0 0 END 580 644 BEGIN Red 1 0 Green Wavelength 645 645 580 Blue 0 0 END 645 780 BEGIN Red 1 0 Green 0 0 Blue 0 0 END ELSE Red 0 0 Green 0 0 Blue 0 0 END Let the intensity fall off near the vision limits CASE TRUNC Wavelength OF 380 419 factor 0 3 0 7 Wavelength 380 420 380 420 700 factor 1 0 701 780 factor 0 3 0 7 780 Wavelength 780 700 ELSE factor 0 0 END R Adjust Red Factor G Adjust Green Factor B Adjust Blue Factor END WavelengthToRGB The product of wavelength and frequency gives the speed of light c 2 9979 x 10 8 m sec Given the linear wavelengths in nanometers on the Visible Light tabsheet the frequency values are computed in TeraHertz While the wavelengths in the spectrum bitmap are linear the frequencies are not The wavelengths of the Balmer series for n 3 to 9 are calculated once by the FormCreate method The Balmer emission spectra for hydrogen is computed from this formula where n 3 4 5 The general hydrogen emission series can be computed from this formula where R H Rydberg Constant for Hydrogen 10 967 757 6 m 1 n k 1 k 2 k 3 and k is defined in the following table k Name Wavelenth Range 1 Lyman ultraviolet 2 Balmer near ultraviolet and visible 3 Paschen infrared 4 Brackett infrared 5 Pfund infrared For additional details of the hydrogen spectra and other atoms and molecules see a good physics book such as Quantum Physics of Atoms Molecules Solids Nuclei and Particle s 2nd edition or Quantum Chemistry 5th edition Most changes in the user interface

    Original URL path: http://www.efg2.com/Lab/ScienceAndEngineering/Spectra.htm (2016-02-14)
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  • efg's KCK Tornado (4 May 2003)
    2003 Fox 4 special Tornado Tales CNN put a contrast enhanced still from near the end of Brian s video 1 above on their web site for a few hours that evening I m told CNN aired the video but I did not see it Sequence of stills using a Nikon CoolPix digital camera Images are slightly contrast enhanced By the last frame the tornado is nearly out of Kansas and in Missouri Note Click on map to explore position with respect to other Kansas City Landmarks This map shows the approximate path of the tornado and the location from where the video was made as well as some locations of damage shown below Brian and I were at the point X near the I 70 East exit on I 435 when we took the pictures From driving through the area the tornado crossed I 435 near Parallel Parkway and definitely hit very near the south entrance to Wyandotte County Park along Leavenworth Road see B pictures below Images from damage at locations A B and C are shown below Point C is really off the map a little more to the east however A week later 11 May 2004 A Looking west on 99th Street north of Parallel These houses were hit after the tornado crossed I 435 Access to most damaged residential areas was limited to residents so I couldn t get any better pictures B The two pictures above are of three houses in a row along 91st Street in Kansas City KS near the intersection with Leavenworth Road The houses took a direct hit from the tornado The damage at this location is likely rated F4 on the Fujita Tornado Damage Scale The only fatality from this storm was near this location House on north side of Leavenworth Road east of 91st Street near Townsend Court about a block east and a block north of the pictures above Most of the houses around Townsend Court circle were heavily damaged These two pictures were taken 17 May 2003 Bricks helped only a little in keeping walls C 200 MPH winds damaged American Body Company in Riverside MO across Missouri River to the northeast of Kansas City KS Unfortunately this company with 40 employees does not plan to re open Background Information I arrived home from an errand and my son suggested we go storm chasing since there were reports of funnels in Leavenworth County we had talked about chasing storms but had never done this before With about 15 seconds of planning we grabbed a TV a power inverter and some rabbit ears I accidentally smashed the 12 V power connector to the inverter in the car door so while my son drove he s 17 now I used some electrician s tape to fix the 12 V power connector and coax the antenna into staying connected a bit more planning would have helped We stopped on the Kansas Avenue exit off I 435 north of the

    Original URL path: http://www.efg2.com/Lab/ScienceAndEngineering/KCKTornado03/index.htm (2016-02-14)
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