谢氏地毯分形几何构造
2020-07-12 10:58
本范例展示了谢氏地毯（谢尔宾斯基地毯）的迭代构造过程。通过多边形、重心、缩放、数列和迭代列表等GeoGebra命令，演示了分形几何中自相似图形的生成原理。该构造涉及几何变换与序列迭代，直观呈现了无限细分的数学之美，适用于探索分形维数与极限概念。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="d" id="FC971A33-6CC6-4655-B69E-7F9F14E056D3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1299" height="664" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4888375673595073" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="273" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="643" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="635" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="643" height="456"/>
<coordSystem xZero="282.50000000000006" yZero="281.5" scale="49.99999999999998" yscale="49.999999999999986"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="9" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.49" y="-3.53" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.53" y="-3.63" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="4"/>
<output a0="poly1" a1="f" a2="g" a3="h" a4="i" a5="C" a6="D"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.9900000095367432"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.10000000000000009" y="8.02" z="28.6596"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-8.019999999999996" y="0.10000000000000142" z="36.69359999999999"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="h">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.1000000000000032" y="-8.02" z="35.670799999999986"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="i">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="8.02" y="-0.10000000000000231" z="27.636799999999994"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="4.630000000000002" y="4.389999999999997" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-3.389999999999998" y="4.49" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Centroid">
<input a0="poly1"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5700000000000001" y="0.4299999999999988" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Dilate">
<input a0="poly1" a1="1 / 3" a2="E"/>
<output a0="t1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="255" b="255" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Sequence">
<input a0="Point[poly1, k]" a1="k" a2="0" a3="7 / 8" a4="1 / 8"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="IterationList">
<input a0="Flatten[Zip[Dilate[p, 1 / 3, q], q, l1]]" a1="p" a2="{{t1}}" a3="3"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="1.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
</construction>
</geogebra>