Rotational Tiling of Pentagons (Iteration and Sequence)
2019-03-16 13:29
This example demonstrates the rotational tiling of pentagons. By combining iteration and sequences with vector and polygon properties, it showcases the rotational and translational symmetries of pentagons on a plane. The construction explores geometric transformations, sequences, and complex numbers in plane geometry, making it ideal for investigating polygon tessellations and symmetry groups.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.529.0" app="classic" platform="d" id="0d2ee116-5ba7-4cbc-a390-2c047bb8a705" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1304" height="768" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.27300613496932513" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="302,109,700,550" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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 , 66 || 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 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="916" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="356" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 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 | 100004 100005 100006 100008 100009 100010 100003" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="916" height="575"/>
<coordSystem xZero="456.54217590674614" yZero="306.40335758042124" scale="16.161336399037477" yscale="16.16133639903748"/>
<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="1"/>
<auxiliary show="true"/>
</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="numeric" label="n">
<value val="12.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="12" absoluteScreenLocation="true" width="200.0" x="52.0" y="34.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="10"/>
<output a0="poly1" a1="f" a2="g" a3="h" a4="i" a5="j" a6="k" a7="l" a8="m_1" a9="p" a10="q" a11="C" a12="D" a13="E" a14="F" a15="G" a16="H" a17="I" a18="J"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.800000011920929"/>
<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="-0.0" y="1.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<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="-0.5877852522924729" y="0.809016994374947" z="0.5877852522924729"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<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="-0.9510565162951532" y="0.3090169943749479" z="1.5388417685876254"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<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="-0.9510565162951532" y="-0.30901699437494745" z="2.4898982848827793"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="j">
<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="-0.5877852522924734" y="-0.8090169943749472" z="3.077683537175253"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="k">
<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="0.0" y="-0.9999999999999999" z="3.0776835371752527"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="l">
<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="0.5877852522924729" y="-0.8090169943749473" z="2.4898982848827793"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="m_1">
<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="0.951056516295153" y="-0.30901699437494745" z="1.5388417685876254"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p">
<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="0.9510565162951534" y="0.309016994374947" z="0.5877852522924732"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="q">
<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="0.5877852522924734" y="0.8090169943749475" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="C">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1.809016994374947" y="0.5877852522924729" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="2.118033988749895" y="1.5388417685876261" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="E">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1.8090169943749475" y="2.4898982848827793" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1.0000000000000002" y="3.0776835371752527" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="G">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="3.3306690738754696E-16" y="3.0776835371752527" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="H">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.809016994374947" y="2.4898982848827798" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="I">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-1.1180339887498945" y="1.5388417685876268" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="J">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.8090169943749475" y="0.5877852522924734" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="G" a1="F" a2="5"/>
<output a0="poly2" a1="r" a2="s" a3="t" a4="a" a5="b" a6="K" a7="L" a8="M"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="r">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="0.9999999999999999" z="-3.0776835371752527"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="s">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.9510565162951532" y="0.30901699437494723" z="8.881784197001252E-16"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="t">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.5877852522924734" y="-0.8090169943749472" z="4.028740053470406"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.5877852522924734" y="-0.8090169943749473" z="3.4409548011779325"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.9510565162951532" y="0.30901699437494745" z="-0.9510565162951538"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="K">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1.3090169943749475" y="4.028740053470406" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="L">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.5000000000000002" y="4.616525305762879" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="M">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.3090169943749471" y="4.028740053470406" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<command name="Join">
<input a0="Sequence[Sequence[k, i, 1, floor((k + 2) / 2)], k, 0, n]"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Vector">
<input a0="F" a1="M"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.3090169943749475" y="0.9510565162951532" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="F"/>
</element>
<command name="Length">
<input a0="l1"/>
<output a0="c"/>
</command>
<element type="numeric" label="c">
<value val="49.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Centroid">
<input a0="poly1"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5" y="1.5388417685876263" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="m">
<value val="17.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="c" absoluteScreenLocation="true" width="200.0" x="55.0" y="66.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Zip">
<input a0="Element[t, 1]" a1="t" a2="IterationList[{Translate[Element[p, 1], Vector[Rotate[u, (Element[l1, Element[p, 2]] * π / 5)]]], Element[p, 2] + 1}, p, {{poly2, 1}}, m]"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="true" label="false"/>
<objColor r="75" g="0" b="130" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Rotate[l2, ((2 * π / 5) * i), N]" a1="i" a2="1" a3="4"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="true" label="false" ev="4"/>
<objColor r="75" g="0" b="130" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Rotate[l2, ((2 * π / 5) * i) - π / 5, N]" a1="i" a2="1" a3="5"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.800000011920929"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
</construction>
</geogebra>