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206 lines
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HTML
206 lines
9.7 KiB
HTML
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<title>Tessellation</title>
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<h1 align="center">Tessellation</h1>
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<p align="center"><i>A pattern of shapes that fit perfectly together!</i></p>
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<p>A <b>Tessellation</b> (or <b>Tiling</b>) is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps.</p>
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<h3>Examples: </h3>
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<div class="centerfull">
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<div class="boxa"><img src="images/tessellation-2.gif" width="170" height="110" alt="tessellation rectangles" /><br>
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Rectangles</div>
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<div class="boxa"><img src="images/tessellation1.gif" width="170" height="110" alt="tessellation octagons and squares" /><br>
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Octagons and Squares</div>
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<div class="boxa"><img src="images/tessellation-5.gif" width="170" height="110" alt="tessellation pentagons" /><br>
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Different Pentagons</div>
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</div>
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<div style="clear:both"></div>
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<h2>Regular Tessellations</h2>
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<p>A <b>regular</b> tessellation is a pattern made by repeating a <a href="../shape.html">regular polygon</a>. </p>
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<p><i>There are only 3 regular tessellations:</i></p>
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<div class="centerfull">
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<div class="boxa"><img src="images/tessellation-regular-triangle.gif" width="170" height="110" alt="tessellation regular triangles" /><br>
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<b>Triangles</b><br>
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3.3.3.3.3.3 </div>
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<div class="boxa"><img src="images/tessellation-regular-square.gif" width="170" height="110" alt="tessellation regular squares" /><br>
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<b>Squares</b><br>
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4.4.4.4 </div>
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<div class="boxa"><img src="images/tessellation-regular-hexagon.gif" width="170" height="110" alt="tessellation regular hexagons" /><br>
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<b>Hexagons</b><br>
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6.6.6 </div>
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</div>
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<div style="clear:both"></div>
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<h2>Look at a Vertex ...</h2>
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<table border="0" align="center">
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<tr>
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<td align="center"><table border="0">
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<tr>
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<td><img src="images/tessellation-vertex.svg" alt="tessellation vertex" /></td>
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<td>A vertex is just a "corner point". <br />
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<br />
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What shapes meet here?</td>
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</tr>
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</table></td>
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</tr>
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<tr>
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<td align="center"><table border="0">
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<tr>
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<td align="right"><p>Three hexagons meet at this vertex,<br />
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and a hexagon has 6 sides. </p>
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<p>So this is called a <b>"6.6.6"</b> tessellation.</p></td>
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<td><img src="images/tessellation-notation.svg" alt="tessellation notation 6.6.6" /></td>
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</tr>
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</table></td>
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</tr>
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<tr>
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<td> </td>
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</tr>
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<tr>
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<td><h3>For a regular tessellation, the pattern is identical at each vertex!</h3></td>
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</tr>
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</table>
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<h2>Semi-regular Tessellations</h2>
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<p>A <b>semi-regular</b> tessellation is made of two or more regular polygons. <b>The pattern at each vertex must be the same!</b> </p>
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<p><i>There are only 8 semi-regular tessellations:</i></p>
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<div class="centerfull">
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<div class="boxa"><img src="images/tessellation-3-3-3-3-6.gif" width="170" height="111" alt="tessellation 3.3.3.3.6" /><br>
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3.3.3.3.6</div>
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<div class="boxa"><img src="images/tessellation-3-3-3-4-4.gif" width="170" height="110" alt="tessellation 3.3.3.4.4" /><br>
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3.3.3.4.4</div>
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<div class="boxa"><img src="images/tessellation-3-3-4-3-4.gif" width="169" height="109" alt="tessellation 3.3.4.3.4" /><br>
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3.3.4.3.4</div>
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<div class="boxa"><img src="images/tessellation-3-4-6-4.gif" width="170" height="110" alt="tessellation 3.4.6.4" /><br>
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3.4.6.4</div>
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<div class="boxa"><img src="images/tessellation-3-6-3-6.gif" width="170" height="110" alt="tessellation 3.6.3.6" /><br>
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3.6.3.6</div>
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<div class="boxa"><img src="images/tessellation-3-12-12.gif" width="170" height="110" alt="tessellation 3.12.12" /><br>
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3.12.12</div>
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<div class="boxa"><img src="images/tessellation-4-6-12.gif" width="170" height="110" alt="tessellation 4.6.12" /><br>
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4.6.12</div>
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<div class="boxa"><img src="images/tessellation-4-8-8.gif" width="170" height="110" alt="tessellation 4.8.8" /><br>
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4.8.8</div>
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</div>
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<div style="clear:both"></div>
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<table width="90%" border="0" align="center">
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<tr>
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<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><p>To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12".</p></td>
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</tr>
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<tr>
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<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td>And always start at the polygon with the least number of sides, so "3.12.12", not "12.3.12"</td>
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</tr>
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</table>
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<p> </p>
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<table border="0" align="center">
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<tr>
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<td align="center"><img src="../images/style/question.gif" width="26" height="47" alt="question" /></td>
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<td><i>Question 1: For the tessellations above, is the pattern the same at each vertex? </i></td>
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</tr>
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<tr>
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<td align="center"><img src="../images/style/question.gif" width="26" height="47" alt="question" /></td>
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<td><i>Question 2: One of those patterns becomes different when we make a mirror-image of it ... which one?</i></td>
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</tr>
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</table>
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<h2>Other Tessellations</h2>
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<p>There are also "demiregular" tessellations, but mathematicians disagree on what they actually are!</p>
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<p>And some people allow curved shapes (not just polygons) so we can have tessellations like these:</p>
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<div class="centerfull">
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<div class="boxa"><img src="images/tessellation-3.gif" width="170" height="110" alt="tessellation curvy" /><br>
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Curvy Shapes</div>
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<div class="boxa"><img src="images/tessellation-7.gif" width="170" height="110" alt="tessellation circles lines" /><br>
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Circles</div>
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<div class="boxa"><img src="images/tessellation-6.gif" width="170" height="110" alt="tessellation eagles" /><br>
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Eagles?</div>
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</div>
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<div style="clear:both"></div>
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<h2>Tessellation Artist</h2>
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<p>All these images were made using <a href="tessellation-artist.html">Tessellation Artist</a>, with some color added using a paint program.</p>
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<p>You can try it too - maybe you will invent a new tessellation!</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(5482, 5483, 5484, 5485, 5486, 853, 854, 3368, 3369, 5487);</script> </div>
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<div class="related"> <a href="tessellation-artist.html">Tessellation Artist</a> <a href="symmetry-artist.html">Symmetry Artist</a> <a href="index.html">Geometry Index</a> </div>
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Copyright © 2018 MathsIsFun.com
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