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210 lines
6.9 KiB
HTML
210 lines
6.9 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<title>Hypercubes</title>
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<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For K-12 kids, teachers and parents.">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<nav>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<article id="content" role="main">
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<h1 class="center">Hypercubes</h1>
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<p class="center"><i>In Geometry we can have different dimensions.</i></p>
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<p>The general idea of a <a href="hexahedron.html">cube</a> in any dimension is called a <b>hypercube</b>, or <b>n-cube</b>.</p>
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<p class="center"><img src="images/dimensions.svg" alt="dimensions 0, 1, 2 and 3" height="148" width="305"><br>
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A 0-cube is a point, a 1-cube is a line,<br>
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a 2-cube is a square, a 3-cube is a cube, etc</p>
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<h2>Points, Lines, Surfaces, ...</h2>
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<p>The magic binomial <b>x+2</b> can tell us how many points, lines, surfaces etc for each dimension:</p>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/dim-0.svg" alt="0 dimensions" height="42" width="138">
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</div>
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<div class="boxb">
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<p>(x+2)<sup>0</sup> = <b>1</b></p>
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<p>For zero dimensions we have 1 point</p>
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</div>
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</div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/hypercube-1.svg" alt="1 dimension" height="10" width="97">
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</div>
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<div class="boxb">
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<p>(x+2)<sup>1</sup> = <b>x + 2</b></p>
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We have <b>x</b> (a line of length "x") and <b>2</b> points</div>
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</div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/hypercube-2.svg" alt="2 dimensions" height="97" width="97">
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</div>
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<div class="boxb">
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<p>(x+2)<sup>2</sup> = (x+2)(x+2) = <b>x<sup>2</sup> + 4x + 4</b></p>
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<p>We have <b>x<sup>2</sup></b> (representing the area), <b>4</b> lines each of length <b>x</b>, and <b>4</b> points</p></div>
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</div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/hypercube-3.svg" alt="3 dimensions" height="130" width="122">
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</div>
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<div class="boxb">
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<p>Now (x+2)<sup>3</sup> = (x+2)(x<sup>2</sup> + 4x + 4) = <b>x<sup>3</sup> + 6x<sup>2</sup> + 12x + 8</b></p>
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<p>We have an <b>x<sup>3</sup></b> (the volume), <b>6</b> surfaces, <b>12</b> lines and <b>8</b> points.</p>
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<p>Verify it for yourself ... how many faces are there on a cube? How many lines, how many points?</p></div>
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</div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/dim-4.jpg" alt="4 dimensions" height="160" width="150">
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</div>
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<div class="boxb">
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<p>But we can go further ... into higher dimensions!</p>
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<p>A Tesseract is the 4D version of a cube: a <b>4-cube</b>.</p>
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<p>(x+2)<sup>4</sup> = (x+2)(x<sup>3</sup> + 6x<sup>2</sup> + 12x + 8) = <b>x<sup>4</sup> + 8x<sup>3</sup> + 24x<sup>2</sup> + 32x + 16</b></p>
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<p>So a 4-cube has:</p>
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<ul>
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<li>1 4D space</li>
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<li>8 cubes</li>
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<li>24 surfaces</li>
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<li>32 lines</li>
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<li>16 points</li></ul>
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<p>We may have trouble imagining what it looks like, but we can know its facts!</p>
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</div>
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</div>
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<div style="clear:both"> </div>
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<h2>How on Earth does this Work?</h2>
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<p>It is pure magic ... and the fact that <b>x+2</b> describes a line with two points.</p>
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<p>Maybe if we get more general it would help?</p>
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<h2>More General</h2>
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<p>Let us step away from pure cubes and allow different sizes:</p>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/dim-1-pts.svg" alt="1 dimension" height="16" width="141">
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</div>
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<div class="boxb">
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<p>(x+2)<sup>1</sup> = <b>x + 2</b></p>
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We have a line of length "x" and 2 points</div>
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</div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/dim-2-pts.svg" alt="2 dimensions" height="56" width="140">
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</div>
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<div class="boxb">
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<p>(x+2)(y+2) = <b>xy + 2x + 2y+ 4</b></p>
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<p>We have a <a href="rectangle.html">rectangle</a> of area xy, with 2 lines of x, 2 lines of y, and 4 points</p>
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</div> </div>
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<div class="dotpoint">
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<div class="boxa">
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<img src="images/dim-3-pts.svg" alt="3 dimensions" height="90" width="149">
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</div>
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<div class="boxb">
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<p>(x+2)(y+2)(z+2) = <b>xyz + 2xy + 2yz + 2xz + 4x + 4y + 4z + 8</b></p>
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<p>We have a <a href="cuboids-rectangular-prisms.html">cuboid</a> with volume of xyz, 2 surfaces each of xy, yz and xz, 4 lines each of x, y and z, and 8 points.</p>
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</div>
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</div>
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<div style="clear:both"> </div>
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<p><br></p>
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<p><br></p>
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<div class="related">
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<a href="dimensions.html">Dimensions</a>
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<a href="hexahedron.html">Cube</a>
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<a href="index.html">Geometry Index</a>
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