ÿþ<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>Mathematical Brooding over an Egg</title> <meta name="author" content="Andr&eacute; Heck"> <meta name="keywords" content="digital images, analytic geometry, GeoGebra"> <meta name="description" content="This article discusses how algebra and geometry can be brought to life with a dynamic geometry package that allows the use of digital images."> <link rel="stylesheet" href="./extras/ScreenStyles.css" type="text/css" media="screen"> <link rel="stylesheet" href="./extras/PrintStyles.css" type="text/css" media="print"> </head> <body> <h2>5. Computing with ellipses and eggs</h2> <p>The main reasons to prefer approximations of the egg curve with semi-ellipses of which the mathematical formulas are in canonical form are: <ul> <li>in this case you easily read off the semiminor and semimajor axes of the ellipses;</li> <li>you can apply mathematical formulas for the area and volume of the surface of revolution obtained by rotating an ellipse about its major axis.</li> </ul> The following formulas hold for the surface of revolution obtained from an ellipse with <var>a</var> = the length of the semimajor axis and with <var>b</var> = the length of the semiminor axis:</p> <p><div class="math"> <img alt="volume formula" src="formulas/eq03.png" /></div></p> <p>and</p><p> <div class="math"> <img alt="area formula" src="formulas/eq04.png" /> </div> </p> <p> with eccentricity <var>e</var> given by</p><p> <div class="math"> <img alt="eccentricity formula" src="formulas/eq05.png" />.</div></p> <p>The major perimeter of the surface of revolution is given by:</p> <div class="math"> <p><img alt="perimeter formula" src="formulas/eq06.png" />,<p></div> <p>where E is the <a href="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html" class="external" target="external" title="Open in a new window">complete elliptic integral of the second kind</a>. These formulas can be found on Internet (e.g. <a href="http://mathworld.wolfram.com/Perimeter.html" class="external" target="external" title="Open in a new window">http://mathworld.wolfram.com/Perimeter.html</a>) and I do not expect that many a student is able to derive the formulas or even knows what an elliptic integral actually is. However, in principle, the formulas can be computed via integration:</p> <div class="math"> <p><img alt="volume integral formula" src="formulas/eq07.png" />,</p></div> <div class="math"> <p><img alt="area integral formula" src="formulas/eq08.png" />,</div></p><p> and</p><p> <div class="math"> <img alt="perimeter integral formula" src="formulas/eq09.png" />, </div></p><p> where</p><p> <div class="math"> <img alt="egg function" src="formulas/eq10.png" />.</div></p> <p>With the exception of volume, and then only in case of a simple function <var>f</var>, you may need a computer algebra system to find the correct mathematical expressions for the integrals. This is not a serious problem because useful approximations of the perimeter of an ellipse exist and can be applied (e.g., see <a href="http://www.numericana.com/answer/ellipse.htm" class="external" target="external" title="Open in a new window"><cite>Final Answers</cite></a>). The following approximation of the perimeter of an ellipse originates from the great mathematician <a href="References.html#Ramanujan" class="external" target="external" title="Open in a new window">Ramanujan</a> (1913-1914):</p> <div class="math"> <p><img alt="Ramanujan formula" src="formulas/eq11.png" /></p></div> <p>The following combination of arithmetic and geometric mean (<a href="References.html#Bronshtein" class="external" target="external" title="Open in a new window">Bronshtein &amp; Semendyaev, 1985</a>) is also very attractive to use in an egg investigation:</p> <div class="math"> <p><img alt="Bronshtein formula" src="formulas/eq12.png" /></p></div> <p> When the above formulas are applied to the approximation of the egg curve by two semi-ellipses with parameters <var>a</var><sub><var>s</var></sub>&nbsp;=&nbsp;2.75 (small ellipse), <var>a</var><sub><var>b</var></sub>&nbsp;=&nbsp;3.43 (big ellipse) and <var>b</var>&nbsp;=&nbsp;2.30 (both ellipses), then one gets a volume of 68.5&nbsp;ml, an area of 82.2&nbsp;cm<sup>2</sup>, a minor perimeter of 14.5&nbsp;cm and a major perimeter of 17.0&nbsp;cm. The computed volume and surface area are in agreement with the measured quantities. The computed area is in agreement with the estimated value according to the following allometric power-law between weight (in gram) and area of avian eggs (in cm<sup>2</sup>) (<a href="References.html#Paganelli" class="external" target="external" title="Open in a new window">Paganelli <i>et al</i>, 1974</a>):</p> <div class="math"> <p><img alt="Paganelli formula" src="formulas/eq13.png" /></p></div> According to this formula and the measured weight of the hen s egg, its surface area would be 83.5&nbsp;cm<sup>2</sup>. Another beautiful allometric law for hen s eggs is (<a href="References.html#Besch" class="external" target="external">Besch <i>et al</i>, 1968</a>):</p> <div class="math"> <p><img alt="Besch formula" src="formulas/eq14.png" /></p></div> <p>This gives the following value of surface area: 81.7&nbsp;cm<sup>2</sup>. My computer values are between these two estimates. By the way, there also exist experimentally found relationships between the volume and area of an egg. For example (<a href="References.html#Hoyt" class="external" target="external" title="Open in a new window">Hoyt, 1976</a>):</p> <div class="math"> <p><img alt="Hoyt formula" src="formulas/eq15.png" /></p></div> <p>If you fill out the measured volume of 68&nbsp;ml, then the estimated surface area is also 81.7&nbsp;cm<sup>2</sup>. In summary, the above surface area computation and the allometric laws found in the literature are in good agreement. </p> <p>The equation for the volume <var>V</var> of an egg can also be written as</p> <div class="math"> <p><img alt="volume formula" src="formulas/eq16.png" />,</p></div> <p>where <var>&lambda;</var>&nbsp;=&nbsp;<var>a</var><sub><var>s</var></sub>&nbsp;+&nbsp;<var>a</var><sub><var>b</var></sub> and &beta;&nbsp;=&nbsp;2<var>b</var> represent the length and width of the egg, respectively. This formula for the volume can be used to estimate the thickness of the eggshell as a root of a polynomial. I outline the method after (<a href="References.html#Narushin" class="external" target="external" title="Open in a new window">Narushin, 1998)</a>:</p> <p>To my purpose, an egg consists of two components: the shell and the contents of the egg. The mass <var>m</var> of the egg is the sum of the mass <var>m</var><sub><var>s</var></sub> of the eggshell and the mass <var>m</var><sub><var>c</var></sub> of the egg s contents. Let <var>V</var>, <var>V</var><sub><var>s</var></sub> and <var>V</var><sub><var>c</var></sub> be the volumes of the complete egg, of the shell only, and of the contents only, respectively. Let <var>&rho;</var><sub><var>s</var></sub> and <var>&rho;</var><sub><var>c</var></sub> be the density of the shell and the contents of the egg, respectively. It holds:</p> <p><div class="math"> <var>m</var>&nbsp;=&nbsp;<var>&rho;</var><sub><var>s</var></sub><var>V</var><sub><var>s</var></sub>&nbsp;+&nbsp;<var>&rho;</var><sub><var>c</var></sub><var>V</var><sub><var>c</var></sub></div> </p><p>and</p><p> <div class="math"> <var>V</var>&nbsp;=&nbsp;<var>V</var><sub><var>s</var></sub>&nbsp;+&nbsp;<var>V</var><sub><var>c</var></sub>. </div> </p> <p>Rewriting leads to the following expression for the volume of the contents of the egg:</p> <div class="math"> <p><img alt="volume formula" src="formulas/eq17.png" />.</p></div> <p>Assume that the complete egg can be approximated as a surface of revolution of two semi-ellipses with canonical parameters <var>a</var><sub><var>s</var></sub>, <var>a</var><sub><var>b</var></sub> and <var>b</var>. The same assumption can be made for the contents of the egg, but in this case with the canonical parameters</p> <div class="math"><p> <var>A</var><sub><var>s</var></sub>&nbsp;=&nbsp;<var>a</var><sub><var>s</var></sub>&nbsp;-&nbsp;<var>&delta;</var>,&nbsp;&nbsp;&nbsp; <var>A</var><sub><var>b</var></sub>&nbsp;=&nbsp;<var>a</var><sub><var>b</var></sub>&nbsp;-&nbsp;<var>&delta;</var>&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;<var>B</var>&nbsp;=&nbsp;<var>b</var>&nbsp;-&nbsp;<var>&delta;</var>. </p></div> <p>So, the existence of an air cell in the egg and the variation in shell thickness, which depends on the particular spot on the egg, are neglected. The formula for the volume of the contents of the egg is:</p> <div class="math"> <p><img alt="volume formula" src="formulas/eq18.png" />.</p></div> <p>Working out leads to:</p> <div class="math"> <p><img alt="volume formula" src="formulas/eq19.png" />.</p></div> <p>Rewriting in terms of length <var>&lambda;</var> and width <var>&beta;</var> of the egg leads to:</p> <div class="math"> <p><img alt="delta formula" src="formulas/eq20.png" />.</p></div> <p>Thus the eggshell thickness <var>&delta;</var> must satisfy the following third degree polynomial equation:</p> <div class="math"> <p><img alt="delta polynomial formula" src="formulas/eq21.png" />.</p></div> <p>In the standard work (<a href="References.html#Romanoff" class="external" target="external" title="Open in a new window">Romanoff &amp; Romanoff, 1947</a>) you can find the required densities for shell and contents of avian eggs: <var>&rho;</var><sub><var>s</var></sub>&nbsp;=&nbsp;2.3&nbsp;g/cm<sup>3</sup> and <var>&rho;</var><sub><var>c</var></sub>&nbsp;=&nbsp;1.037&nbsp;g/cm<sup>3</sup>. Other data of the hen s egg under consideration are: <var>m</var>&nbsp;=&nbsp;74&nbsp;g, <var>V</var>&nbsp;=&nbsp;68&nbsp;cm<sup>3</sup>, <var>&lambda;</var>&nbsp;=&nbsp;6.21&nbsp;cm, and <var>&beta;</var>&nbsp;=&nbsp;4.62&nbsp;cm. The shell thickness <var>&delta;</var> of the hen's egg under consideration in this article must satisfy the following polynomial equation: <div class="math"><p> <var>&delta;</var><sup>3</sup>&nbsp;&minus;&nbsp;7.725<var>&delta;</var><sup>2</sup>&nbsp;+&nbsp;19.6812<var>&delta;</var>&nbsp;&minus;&nbsp;0.6585&nbsp;=&nbsp;0. </p></div> <p>The only real solution is <var>&delta;</var>&nbsp;=&nbsp;0.034&nbsp;cm. This is 2.5&nbsp;&mu;m less than the thickness according to the following allometric law (<a href="References.html#Ar" class="internal">Ar <i>et al</i>, 1974</a>):</p> <div class="math"> <p><var>&delta;</var>&nbsp;=&nbsp;5.126&middot;10<sup>-3</sup>&nbsp;&middot;&nbsp;<var>weight</var><sup>0.456</sup>. </p></div> <p>Quite a nice result of the mathematical modeling of the hen's egg.<p> <div class="footer"> <p><a href="index.html" class="internal">Home</a> | <a href="Geometric.html" class="internal">Previous</a> | <a href="Algebraic.html" class="internal">Next</a> </p> </div> </body> </html>