Mathematical Brooding over an Egg, André Heck
For students, it's clearer what is expected from them when the research question "What is the volume and surface area of a given hen's egg?" is reformulated and split into the following tasks:
This reformulation takes nothing away from the open character of the task, but it forces students to develop and carry out at least one mathematical method and it invites them to compare various methods. Assuming that students will indeed develop a variety of methods, an evaluation of their techniques and a classroom discussion about the descriptive qualities of the mathematical models is appropriate.
In this article, I present the following mathematical model: the egg as a surface of revolution of some function, with the rotation axis in the longitudinal direction. The exact mathematical function that describes the egg curve can be freely chosen, but a preference is given for functions that keep the computations as simple as possible. Circles, ellipses and parabolas are for this reason good candidates. In the process of finding a suitable mathematical function I use a picture of the egg that was created with a digital camera, and I embed it in a worksheet made with the dynamic mathematics software package GeoGebra. In GeoGebra you can place an image as background on the drawing pad: select the
Insert image button in the toolbar, left-click on the drawing pad, and set the image to be in the background so that it is behind the coordinate axes. If you connect the image with the absolute position on the screen, it will remain the same whatever you do with the scaling of the coordinate system. Hereafter an egg curve can be determined by geometric constructions, via an algebraic approach, via regression, or a combination of these techniques. I will discuss all these mathematical approaches to the problem, but not before I point out a possible false start of the investigative work.
Figure 3 is a dynamic Geogebra figure in which the background image is a digital photograph made with a web-cam of an egg that was positioned on top of a piece of graph paper. The coordinate system has been scaled so that it matches the scaling of the graph paper and has its origin placed such that the horizontal axis matches the longitudinal axis of the egg and the vertical axis intersects the egg at its widest points. This positioning and scaling of the coordinate system is done with the tool button . To translate the drawing pad, drag with the left mouse a non-specific point of the drawing pad. Scaling of the horizontal and/or vertical axis can be achieved be dragging a point on the axis. The vertical scaling can be linked with the horizontal scaling by right-clicking on a point of an axis, followed by selecting in the roll menu the wanted
xAxis:yAxis proportionality. On top of the background image two ellipses have been constructed. Each ellipse describes well a particular part of the egg curve and the two ellipses are smoothly connected at the widest part of the egg. Each ellipse is a conic section defined by five points. The formulas for the construction points and the ellipses are displayed in the algebra window at the bottom of the figure.
You can replay or step through the geometric construction of Figure 3. If you want to give it a try yourself, rewind the construction to its first step and choose the button to construct an ellipse defined by five points (clicking five points of which no four points lie on a straight line will produce a conic). You can always reset the construction to its original state by pressing the reset icon in the upper right corner of the drawing pad.
So far so good, but when you look closer you will notice that the photograph was taken from distance that was too close. By counting the squares on the graph paper or by looking at the coordinate axes you quickly find out that perspective distortion (the egg seems to be 5.6 cm wide and 7.5 cm long) and lens distortion (the graph paper does not look like a nice rectangular grid) cause problems. The best thing to do is to shoot a picture of the egg from a sufficiently large distance with a digital camera with good zooming facilities and with no or negligible lens distortion. Also cut a hole in the graph paper so that the egg fits into it and the paper can be positioned halfway up the egg, as shown in Figure 4. An alternative for a digital camera or web-cam could be a good quality scanner.