It is unquestionably evident that the peanut butter toast must be bisected twice along the diagonal.
This produces four (4) isosceles right triangles, in the case of square toast. (Applying this method to rectangular toast produces two obtuse and two acute triangles, which does not affect the following analysis.) Experiments indicate that the triangles are, predominantly, eaten beginning with the vertex opposite the two equal angles. Current theory holds that the driving force for this behavior is the phenomenon of crust rejection, which is not universal among toddlers but is nevertheless common enough to warrant its consideration as a general case. Then crust acceptors can be regarded as a special case of the generic toddler, which will be treated at the end of this report.
By the time the crust (which I define self-consistently as that part of the peanut butter toast that is inedible to a crust rejector) is reached, and laid down on the plate, its configuration is approximately linear.
Since the toddler's face is a convex curve (which can be approximated as a paraboloid for the purposes of contact mechanics, viz. the theory of Hertz), it is tangent to the idealized crust at a single point. Compression of the curve against the crust (as if to obtain the last bit of jelly) can expand this point into an ellipse, of course, but any such ellipse is finite and limited, as the force of compression increases with the 3/2 power of the contact area. (The value of 3/2 assumes elastic contact, which is, of course, preposterous, but can be excused on the basis of the fact that my textbook on non-Hertzian contact mechanics has gone missing.)
The competing technique that has the most support is to bisect the toast twice, parallel to the sides. But this practice, though well-grounded in theory (chiefly because it produces four congruent quadrilaterals that may be conveniently stacked, regardless of the aspect ratio of the original toast, and thus appeals to the toddler who likes making little sandwiches), fails miserably in the laboratory setting.
For the toast is highly likely to be consumed along the path of least resistance, which clearly begins at the only crust-free vertex of each quadrilateral and proceeds across the toast until the crust is reached. But the crust of such a quadrilateral extends along two adjacent sides of the quadrilateral, subtending a right angle. Whereas the linear crust produced using the double-diagonal-bisection method is tangent to the toddler face in the absence of compression force, the crust produced in the double-parallel-bisection method conforms to the toddler face even in the case of very small compression force, and even in the most idealized situation is guaranteed to make contact in at least two separate locations. At each of these two locations the situation is comparable to that of the single contact in the double-diagonal-bisection method.
Therefore, the double-parallel-bisection method can be expected to result in toast-face contact over approximately twice the surface area that would result from the double-diagonal-bisection method. Accordingly, twice the amount of peanut butter will be transferred from the toast to the face.
The author of this report, therefore, recommends that the double-diagonal-bisection method be employed for all crust-rejecting children. Crust acceptors are exempt from this recommendation.
I love it! I had noticed a similar phenomenon regarding amount-of-sandwich eaten. Bisecting only once parallel to either side results in only about half of the sandwich being eaten, with the other half designated as "crust-like". As I thoughtfully cut the sandwich into four pieces, it occured to me that triangles would be best.... But I never thought to put it in such a cogent argument--I'm still working up to a thesis :)
Posted by: mandamum | 16 August 2005 at 07:51 PM