Physical intuition can also be used to deduce mathematical truths. We will touch on a few such examples in this section. 1
Let us revisit Torricelli’s theorem, which was briefly alluded to (in a footnote) in Chapter 3 (as part of finding the shortest highway system for four cities on the corners of the square): Suppose we have three points on a table and want to connect them in such a way as to minimize the sum of the distances between them. We argued mathematically that the path must form a “trivalent” graph–a collection of three lines meeting at a vertex–with three angles of 120^{\circ} . Here is a physics-motivated way of seeing this 2 : Imagine cutting holes in the table at those same three points and then attaching three identical balls to the tips of three strands of string (of fixed length), connected at a common vertex above (Fig. 57 ). Assuming the masses of the balls are equal to m , the potential energy of the total system is -mg times the sum of the lengths of the strings hanging beneath the table. This is minimized when the lengths of the strings hanging beneath the table are maximum. At such a point, the length of the connecting strings on the table is minimized, because the total length of the strings is fixed. By applying the equilibrium condition, we see that the tensions at the vertex must balance, and since the tensions are all equal (being pulled by equal weights), the angles must be equal too and, hence, must be 120^{\circ} . This shows that bringing physics to bear on what was originally a mathematics problem can give us insights and lead to simple solutions that appeared to be difficult in the original mathematical formulation.
Line of Best Fit
Imagine a graph of data points (x_i, y_i) . Suppose we want to find the line of best fit (for the purposes of linear regression), meaning that we want to minimize the sum of the squares of the vertical distance from the data points to the line. We can treat this as a physical situation by imagining that the points are nails on a table with the horizontal direction denoted as the x-axis and the vertical direction as the y-axis, as in Fig. 58 . The nails, in turn, are attached to springs that are connected to rings on a straight rod. The springs are confined to vertical tubes that allow them to move only in a vertical direction. The distance squared between the nails to the rod thus represents the potential energy of the springs, so finding the best linear fit is exactly the same as finding the configuration of the rod that minimizes the springs’ potential energy.
