From magic triangles to vector spaces to magic tetrahedrons

Extract

It has been my practice for many years to introduce the concept of a vector space to students by looking at families of magic squares and triangles. Consider, for example, all possible magic triangles like that shown in figure 1(a), where each circle is to be filled by a real number so that the totals along each edge of die triangle are constant. Then scalar multiples of such triangles, together with sums of them, will also be magic. Once four numbers are put into the circles, such as a, b, c, and d in figure 1(b) then the other two entries are uniquely determined. Because the other four numbers can always be freely chosen, there are four degrees of freedom.