For our second installment in the series, I want to look for an embodiment of the idea of multiplication that’s quite different, at least on the surface, from the idea in the previous post of multiplication as 3 rolls of 14 Mentos candies each.
So let’s see what we got. On top we have a line with markings 0, 1, 2, …, 18. Lower, we have a line with similar markings. The “0” on the top line connects to “0” on the lower line, and “3” on the top line corresponds to “1” on the lower line. A number of red lines are shown that all originate in a common point. If I draw a red line from the common point through “2” on the bottom scale, that line “hits” the top line at exactly the marker 6. A line through “6” on the lower line likewise hits the top line at exactly 18. You can play with this yourself, all you need is a quadrille pad and a straight edge. We’ve got ourselves a little multiplication machine here, capable of multiplying by 3.
Assuming you were to agree that this machine does multiplication, it doesn’t follow that we have any insight in how this picture shows multiplicative structure – or what makes the structure multiplicative. Well, maybe the following pair of pictures help:
They are really the same picture as above, different only in that two shapes are emphasized. In the top picture, a small triangle is emphasized in orange, in the lower picture a larger triangle is emphasized. Both triangles have the same shape, and one looks like a scale drawing of the other. We could talk in terms of the large triangle having been scaled up from the small triangle, and it won’t surprise us that all the distances in the large triangle are scaled up in the same way, by the same amount. And how much do you think the amount of scaling up is? It is 3, of course, the big triangle having sides that are triple those of the small triangle.
In considering a picture that we have scaled up, we have found an embodiment for the idea of multiplication that doesn’t at first glance looks much like rolls of Mentos – a number of groups with an equal number of items in each group. On a closer look, it isn’t terribly hard to connect the two notions together.