Why Celebrate 125 Years?

A Mathematician's View of Our Quasquicentennial Anniversary

By John J. Watkins, Professor of Mathematics

This year Colorado College is celebrating its 125th anniversary -- but why? One hundred and twenty- five years is so awkwardly placed between a centennial (100) and a sesquicentennial (150) that even the eminent lexicographer William Saffire was stumped when he was asked what it should be called, although he did manage to track down quasquicentennial, meaning one-fourth plus one-hundred. But 125 is a number that is much more deserving of our interest than that absurd Latinate term would suggest.

Try thinking about 125 this way. Imagine yourself back in 1875 watching General Palmer direct the placement of a huge one-ton block of Colorado sandstone right smack in the middle of the CC quad to commemorate the College’s first year. Then imagine yourself coming back the following year to watch an identical block being placed next to the first, and so on year after year for twenty-five years until an absolutely perfect 5x5 square is formed. Then, in 1900, you come back just to see what they are going to do with the 26th block of sandstone and watch with approval as they begin constructing a second layer on top of the first. Now, here we find ourselves in 1999 about to put the 125th block of sandstone into place and thus complete an absolutely perfect 5x5x5 cube. Wouldn’t you want to be there to watch that?

Not quite convinced? Well, here is another way to look at 125. Surely it is an accident that we work with a number system that is based upon 10, even granting Aristotle’s obvious point that we have 10 fingers. It is therefore quite natural that numbers such as 100 or 1000 or 2000 take on special significance for us. But what if at Colorado College we had instead decided to base our numerical reckonings upon 5? After all, there are other cultures who have done just that. Moreover, Aristotle’s argument works for 5 as well as it does for 10. Then, imagine the excitement, not to mention the headlines around the world, as we approached this particular anniversary. We would now be celebrating our 1000th anniversary!

That’s right, our 1000th anniversary, because, 1000(base 5) means precisely 1x5(3) + 0x5(2) + 0x5(1) + 0x5(0). In other words, 1000(base 5) is just another way of writing five-cubed, and another way of representing the number of sandstone blocks stacked up in the CC quad. Call it 1000, or call it 125, it is still the same remarkable number.

A key feature of Greek mathematics -- and hence, of our own mathematics -- is that they attached special importance to numbers that represented fundamental geometric shapes such as cubes and squares and triangles. There are many remarkable ways in which these kinds of numbers interact with each other and with other numbers. For example, think about square numbers themselves as a set of "building blocks" and that the object is to construct other numbers by adding these square "blocks" together. A famous theorem, proved in 1770 by Joseph Louis Lagrange, tells us that we never need more than four squares to construct any number whatsoever. Of course, some numbers, like 125, can be written as a sum of only two squares: 125 = 10(2) + 5(2) . In others words, we could disassemble Palmer’s sandstone cube and rearrange the blocks into two square formations, one a 10x10 square and the other 5x5. What is particularly interesting about 125 is that there is yet another way to do this, since 125 also equals 11(2) + 2(2).

If you are willing to allow subtraction as a basic process of construction with our building blocks, then 125 can also be written as the difference of two squares, since 125 = 15(2) - 10(2). In fact, the numbers 10 and 15 are themselves quite special since they are consecutive triangular numbers. Triangular numbers are numbers, such as 10 and 15, that represent how many billiard balls it takes to form perfect equilateral triangles of various sizes. The triangular numbers are: 1, 3, 6, 10, 15, 21, 28, . . . . Disappointingly, however, it is not possible to write 125 as the sum of two triangular numbers. But, it is possible to write 125 as a sum of three: 125 = 6 + 28 + 91. In truth, this is not terribly remarkable, because another famous theorem -- this one is due to Carl Friedrich Gauss -- tells us that any number can be written using at most three triangular numbers. But wait! We can write 125 as a difference of two triangular numbers, and this can be done in two ways: 125 = 153 - 28 and 125 = 378 - 253.

Another fundamental geometric shape is the hexagon. By forming billiard balls into perfect hexagons of various sizes we get what are called hex numbers. The hex numbers are: 1, 7, 19, 37, 61. . .

Now we can stack up these hex numbers -- however, since it is almost impossible to stack one billiard ball on top of another billiard ball, I prefer to imagine stacking Campbell soup cans on top of one another as I do this -- into a pyramid shape. So, for example, we could begin on the bottom with 61 soup cans formed into a perfect hexagon. Then, on top of that layer, we could place a slightly smaller layer of 37 cans. Next, 19, and then 7, and finally, 1 can on the very top of our pyramid. Guess what? We used exactly 125 cans!

Actually, one of my favorite ways of thinking about 125 has nothing to do with Greeks or with geometry, but with the fact that 125 really does represent a block. No, not a block as in a 5x5x5 cube, but a BLOCK as in that most sacred unit of measure at CC. The rest of the world may divide academic years into semesters and quarters. How boring. We bravely go where none has gone before and divide our academic year into 8 fundamental units. Could this be a mere coincidence? Or is the cosmos trying to tell us something?
one block = 1/8 of a year = .125

A serious question may have occurred to you by now. What do we do next year with the 126th block of sandstone? My best suggestion is to place it on the ground next to one corner of the 5x5x5 cube and begin the construction of a 6x6x6 cube, which we will complete with suitable fanfare in the year 2090. Or, if you really wanted to be bold, we could begin the construction of a 4-dimensional cube (analogous to what we did in 1900 with the 26th block when we broke out of 2-dimensions and moved up into the 3rd dimension). We’d finish in 2499. Another approach, although highly labor intensive, would be to rearrange the blocks each year into some particularly pleasing geometric form. One problem with this idea is that some numbers, like 126, are just plain dull. Then again, 127 is a really interesting number.

Meanwhile, William Saffire undoubtedly will be working on what to call our 175th anniversary. I do hope he can come up with something better than quasquisesquicentennial.

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