Opinion | The more things change, the more they’re invariant7 min read . Updated: 17 Oct 2019, 10:20 PM IST
The search for an invariant gives us a different perspective on a problem we’re faced with, and that new perspective often takes us closer to a solution
Remember the Hummer shuffle, which made an appearance in this column a couple of years ago? (“The fixed point of it all"). It’s simply this: On a pack of cards, turn over the top two. If they were all face-down to start with, the Hummer shuffle turns the top two cards face-up.
As I wrote then: “Now cut the deck however you choose; do the Hummer shuffle again—in fact, do both moves as many times as you like, in any order. When you stop, you will have a deck that has some unpredictable number of face-up cards. But even if that’s unpredictable, there is one feature of your card deck that has stayed invariant through all your shuffling and cutting: the number of face-up cards is even."
That small but significant truth is the basis for various intriguing card tricks. With a roomful of people a few days ago, I even tried a slight variation: I turned only the top card face-up before doing the Hummer shuffle on the pack. Now, through any number of cuts and further Hummers, the number of face-up cards stayed odd—and that was the basis for the trick I played on them. I have to confess, though, that it left them singularly unimpressed.
Still, there’s a point here: About invariance. This is a principle that applies all the way from card tricks to logical puzzles and on into realms of mathematics and physics, and is of immense value every time. Given some objects, possibly mathematical, that we’re interested in and an operation we want to subject them to, we look for some property of the objects that doesn’t change once we apply the operation. With the cards I mentioned above, the operations are the Hummer shuffle and the cut. Do them once, twice or a hundred times, and the properties I mentioned—the oddness or the evenness of the number of face-up cards—is invariant.
So, let me throw some other invariants at you, to give you a flavour of why the idea is useful.
* Hold a tennis ball in your hand and rotate it. You will agree that neither its surface area nor its volume changes: They are invariant under such rotation.
* Take any two numbers and consider the difference between them. For example, 871 and 47: The difference is 824. Add the same quantity to both numbers—say 131, to produce 1,002 and 178—and the difference between the two new numbers remains 824: That difference is invariant under such mutual addition. Trivial, you think? But the study of number theory begins with apparently trivial fundamentals like this.
* Draw a triangle on a sheet of paper. Now change the lengths of its three sides any which way, drawing a new triangle each time. Whatever triangle you come up with, the sum of its angles is always 180° (degrees): That sum is invariant regardless of what triangle you draw. This is, of course, basic trigonometry.
* Fashion a teacup out of clay or plasticine. Now stretch and reshape it—but no tearing!—so it becomes doughnut-shaped. Both teacup and doughnut have one hole all the way through the clay, and no more: That one hole is invariant under any amount of stretching or reshaping you do to the clay. The study of properties that are invariant under this kind of reshaping is at the foundation of the entire mathematical field of topology.
Intriguingly, the invariant I mentioned about the angles of a triangle does not necessarily hold on three-dimensional surfaces—like teacups or doughnuts or even planets. Consider, for example, the triangle formed on the surface of the Earth by the 0° and 90° meridians (lines of longitude) and the Equator. All three are straight lines. The two meridians meet at the North Pole, and of course each meets the Equator. All three angles are 90°, meaning the triangle’s total is 270°. Not 180°, like you’ve always believed about triangles: That invariance is only for two-dimensional triangles.
The point, for someone like me who likes dipping into mathematics, is that invariance is about seeing things in different ways.
For example, give this puzzle a thought. I have a bag with 100 marbles, each with a random number on it. You remove two marbles at random, add their numbers, write that on a new marble, put it in the bag and throw away the two you took out. Now there are 99 marbles in the bag. Repeat until there’s only one marble left in the bag. What’s the number on it? (Stop reading here if you’d like a few minutes to think about this without hints.)
You might try solving this by writing down 100 numbers, then doing the two-out-one-in operation over and over… tedious at best. Then you might try it with just 10 numbers; probably still tedious.
But instead, you could ask: What about this set of numbers stays invariant under this particular two-out-one-in operation? I suspect you’ll get an answer to that if you work out what happens if I had only two numbers. But try it, nevertheless, with 100.
What can you do with 100 random numbers? Maybe you can add them all up. Hold on to that total. When you remove two, that figure in your mind decreases by their sum and the bag is down to 98 marbles. But then you put their sum back as a single number, going up to 99 marbles. Which means the total of the set of 99 is the same as the total of all 100. Aha! Are you on to something here? Do the operation again, and you realize the total of the resulting 98 numbers is the same as the total of 99, the same as the total of the original 100.
Indeed you’re on to something: The invariant here is the sum of all the numbers in the collection. And this means that when you’re down to one marble, voilà, the number on it is the total of the original 100.
The search for an invariant gives us a different perspective on a problem we’re faced with, and perspective often takes us closer to a solution. Put it another way: Invariance gives us a deeper understanding, and that’s why it applies in so many areas of mathematics.
In physics, invariance is embedded in the conservation laws, like the conservation of momentum. Momentum is the product of an object’s mass and its velocity. The law tells us that when two objects collide, the sum of their respective momentums stays the same. This will make immediate sense to you if you think of standing on a street and being hit by a speeding car. Slamming into you will slow the car down, certainly. It will also speed you up. You may even have time to calculate that the sum of the car’s momentum and yours before the collision remains the same after, given that the car has slowed and you are suddenly flying rapidly through the air. This applies in reverse, too. Imagine going out for a run, turning a blind corner and smacking hard into a stationary SUV. As you fall to the ground in agony, your momentum sinks to zero. But console yourself by remembering that you have in effect transferred all of your momentum to the SUV, thus moving it a tiny bit.
In either case, momentum is conserved. The sum of your momentum and the car’s isn’t changed by the collision. In other words, it’s invariant.
Similarly with angular momentum, which applies to objects that rotate. It’s defined as the momentum (as above) of the moving object, multiplied by the radius of the circle it traces as it rotates. The law explains why a figure skater doing a pirouette might start by stretching out her arms, then pull them in so she can whirl faster and faster. It explains why Venus travels faster on its path around the Sun than our Earth does, even though both planets are nearly the same size. Or why a spinning top stays upright. Or why, if you tie a stone to a string and whirl it around your head, it moves faster as you shorten the string.
In each case, again, angular momentum is conserved. Invariance, again.
Understanding such invariance about objects in motion helps us understand the working of our solar system, or the flight of an aircraft, or any number of other phenomena around us. It was crucial in planning the paths of Indian Space Research Organisation’s Chandrayaan and Mangalyaan missions. (Regular readers of this column will remember I used the analogy of the whirling stone to explain their orbits). This is why invariance is so fundamental, so invariably fascinating.
Though I don’t know if saying all this to the folks who were so indifferent to my card trick the other day would have changed anything. Invariantly unsmiling, all of them.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun