2 + 2 = 5

I have no mathy pictures, so here is a lobster instead

A segment on the news recently disturbed me. It was about a new way to teach math in elementary schools (I’ve become interested in these stories as my daughter is now in the world). The new way is a creative approach where there are no wrong methods, and they may also have reported that there are no wrong answers - I’m really hoping I heard the last part wrong. There may be more to the story than the news presented (as is often the case), however, if it is this way, I’m quite worried about our mathematical future.

I’m all for creative approaches, however, in math there are methods that will take you to the answer quickly. One of the nicest things about basic math is that there are right answers to be found and, they can be verified as correct. Creative methods may get the right answer eventually but are not necessarily effective tools for everyday math. It may be dull to memorize mathematical basics - never the less, one needs to get the basics down. No one ever suggests that we take a creative approach to learning how to read. We are expected to learn the grammar rules necessary to understand written text - so why is math different?

Technology takes away the necessity to do math in some circumstances, but technology doesn’t always work. Understanding some math is necessary - how would you determine if you can afford something? How about telling if you tax return is reasonable? Did you get the right amount of change back on a purchase? Or when is a sale at a store actually a good deal? Recently, I found a man looking at peanut butter in the grocery store trying to determine if it was a better deal to get a smaller jar on sale or a larger one - in this case the smaller jar was the better deal (he seemed relieved when I told him).

A lot of people have trouble with math, but I wonder if this is due to our society’s portrayal of math as scary. Math isn’t scary, it’s simply a set of rules to manipulate numbers. Since, people’s brains work slightly differently making picking up math harder for some, if someone is struggling to learn a mathy technique help should be available to coach them towards the right answer.

I’m biased about math because I use it all the time and am comfortable with it. Although, I’m not very good at doing math in my head but, I’ve practiced tricks allowing me to do everyday computations. With a pen and paper I can work out most things - calculators make it even easier. At higher level university courses, math becomes more abstract and harder to intuitively grasp. One needs to use this type of math regularly, or be mathematically gifted (which isn’t me), to apply it. As a scientist, I understand the math that describes my field but the pure abstract math is often baffling to me as I haven’t spent time working with it.

Picture is from here

Origami Shrimp

In my aquarium I have a number of Amano Shrimp who keep the place clean. Amano Shrimp originate from South Eastern Asia and have clear bodies about a knuckle long with wine-red spots. They have an interesting life cycle in that they are a fresh water shrimp whose larvae require salt water to live. In the wild they must migrate up and down rivers throughout their lives. When I give my fish flake food, these shrimp always dart forward and snatch the largest flakes. They then fold up the flakes into what looks like origami shapes before munching on them. I assume they fold their food this way to make a large flake less cumbersome to move with – or perhaps they just like origami.

Origami is a Japanese art of folding paper. According to wikipedia: The goal of this art is to transform a flat sheet of material into a finished sculpture through folding and sculpting techniques, and as such the use of cuts or glue are not considered to be origami. I have a number of origami how-to books from which I could make creatures from sea stars to giraffes – I don't do a lot of folding, I just have some books.

Origami is an applied geometry that has practical applications beyond making pretty cranes. Origami folds can be planned mathematically as there are a limited number of ways a piece of paper can be folded. Computational origami extends the math to optimize folds for practical like folding an airbag for car or finding an efficient way to fold solar panels to make the journey to space.

Interesting links here and here.

Flow over bumps

As Fluid flows over a bump surface disturbances, such as waves, develop potentially extending both up and downstream. The form these disturbances take depends on fluid velocity and fluid depth which can be combined together in the Froude (F) number. The Froude number equals the fluid velocity over the square root of gravity times fluid depth. Because it has no dimensions, the Froude number allows flow in dramatically different circumstances to be compared, for example, atmospheric flow over a mountain range could be compared to tap water flow from your kitchen sink.

Naval architecture provided the original application for the Froude number. Here the hull length of a ship is used instead of fluid depth. It's important because, the Froude number relates to the ship's drag or resistance to moving through the water. This number is named after William Froude (1810-1879) who experimented on ship's hulls in his large fluid tank. William developed towing tank techniques in his efforts to model frictional drag on ships. However, he didn't discover the dynamical relationship between the fluid velocity and hull length. It was Ferdinand Reech (1805-1880) who first described this relationship and used it for testing ships and propellers around 1852. It's likely that this relationship's roots reside with even earlier French mathematicians.

So what does the Froude number tell us? When F is smaller than one, flow over the bump is 'subcritical'. Waves on the surface can travel upstream, meaning that downstream conditions affect the flow upstream. For example, when a pebble is tossed into the water of a flowing stream, the resulting ripples propagate both upstream and downstream. When F is larger than 1, flow is 'supercritical'. In this case, no surface disturbance can travel upstream. The ripples created by a pebble tossed in downstream cannot overcome the speed of the water. The flow upstream is not changed. When F is equal to one the flow is 'critical'. This is the point of transition from subcritical to supercritical effects.

Now, back to flow over a bump. As subcritical water is pushed over the bump, squeezing takes place because the water is now shallower and the same amount of water is flowing through. This forces the water to speed up over the bump and transition to supercritical. This faster water crosses over to the other side of the bump, where it's again deeper and slower moving. When the fast flowing water reaches the slower water it abruptly slows and waves form. Since the water is moving too quickly to allow waves to propagate upstream, (because it is supercritical) these waves build up, forming a sudden water level increase that can be standing still in the flowing water. This is called a hydraulic jump, a non-linear effect and can be observed in a kitchen sink or in water passing over a weir. Mathematically, a hydraulic jump is a discontinuity, however in the real world viscosity makes it a region of rapid change instead.

The greater the Froude number is, the more pronounced the jump will be. For initial flow speeds slightly above the critical speed, the transition appears as an undulating wave. As flow speed increases, the Froude number also increases and the transition becomes stronger eventually developing a more abrupt shape. When the speed is high enough, the transition front will break and curl back upon itself. At this point, the jump may contain violent turbulence, eddying, air entrainment, and surface waves. Turbulence removes the extra energy, allowing the flow to transition from supercritical back to subcritical.

The day math nearly killed me – a cautionary tale about checking your calculations

I was once a soldier, what seems like an eon ago now. As a newly trained junior officer I was given my first command, a troop (platoon) of about 25 soldiers. A week after I took command, I was deployed along with my troop for an exercise on the demolition range. I had run demolition ranges in training, but always someone was there to watch me and catch my mistakes. This day I was in charge, I knew what to do and I was naive.

We were cutting metal with C4 explosives – it was a bit more complex than that, but cutting metal was essentially what we were doing. It was early spring, not yet warm enough to want to spend the day dilly-dallying in the sun, so I wanted to get what I had to be done completed without unnecessary delay. As soon as we arrived, I briefed my soldiers on how the day would go and assigned tasks. I put Sergeant 1 in charge of cutting the time fuse, while Sergeant 2 and myself supervised the soldiers laying out the C4 to cut the metal.

Time fuse is a tricky thing, because a roll of time fuse can't be counted on to burn at the same rate as any other role. So, for every roll someone needs to time how long it takes to burn over a known distance, let's say 1 meter. Then that someone needs to figure out how long it takes to walk from where the charge was to be detonated to the safe area. On this day we had a lovely concrete bunker to hide in for safety, about 200 meters from our detonation spot.

It's important to get this right because shrapnel from cutting metal with C4 can travel up to a kilometer.

Sergeant 1 carefully measured the time it took the burn 1 meter of time fuse. He then took out a stopwatch and walked at a brisk but not hurried pace from where we were working on the charges to the bunker, adding about 30 seconds to his result as an extra safety measure. We wanted the explosives to go off shortly after we got into the bunker. In addition to wanting a nice count down for dramatic effect, it is important that we know precisely when the detonation was to occur because a detonation that doesn't occur when it is supposed to is the kind of thing that can wreck a day.

The next step is to figure out how much time fuse is needed to set off the charge at the appropriate time. So this is what we have:

time taken to burn one meter of time fuse = time taken to walk to bunker / length of time fuse

Which can be rearranged to give:

length of time fuse = time taken to walk to bunker / time taken to burn one meter of time fuse

Sergeant 1 figured out how much time fuse was needed, cut and delivered it to where we were setting up the charges. I should have checked his math – math with times can be tricky.

Sergeant 2 and I waited until everyone was safely in the bunker before we lit the time fuse. We walked at a brisk but not hurried pace towards the safety of the bunker. The bunker door was situated so it was facing away from the explosions so we would have to walk around the building to get inside. When we were about 5 m from the bunker door at the edge of the bunker, the explosives detonated.

We ran into the bunker and slammed the door shut. We could have been riddled with little slivers of flying metal, but neither of us were. And those injuries which didn't happen, they would have been fully my responsibility. Sergeant 1 was very apologetic, I think he expected me to punish him profusely – but I didn't. My squadron (company) commander and the squadron sergeant major had shown up just before we detonated, so I had a little chat with them. Since I accepted responsibility and would never ever make that mistake again, I was allowed to continue. From that day on, I always checked the time fuse calculations. If I did them myself, I had someone check my work. I never had a mistimed explosion again – instead I had to deal with a whole whack of new problems.