# Tag Archives: math

## Flatland and Extra Dimensions

What would life be like if you lived in two dimensions instead of three?

Back when I posted about popular science books for non-scientists, one of the suggestions I got after the fact was Flatland: A Romance of Many Dimensions, the 19th century classic by Edwin A. Abbott. Which is absolutely worth reading, and a great example of what I love in science writing (or science fiction): an idea that makes you change your whole perspective on the world and reimagine it from a different point of view.

The idea behind Flatland is this: what would it be like if the world we inhabited were flat instead of 3D? You can imagine it as living within a piece of paper, or on the surface of a table. The notion of up and down would be meaningless; we’d only have left and right, and front and back. So we’d be moving in two dimensions rather than three, and we’d also perceive everything around us to have only two dimensions. There wouldn’t be any going over a fence, or peeking under a door. If a thing blocked your way, it would block it completely, and everything behind it would be completely invisible. Of course, you wouldn’t be able to pass through things in Flatland, the same way you can’t in the real world. So if a person stopped directly in front of you, you’d have to pass to either side, or not at all.

There’s a lot of social commentary in Flatland as well, satire aimed at Victorian England that comments on gender divisions, class hierarchies, and dogmatism against new ideas. It’s worth a full read for that, though its examination of spatial dimensions is what’s kept it famous.

Life in Flatland may seem like an academic abstraction. But actually, while our world is three-dimensional, there are some things in it which effectively have only two dimensions, especially in the world of nanoscience. The touted wonder material graphene is effectively two-dimensional, because in the third dimension it’s only one atom thick. That means that electrons moving through graphene are effectively in a two-dimensional environment, a Flatland, and can’t use the third dimension to go around each other. More two-dimensional materials are being discovered every day, and taking one dimension of a material to the nanoscale while leaving the others large changes the physical laws in that material significantly!

And what if there were more dimensions to the world? What if instead of three dimensions to space, there were a fourth, or a fifth? In that case, life here in three dimensions would seem like Flatland, without the fourth dimension to move through. Some physicists studying string theory think there may in fact be additional spatial dimensions, but that they must be curled up within the three we know in order to be undetectable.

So the idea of Flatland, a world where there are only two dimensions instead of the usual three, isn’t just a science fiction classic, it’s also a valuable thought experiment that ties into both nanoscience and string theory!

For the most part I don’t write that much about science communication here, because my posts on this blog are one demonstration of what I feel science communication can be! But I spent the end of last year thinking a lot about outreach, and seeing how my outreach philosophy is different from other communicators who are doing great work, and I wanted to explain that a little more.

I’ve always found science fascinating as a lens for understanding the world and appreciating its beauty. But I think that in science and engineering, and especially my field of physics, there’s an inherent tension. On the one hand, you have the beauty and awe that science help illuminate, and the excitement of increasing your own realm of knowledge, or even pushing the boundaries of the knowledge of mankind. That is all exciting and lofty and many people who aren’t into science still see the appeal, because curiosity about the world around us is something every child starts out with. But on the other hand, there’s often an elitism in science, a sense of scientists as gatekeepers of truth high up in a hierarchy, which is encouraged by the media at times and even some scientists.

When I tell people I’m a scientist, or a physicist, a lot of times they tell me a story about the one bad physics teacher they had, who ruined all of science for them. This apparently happens a lot, and I do get that teachers can make or break a subject at times. (My first physics teacher was not stellar.) But it’s not like bad English teachers ruin reading and writing for anyone. “If it weren’t for that middle school teacher harping on verb tenses all the time, I would probably be a Proust scholar by now, but as it is I don’t even remember how to read.” But I think culturally, communication and language and the arts derived from those things are considered fundamental, in a way that science and math used to be but no longer are. It should be as much a mark of education to know some basic science as it is to have read some of the classic novels or to know the Beethoven symphonies! I’m never going to be one of those people who makes the argument that science literacy is more important than other forms of cultural literacy, but why isn’t it at least equivalent? I think that’s a direct result of our having tried to set science apart as a better, higher thing. When you put something up on a pedestal, it gains status but loses accessibility. Science is now considered less relevant for everyone to know, even though it’s just as foundational as it ever was.

But I don’t fundamentally believe that scientific ideas are out of reach for a layperson. There’s no insurmountable math barrier or smartness barrier, science is a topic like many other topics. And I mean that a layperson can understand basically any scientific idea, not just the vague and descriptive ones. Math is a great language for explaining science, if you know how to speak it. But actual language also does the trick! You just have to be willing to think about the best way to use it.

Only being willing to explain physics using math is a failure of imagination. And sure, maybe an explanation that doesn’t use math is going to be missing some things, but so is a math explanation that gives no qualitative interpretations. If you have no science background, and I’m telling you about electrons, you may not come to understand electrons in exactly the way that I do. But that’s as much because we are different people with different experiences and conceptual ways of thinking as it is because I have spent time studying physics.

There is a saying that you can’t teach someone physics, you can only help them to learn it for themselves. And while I agree that it’s the student who has to mentally grapple with and eventually accept the tricky topics in science (and life), that doesn’t mean there’s no point trying to teach! Each person comes to understand concepts, whether it’s particle-wave duality or mind-body duality, on their own terms. If someone is asking me to help them find those terms for a concept I know a little about, I can’t make the leaps for them, but I can try different approaches to facilitate that understanding. And I love doing that; it usually expands and reforms my own understanding as well.

## Boolean Logic

We talked last time about the conveniences in implementation that can be had by using digital electronic signals for calculations rather than analog signals. But I want to get a little bit more into what the digital system means for how calculations are performed, which means talking about what Boolean algebra and logic are.

But first let’s talk about algebra. Fundamentally, algebra is taking a set of elements and applying operations to those elements. A straightforward example would be the arithmetic operations of addition, subtraction, multiplication, and division on all integers. If you take algebra in school, you study many more complex functions that can operate on various numbers, and you get into the symbolic representation of numbers (i.e. solving for x, where x is a number whose value you want to know).  So algebra is essentially an abstract framework for mathematics, one that allows you to see and describe rules and patterns more easily.

But it’s also possible to define specific sets of numbers or elements to work with, or to define a limited number of operations, and construct specialized algebras. That’s the basis of abstract algebra, a really cool math topic that I strongly suggest learning more about! And Boolean algebra is essentially a form of abstract algebra that developed in parallel to abstract algebra as a formal field. It was developed by George Boole, a mathematician and logician who wanted to establish an algebraic way of performing logical operations. Boole examined the operations that were possible for a set of only two values, the so-called ‘truth values’ of 0 and 1 (or false and true). His 1854 book examined truth tables for these operators, and we’ll look at a couple to get a feel for them.

 x y xANDy 0 0 0 0 1 0 1 0 0 1 1 1

Above is the truth table for an operation called AND. Given inputs x and y, which can have the values 0 or 1, the output xANDy is 1 only if both x and y are 1. So the left two columns list the possible input values for x and y, and the rightmost column shows the output xANDy for each of those combinations. For example, the last line is a way of displaying the equation (1)AND(1) = 1.

 x y xORy 0 0 0 0 1 1 1 0 1 1 1 1

This truth table is for OR, an operation that returns 1 if either or both inputs have the value 1.

The real technological relevance of Boolean algebra came when Claude Shannon realized in 1937 that it could be used to analyze and predict the operation of electrical switches. This can be used to develop a combinatorial logic to analyze very complex circuits whose output depends solely on their inputs, which is the main theoretical tool behind digital electronic circuits today!

## The Language of Math

Thus far I have avoided bringing math into my posts about the physical phenomena that underpin modern electronics. Math is one language we can use to discuss physics, and it’s certainly a language that lends elegance and precision. I have an undergraduate degree in math, and partly chose physics because it seemed to me to be the most math-heavy of the sciences, so certainly I see the appeal! However, I think that verbal descriptions are very valuable as well, both for people with a strong math background and for people who’d rather avoid it. The best undergraduate textbooks I used had a combination of verbal and math explanations (for example, the excellent Electromagnetism and Quantum Mechanics textbooks by David Griffiths).

So in general, I think that if you want a really thorough understanding of science, you have to dig into some math. You also have to dig into some physics, and you have to dig into some chemistry, and you have to dig into some biology. You even have to dig into quantum mechanics, which is not as bad as many people think. It’s all interconnected, and the many different languages of science play off each other in interesting and unexpected ways. But if you can only explain something in math, and not using words, then your understanding is incomplete in my view. And for those with an interest in science who don’t need to dig in to all the details, the verbal explanation is usually the best place to start!

On top of which, there are so many ways to explain a concept verbally (or using video or other media, as Erin recently posted). The simplicity of math-based explanations is wonderful, but can feel like a dead end if the math doesn’t click mentally. But if you are open to a diverse range of explanations, it’s possible to try many different approaches until one finally puts all the others into perspective. I have found this to be the best way to get my own head around really bizarre scientific ideas, to approach them from many sides until they finally make sense.

That said, sometimes the mathematical interpretation of an idea leads somewhere new, and thinking about math becomes necessary to understanding the new idea! I plan to keep approaching these topics from the verbal side, and we’ll see how it goes with the subject of the next post!