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Degrees of Freedom

Degrees of Freedom


The boundless dimensions of math and physics
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I Am Hyperspace, and So Can You

The views expressed are those of the author and are not necessarily those of Scientific American.


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sqrt(i) heart mathMy first encounter with higher dimensions was somewhat of an anticlimax. It was one of my very first days in university, back in Rome, and I was sitting in an amphitheaterlike classroom, in one of the classes required of all first-year math students.

I don’t remember the exact words the professor said, but it must have been something like this.* “Let x1, x2, … xn be n real numbers. The ordered set (x1, …, xn) is an n-dimensional vector. The set of all n-dimensional vectors is called the n-dimensional vector space Rn.”

Just like that? No spacetime warps to cross, no Captain James T. Kirk lost at sea, no Doctor Who to be our guide? Just this rather unassuming man dishing out formal definitions the way a tax preparer would list the documents to attach to your 1040. (Itemized deductions … W2 … and oh, I almost forgot: m-dimensional subspace. Include, but do not staple.)

The professor had introduced us to one of the most mind-blowing ideas humankind has ever imagined, and he had done it with no advance notice, no emphasis in his voice, not even so much as a change of inflection.

In the following years at Rome—La Sapienza, and later in graduate school, I was to spend more time than I could have possibly imagined thinking hard about higher dimensions, the strange objects that inhabit them and the incredible minds who study them. Both my senior thesis and my PhD thesis would be entirely set in the context of n-dimensional spaces. Later, when I left academia and became a journalist and then an editor, I learned just how hard it is to explain those concepts to the “uninitiated.”

In fact, I learned that it was often harder for me to convey ideas with which I was intimately familiar than ones I was just learning about—the phenomenon of being, as reporters say, “too close to the story.” It was one of the reasons why, for several years after my conversion from mathematician to writer, I wrote so few articles about math and concentrated instead on covering physics, cosmology, archeology and other topics.

Although many flavors of higher dimensions accompanied me in different ways for all these years, the basic idea was all in that two-line definition given by my first-year professor. It is the idea of degrees of freedom.

Every additional dimension adds a degree of freedom, which lets you move to places or in ways that were previously inaccessible. (This, by the way, is essentially what it means for the three directions of space to be “linearly independent”.)

The reason why people cannot “see” more dimensions than the usual three is that, well, people cannot see them. Our brains are wired to help us to navigate a three-dimensional world, and there is just no space in there to visualize more dimensions.

Here I shall correct myself: our brains aren’t trained to see more dimensions. In a post I plan to write soon, I will discuss how it seems to me that humans should be able construct mental maps of spaces with more than three dimensions. Short of doing that, mathematicians use various tricks to, in a sense, visualize the extra dimensions without visualizing them, as I will describe in another future post.

But as mathematicians and physicists discovered already in the 1800s, higher dimensions are of fundamental importance both in theory and in practice. They are perfectly legitimate places where to study geometry and all sorts of other mathematical constructs. And they are an indispensable tool for attacking problems that concern our 3-D world.

The motions of the planets in the solar system, for example, are best understood by following the trajectory of a single point in a 60-dimensional space (six degrees of freedom—three for position and the rest for momentum— for each of the nine planets plus the sun). And the modern theory of elementary particles and forces is all based on additional, “internal” degrees of freedom.

Blogs are becoming an important tool for mathematical research itself, and most math-related blogs are oriented to specialists. A few math blogs do cater to a more general public, some in rather innovative ways. Vi Hart, in particular, developed what seems to me an entirely new medium–less a blog than an immersive experience into the author’s mind as she scribbles things on paper. (A scribblog?)

It is difficult for a popular-math blog to avoid collecting evergreen mathematical curiosities—golden ratios, fractals, origamis, sudokus,  Escher panels and other images that are supposed to demonstrate how “math and art are ultimately the same thing.” (As I may discuss in a future post, I happen to think they are not.) To a certain extent, this blog will be no exception.

Also like other blogs, Degrees of Freedom will include, besides a “math angle” on current events, the occasional rant, rave, cool image, breaking news, interview, book review, guest post, list of favorite links and so on. But my pledge to the reader is that the blog will live up to its name and move in new directions—exploring areas at the intersection of math and physics in ways that readers haven’t seen before.

For example, I plan to describe how those tricks to “see” higher dimensions can help you visualize the expansion of the universe; or understand what we look at when we look at the afterglow of the big bang, aka the cosmic microwave background; or the bizarre “Calabi–Yau spaces” of string theory.

One final note, which will not mean much to younger readers but might clarify my intents to subscribers of the old-time Scientific American: Degrees of Freedom is not and could never be Martin Gardner‘s blog. By that, I don’t just mean the obvious fact that I am not the man who for a quarter century delighted the magazine’s readers with the unsurpassed clarity, originality and authoritativeness of his Mathematical Games column. I also mean that although this is in a certain sense a Scientific American math column, I don’t intend for it to imitate, emulate or re-enact the column that inspired some of us to become mathematicians (or physicists, computer scientists, engineers—you name it) in the first place.

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Graeco-Latin SquareAbout Degrees of Freedom’s logo: Martin Gardner’s Mathematical Games column was featured on the cover of the November 1959 Scientific American. (It was not unusual for the math column to make the cover. Those were the days.) The subject was Graeco-Latin squares, a combinatorial game displayed here in terms of colored squares. Each of ten different colors appears exactly once on each row as an outer square and once on each row as an inner square. Similarly, each color appears exactly once on each column as both an inner and an outer square. The oil-0n-canvas depiction of the square was painted by the magazine’s staff artist at the time, Emi Kasai.

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*The definition of vector space won’t be on the test. Actually, nothing will be on the test because there won’t be any test. But if you are curious, here’s what the professor meant by the “n-vector” (x1, … , xn):  it is just a convenient notation mathematicians use when they want to make the point that you have n numbers, and that number n could be any one of the infinitely many natural numbers (1, 2, 3, 4, 5 and so on), so that you better omit the x’s in between and replace them with dot, because you haven’t made up your mind as to how many exactly there are. More importantly, any statements you write that have n in them usually ought to apply to that infinity of possible cases.

About the Author: Davide Castelvecchi is a freelance science writer based in Rome and a contributing editor for Scientific American magazine. Follow on Twitter @dcastelvecchi.

The views expressed are those of the author and are not necessarily those of Scientific American.



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  1. 1. Wilhelmus de Wilde 1:48 pm 07/9/2011

    Our consiousness can deal easily with infinities which lead to controversal incalculable elements in our causal deterministic limited universe.

    Link to this

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