How do higher dimensions work?
A “dimension" is just a geometric way of thinking about a variable: each variable in a problem corresponds to an axis or direction. If you’re studying a phenomenon that involves three variables, then that phenomenon can be understood geometrically in a 3D space. If your situation involves 10 variables, then you can model it in 10D space. Even if you can’t visualize 10D space, the mathematics works just fine.
In most respects, there’s not much difference between the geometry of 3 dimensions and the geometry of 29 dimensions. Indeed, geometers typically work in N dimensions, where N could be any positive integer whatsoever. If you’ve taken a course in linear algebra, then you’ve already seen this in action: linear algebra works just as well for 3 variables as it does for 29 variables.
For this reason, when a particular dimension behaves differently from all the others, it comes as a bit of a shock. In the early 1980’s, for example, mathematicians were stunned by the discovery that 4 dimensional spaces can exhibit bizarre topological features not found in any other dimension.
But 4D isn’t the only special dimension. In the 80’s and 90’s, geometers discovered exotic 7D spaces that curve, bend, and twist in “exceptional” ways --- types of curving that have no counterpart in any other dimension.
That’s really weird. Why are 7D spaces able curve in such strange ways?
One reason has to do with the cross product. If you’ve taken a math or physics class that involved vectors, you may have learned about the cross product: an operation that takes two vectors in 3D space, say v and w, and outputs a vector called v x w that is perpendicular to the first two. It turns out that cross products can only exist in 3D space and 7D space, a fact that has profound implications for the geometry of 3D and 7D spaces.
A second reason has to do with special number systems. Recall that the real number system is 1-dimensional, comprising all numbers on the “number line.” Similarly, the complex number system is 2-dimensional, often visualized as the complex plane. But the story doesn’t stop there. Just as we “doubled the dimension" of the real number line to build the complex plane, we can similarly "double" the complex plane to build a 4D number system called the quaternion numbers — and then double once again to get an 8D number system called the octonion numbers. One can keep going — building a 16D number system, etc. — but every time the dimension doubles, algebraic properties get lost.
What does this have to do with 7D? It turns out that the 4D quaternion number system is the true source of the 3D cross product — and the 8D octonion number system is the origin of the 7D cross product.
How do you study 7D spaces?
If you have a 7D space that you want to understand, you can learn a lot about it by studying special shapes inside of it. This sort of idea crops up everywhere in geometry and topology. For instance, one can study the euclidean plane by considering the straight lines inside of it — or the round sphere by means of the great circles on its surface. In the same way, a topologist might study a space by considering the loops inside of it.
Inside a 7D space, the “special shapes” turn out to be 3D and 4D minimal surfaces — shapes whose area is as efficient (“minimal”) as possible.
What’s a minimal surface?
A minimal surface is a surface that is geometrically efficient, one that has the least possible area relative to its boundary. For example, imagine taking a loop of wire, dipping it in a soapy water, and then removing it. A film of soap will have formed across the wire, and this soap film is a minimal surface. Among all possible surfaces that could have spanned the wire, nature chose the most efficient option.
