Geometry and Coordinates
At the Differential Geometry math art shop we prefer shape to symbolic description. Sometimes, however, symbolic description can guide us where geometric intuition may falter.
For example, Cartesian coordinates in the plane may be viewed as a “perfect dictionary” between points of the plane and “addresses,” ordered pairs \((x, y)\) of real numbers: Each location has a unique address, and each address specifies a unique location.
Because our physical universe appears to permit three mutually-perpendicular axes through each point, we can extend the idea of Cartesian coordinates to Euclidean space. Each point of space now has a unique address, an ordered triple \((x, y, z)\) of real numbers, and each ordered triple \((x, y, z)\) specifies a unique location in space.
Now, we might reasonably balk at a geometric attempt to consider a fourth perpendicular axis: What direction does the fourth axis point? But in our symbolic description, writing \((x, y, u, v)\) is hardly more difficult than writing ordered triples. We can similarly consider complex numbers \(z = x + iy\) and \(w = u + iv\), and identify the ordered quadruple \((x, y, u, v)\) with the complex ordered pair \((z, w)\). In symbolic terms, four-dimensional space is on a nearly equal footing with three-dimensional space!
Broadly speaking there are two kinds of people: Those who are happy to accept the preceding paragraph, and those who are not. Here are a few observations for those of you who are not happy.
First, a coordinate and a spatial dimension are not the same things. A coordinate is a mathematical abstraction, while a spatial dimension is connected to experience, phenomena. Just because we cannot draw four mututally-perpendicular lines does not mean “four-dimensional space does not exist” as a mathematical construct. In a distinct but related vein, even Cartesian three-space is a mathematical abstraction, not the spatial universe where human life occurs.
Second, four-dimensional geometry conforms to our sense of sight, similar to shadows of rotating wire frames. For example, the result of rotating \((x, y, u, v)\) through angle \(\theta\) in the \((u, v)\)-plane is \[ (x, y, u\cos\theta - v\sin\theta, v\cos\theta + u\sin\theta). \] To see this in three-space we can simply discard the fourth coordinate, i.e., place a point light source infinitely far away on the \(v\)-axis and take the resulting shadow in \((x, y, u)\) space. If we restrict this rotation to a surface, perhaps the set of points \[ (u^{2} - v^{2}, 2uv, u, v), \] and then make an animation where the rotation angle is proportional to time, we see a surface rotating about the \(u\)-axis (vertical). At the same time, this is no ordinary rotation: The coordinate grid moves sinusoidally, parallel to the \(u\)-axis.
If we consider the same surface, but rotate the \((x, v)\)-plane, \[ (x\cos\theta - v\sin\theta, y, u, v\cos\theta + x\sin\theta), \] we again see the shadow of a rotating surface, although now an entire plane of points, the \((y, u)\)-plane, visibly remains fixed!
Although four-dimensional space is only a mathematical abstraction, it has a geometry that we can see by casting shadows into three-space, and this geometry perceptibly extends our experience with ordinary plane shadows of spatial objects.
Third, any collection of independent numerical quantities may be usefully viewed and treated as “coordinates.” The position and speed of a car on a straight road are independent, and together form a “phase plane” of possible staes. The positions and speeds of two cars on the same road are four independent numerical quantities, and together constitute a four-dimensional “phase space” of possible positions and speeds.
A single point of this phase space, perhaps written \((x_{1}, v_{1}, x_{2}, v_{2})\), encodes the position \(x_{1}\) of the first car, the speed \(v_{1}\) of the first car, and similarly the position and speed of the second car. A path in this phase space can be used to describe the changing positions and speeds of both cars as time passes.
By pondering these observations, one realizes that spaces of dimension greater than four, maybe much greater than four, arise naturally, and that doing mathematics in these spaces may have descriptive and predictive power. This is, in fact, one way to conceive of statistical mechanics and data science: Euclidean geometry in spaces having vast number of dimensions.
This is not just loose analogy. The concept of statistical correlation may be interpreted literally as the cosine between two directions. Particularly, two data sets are “statistically independent” precisely when the data sets, viewed as \(n\)-space displacements and suitably normalized, are perpendicular. One sometimes hears phrases such as that's orthogonal to our concerns, meaning independent from, or irrelevant to, the matter at hand.
The Algebra of Euclidean Geometry
Consider the four-space displacements \((1, 0, 0, 0)\) and \((1, 1, 1, 1)\). How long is each? What angle do they make?
Length To find the length of \((1, 0, 0, 0)\), we might observe that this displacement lies along the first coordinate axis, and so has length \(1\). That's well and good, but doesn't help with \((1, 1, 1, 1)\). Let's ask easier questions: What is the length of \((1, 1, 0, 0)\)? Of \((1, 1, 1, 0)\)?
The displacement \((1, 1, 0, 0)\) lies in the plane of the first two coordinates, where it amounts to \((1, 1)\). The length \(\ell\) may be viewed as the hypotenuse of a right triangle with legs \(1\). By the Pythagorean theorem, \(\ell^{2} = 1^{2} + 1^{2} = 1 + 1 = 2\), or \(\ell = \sqrt{2}\).
What about \((1, 1, 1, 0)\)? We may view this as a displacement in three-space, \((1, 1, 1)\). To find its length, we might draw the right triangle with corners \((0, 0, 0)\), \((1, 1, 0)\), and \((1, 1, 1)\). The leg between \((0, 0, 0)\) and \((1, 1, 0)\) has length \(\sqrt{2}\) by the preceding paragraph. The leg between \((1, 1, 0)\) and \((1, 1, 1)\) is a displacement of \((1, 1, 1) - (1, 1, 0) = (0, 0, 1)\), which has length \(1\) because it is parallel to the third axis. By the Pythagoream theorem, the length \(\ell\) of \((1, 1, 1)\) satisfies \[ \ell^{2} = (\sqrt{2})^{2} + 1^{2} = 2 + 1 = 3. \] We conclude that the length of \((1, 1, 1, 0)\) is \(\ell = \sqrt{3}\).
We're ready to find the length of \((1, 1, 1, 1)\), reasoning by analogy with \((1, 1, 1, 0)\). We'll consider the right triangle with corners \((0, 0, 0, 0)\), \((1, 1, 1, 0)\), and \((1, 1, 1, 1)\).
The leg between \((0, 0, 0, 0)\) and \((1, 1, 1, 0)\) has length \(\sqrt{3}\) by the preceding paragraph. The leg between \((1, 1, 1, 0)\) and \((1, 1, 1, 1)\) is a displacement of \((1, 1, 1, 1) - (1, 1, 1, 0) = (0, 0, 0, 1)\), which has length \(1\) because it is parallel to the fourth axis. By the Pythagoream theorem, the length \(\ell\) of \((1, 1, 1, 1)\) satisfies \[ \ell^{2} = (\sqrt{3})^{2} + 1^{2} = 3 + 1 = 4. \] We conclude that the length of \((1, 1, 1, 1)\) is \(\ell = \sqrt{4} = 2\).
Consider an arbitrary displacement \((a, b, c, d)\). Similar reasoning with the Pythagorean theorem gives the length of this displacement as \[ \ell = \sqrt{a^{2} + b^{2} + c^{2} + d^{2}}. \] As a consistency check, this formula yields the same lengths we found in the preceding examples.
There is nothing specific to four-space; analogous formulas give the length of an arbitrary displacement in Euclidean spaces of arbitrary dimension. Because length of a displacement generalizes the absolute value of a real number, mathematicians use the same notation, sometimes writing an arbitrary displacement as \(x\) (an ordered \(n\)-tuple of real numbers) and denoting its length, or magnitude, by \(|x|\).
Angle The story with angle ends with a fairly short formula. We will not explain in detail where the formula comes from, but do note that it comes from the law of cosines applied to a non-right triangle in \(n\)-space, and that a proof can be found in a linear algebra book, usually early in the discussion of “inner products.”
The key quantity is the dot product of two displacements. In three-space, the dot product is the real-valued function defined by \[ (a, b, c) \cdot (a', b', c') = aa' + bb' + cc'; \] in words, multiply corresponding components, and sum these products. For example, \[ (3, 4, 5) \cdot (2, -1, 0) = (3)(2) + (4)(-1) + (5)(0) = 6 + (-4) + 0 = 2. \] The dot product in \(n\)-space generalizes this to pairs of displacements with an arbitrary number of components. For example, if \(x = (1, 0, 0, 0)\) and \(y = (1, 1, 1, 1)\), then \begin{align*} x \cdot x &= 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 1, \\ x \cdot y &= 1 \cdot 1 + 0 \cdot 1 + 0 \cdot 1 + 0 \cdot 1 = 1, \\ y \cdot y &= 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 4. \end{align*} Particularly, the dot product of a displacement with itself is the sum of the squares of the components, namely the square of the length. We can express this concisely in formulas: \(|x|^{2} = x \cdot x\), or \(|x| = \sqrt{x \cdot x}\).
Now assume \(x\) and \(y\) are non-zero displacements. In the plane through the origin and containing \(x\) and \(y\), there is a unique real angle \(\theta\) between the displacements \(x\) and \(y\) if we impose \(0 \leq \theta \leq \pi\). The key formula relating the dot product with Euclidean angle is \[ x \cdot y = |x| \, |y| \cos\theta. \] Since the displacements are non-zero, we may divide both sides by the positive real number \(|x|\, |y|\). Doing so, and swapping the two sides in the preceding formula, we obtain \[ \cos\theta = \frac{x \cdot y}{|x|\ |y|}. \]
Let's see what this gives for \(x = (1, 0, 0, 0)\) and \(y = (1, 1, 1, 1)\). We calculated the necessary dot products, and therefore magnitudes, above: \(|x| = 1\), \(|y| = \sqrt{4} = 2\), and \(x \cdot y = 1\). Our formula reads \[ \cos\theta = \frac{1}{2},\qquad \text{or}\quad \theta = \frac{\pi}{3}. \] Despite our inability to maneuver a protractor in four-space, we find a \(60^{\circ}\) angle between the displacements \(x\) and \(y\)!
Most of the images and objects at the math art shop depict geometry of real three-space, and many involve objects projected from real four-space. Our helicoid-catenoid logo may be viewed as a projection from real six-space, complex three-space! This geometry is accessible to our senses only to the extent we can represent it in planar (two-dimensional) images or as spatial objects, but is accessible to calculation thanks to a concise symbolic representation of points as ordered \(n\)-tuples or real or complex numbers, and the remarkable dot product, which computes length and angles where physical measuring instruments cannot reach.