# Can there be a geometry where angles can be infinite?

In Euclidean (and and Riemanian) geometry an angle can be from $$0$$ to $$2\pi$$. The distance in Euclidean geometry is not limited. In spherical geometry the angles are limited as well, but the distance is also limited.

I wonder, can there be a geometry where the angles can be unlimited at any point while the length is limited (or not, I am interested in the both cases).

I think such geometry would not be locally Euclidean? What would be other properties?

• Okay, admittedly this question is very soft and speculative and maybe not very clear. But I think there’s something mathematically interesting to be explored here, so maybe hold off on the downvotes...? – Franklin Pezzuti Dyer Jan 3 at 21:54
• Sure, call "lengths" "angles" and vice versa in your favorite geometry, just as they swap lines and points in projective geometry. Without something identity-making for "angles" it is as Hilbert said, we can call them desks and chairs for all it matters. – Conifold Jan 3 at 23:05
• One of nine Cayley-Klein geometries actually has both unbounded distances and angles. Its points are those outside a non-degenerate conic (absolute), and its lines are those intersecting the absolute. Since any pencil of lines is contained between two lines touching the absolute (which are "at infinity") the angle metric is hyperbolic, and angles are unbounded. The distance metric is also hyperbolic. Some authors call it doubly hyperbolic geometry, see Supplement to Yaglom's Simple non-Euclidean geometry. – Conifold Jan 4 at 7:35
• One would presumably define angles on fractals by embedding them into a manifold. The set of them can then be funny (not an interval, maybe itself a 1D fractal), but they will all be finite. You'll need some non-Archimedean version of doubly hyperbolic geometry to get infinite angles, maybe. – Conifold Jan 4 at 10:22
• There is the notion of a Horn angle (see What is the state of research on Horn Angles?), which can be considered a refinement of zero-measure angles where the sharpness of a curvilinear angular aperture is measured by gauge functions (like is done for generalized Hausdorff measures). I suppose one could define the inverse notion by considering how quickly a curvilinear angular aperture straightens out to $180$ degrees as you zoom in. – Dave L. Renfro Jan 4 at 17:22

First of all, one has to specify what is a geometry. There are many fields of mathematics which regard themselves as geometry, many do not have notions of distances and angles, for instance, symplectic geometry and algebraic geometry; see my answer here for more examples. There are several geometric fields, which do have notions of distances and angles: synthetic geometry, metric geometry, Riemannian geometry, semi-Riemannian geometry.

Next, one has to specify what is locally Euclidean. Again, it depends on the field: In topology, this means locally homeomorphic to a Euclidean space (typically, of fixed dimension). In Riemannian geometry, this means locally isometric to a Euclidean space. (It is a theorem, going back to Riemann, that the latter condition is equivalent to zero sectional curvature.) To make things worse, physicists, sometimes, conflate "local" and "infinitesimal" and refer to Riemannian metrics as "locally Euclidean" since each tangent space is isometric to a Euclidean space (unlike in the semi-Riemannian setting).

Lastly, one has to clarify what properties "angles" are required to satisfy, otherwise, one can make totally random definitions.

All this is absent in your question. I will adopt the viewpoint is that any established field of mathematics called "geometry" qualifies as "a geometry." I will also assume that locally Euclidean is taken in the topological sense. I will also use notions of "angles" established in a particular field of geometry.

1. Lorentzian geometry, i.e. the geometry of $$R^{n,1}$$, the $$n+1$$-dimensional real vector space equipped with a quadratic form of signature $$(n,1)$$.

The notion of an angle between two time-like vectors is defined, for instance, in

G. Birman and K. Nomizu, Trigonometry in Lorentzian Geometry, The American Mathematical Monthly, Vol. 91, No. 9 (1984), pp. 543-549.

Namely, if $$u, v$$ are two time-like vectors then the angle $$\alpha=\angle (u, v)$$ between $$u$$ and $$v$$ is defined by the formula $$cosh(\alpha)= - \frac{\langle u, v\rangle}{|u|\cdot |v|},$$ where $$\langle u, v\rangle$$ is the Lorentzian inner product and $$|w|=\sqrt{|\langle w, w\rangle|}$$.

Thus, the angle equals the (hyperbolic) distance between the projections of $$u$$ and $$v$$ to the hyperbolic $$n$$-space. This makes sense since the angle between two unit vectors in a Euclidean space equals the spherical distance between the corresponding points on the unit sphere.

With this definition, the angle takes arbitrarily large values $$\alpha_k=\angle (u, v_k)$$ diverges to $$\infty$$ (or $$-\infty$$, depending on whether the angle is oriented or not) when $$u$$ is fixed while $$v_k$$ converges to a light-like (null) nonzero vector.

They leave the notion of angles undefined for more general vectors but develop some reasonable trigonometry for their notion. One can extend their angle to allow for general non-null vectors by allowing angles to take imaginary values.

1. Metric geometry, i.e. geometry of general metric spaces. A good modern reference for this field is

D.Burago, Y.Burago, S.Ivanov, A Course in Metric Geometry.

For metric spaces of curvature bounded above or below in the sense of Alexandrov, there is Alexandrov's notion of angle, but it takes values in $$[0,\pi]$$. However, one can define the more general notion of angles on the space of directions $$\Sigma_x X$$ at the given point in a metric space $$X$$ at a point $$x\in X$$ by path-metrizing the space of directions and using Alexandrov's angle only when directions are sufficiently close to each other. (The space of directions $$\Sigma_x X$$ is the space of germs, at $$x$$, of geodesic segments $$xy\subset X$$, with $$y\ne x$$.)

The angle between two directions at $$x$$ is then the distance between the corresponding points in $$\Sigma_x X$$.

This results in metric spaces where the angle can be locally unbounded and is even can take infinite values (if $$\Sigma_x X$$ is disconnected).

Here is a specific example. I will assume that you are familiar with covering spaces, metric spaces and basic Riemannian geometry.

Start with the Euclidean plane $$E^2$$; pick the origin $$o\in E^2$$ and let $$p:Y\to E^2-\{o\}$$ denote the universal covering of $$E^2- \{o\}$$ with the pull-back Riemannian metric. This is an incomplete metric space. Take its metric completion $$X$$. This is a metric space of curvature bounded from above by $$0$$, or a CAT(0) space.

This space will have a distinguished point $$x\in X$$ such that $$p$$ extends to a continuous (actually, 1-Lipschitz) map $$q: X\to E^2$$, sending $$x$$ to $$o$$. Then the space of directions $$\Sigma_x(X)$$ has a path-metric (defined via path-metrization of Alexandrov's angle), so that the projection $$p$$ induces a map $$\Sigma_x X\to S^1=\Sigma_o(E^2)$$ which is a universal covering of the unit circle and the metric on $$\Sigma_x X$$ is the pull-back of the standard (angular) metric on the unit circle. Thus, $$\Sigma_x X$$ is isometric to the real line. In particular, angles between geodesic segments in $$X$$ emanating from $$x$$ can take arbitrarily large values.

However, this example is not locally-Euclidean in the topological sense. It is not even locally compact.

Edit 1. Here is a more hands-on description of the space $$X$$. Start with a Euclidean quadrant $$S\subset E^2$$, i.e. a sector bounded by two orthogonal rays $$l, r$$ emanating from the origin $$o$$. Consider a countably-infinite collection of isometric copies of $$S$$, denoted $$S_k, k\in {\mathbb Z}$$, where each $$S_k$$ has two boundary rays $$l_k, r_k$$. For each $$k$$ consider the unique isometry (isometric bijection) $$f_k: l_k\to r_{k+1}.$$ Now, "glue" the sectors $$S_k$$ using the maps $$f_k$$. More formally, consider the quotient space of $$\coprod_{k\in {\mathbb Z}} S_k$$ by the equivalence relation $$a\sim f_k(a), a\in l_k, k\in {\mathbb Z}$$. This is a topological space $$X$$. It has a projection map $$q: X\to S/\sim$$, where $$\sim$$ is the equivalence relation defined via the unique isometry $$f: l\to r$$. (The space $$S/\sim$$ is homeomorphic to $$E^2$$. It has a natural structure of a locally Euclidean Riemannian manifold away from the projection of the apex $$o$$.)

Metrize the space $$X$$ as follows. Given a piecewise-smooth path $$c: [0,1]\to X$$ (i.e. a path such that the composition $$q\circ c: [0,1]\to S/\sim$$ is piecewise-smooth), define the length $$L(c)$$ of $$c$$ as the sum of lengths of its smooth pieces. Now, define the metric $$d$$ on $$X$$ by $$d(x,y):= \inf_c L(c),$$
where the infimum is taken over all piecewise-smooth paths $$c$$ connecting $$x$$ to $$y$$. One verifies that each geodesic segment $$xy$$ in $$X$$ emanating from the point $$x$$ corresponding to $$o$$, is contained in one of the sectors $$S_k$$. If two such segments belong to the same sector $$S_k$$ then the angle between them is just the Euclidean angle. In general, for two segments $$xy\in S_k, xz\in S_m$$, $$k, we have $$\angle(yxz)= (m-k)\frac{\pi}{2} + \angle(yx, l_k) + \angle(yz, r_m)$$ ("additivity of angles"). In particular, $$\angle(yxz)$$ can be arbitrarily large, depending on $$k, m$$.

Edit 2. Of course, the geometry of $$X$$ is locally Euclidean (in the metric sense!) away from the point $$x$$, hence, for $$x'\ne x$$, all angles $$\angle(y'x'z')$$ belong to the interval $$[0,\pi]$$. One can modify the construction to produce a more complex space which still has nonpositive curvature but angles are unbounded at every point. I will describe two such generalizations. The first one is "cheating" (a "Ponzi scheme"). It is easier to describe but the construction is a bit tedious. The second easier to describe but is harder to understand, it is a bit hideous.

i. Take $$X$$ as above and at every point $$a\in X-\{x\}$$ attach an isometric copy $$X_{a}$$ of $$X$$. Path-metrize the new space in the obvious fashion: The distance between $$y\in X_{a}, z\in X_{b}$$ equals the sum $$d_{X_a}(y,a)+ d_X(a,b)+ d_{X_b}(b, z)$$ The new space $$X^1$$ has the property that $$X$$ is isometrically embedded in $$X^1$$ at every point $$a\in X$$ the angles are unbounded (i.e. each point of $$X$$ "got paid at the expense of the new points"). However, each $$X_a- \{a\}$$ is still locally Euclidean. Now, repeat the step 1 by attaching an isometric copy of $$X$$ at each point of $$X^1- X$$ and thus forming a new space $$X^2$$. Now, each point of $$X^1$$ "got paid at the expense of the new points." Continue inductively. Lastly, let $$X^\infty$$ be the direct limit of the sequence of metric spaces $$X\subset X^1\subset X^2\subset ...$$ The space $$X^\infty$$ has the property that at every point we have unbounded angles. What's worse, however, is that at each point of $$X^\infty$$ some angles even take infinite values. The reason for this is apparent already on step 1: The space of directions at every point $$a\in X\subset X^1$$ is disconnected. One way to resolve this is to introduce fractional intermediate steps 1.5, 2.5, etc. On the step i.5 we glue to $$X^i$$ one Euclidean quadrant per each point $$a\in X^{i-1}$$, connecting a ray in $$X^{i-1}$$ to a ray in $$X_a$$.

ii. This construction requires no Ponzi Scheme, but uses the Axiom of Choice. Namely, let $$T^2$$ be a flat 2-dimensional torus. Pick a point $$t\in T^2$$ and let $$Y'$$ denote the universal covering space of $$T^2-\{t\}$$ with the pull-back Euclidean metric. Let $$X'$$ denote the metric completion of $$Y'$$. Now, this space has a discrete countably infinite subset of points where angles are unbounded, but $$X'$$ is locally flat away from these points. Now, let $$\omega$$ be a nonprincipal ultrafilter on $${\mathbb N}$$ (the existence of $$\omega$$ requires the Axiom of Choice) and let $$X'_\omega$$ denote the $$\omega$$-asymptotic cone of $$X'$$ with respect to the sequence $$\frac{1}{n}$$ of scaling factors. Informally, we take a suitable limit of the sequence of metric spaces $$(X', \frac{1}{n}d_{X'})$$
(with respect to some irrelevant base-point). The resulting space $$X'_\omega$$ is metrically homogeneous, still has nonpositive curvature, and has the property that angles at all points of $$X'_\omega$$ are unbounded.

• Your assumption seems to be too strong. – Anixx Jan 4 at 17:49
• @Anixx: Which assumption and why does it seems to be strong? My suggestion is to clarify your question. – Moishe Kohan Jan 4 at 18:09
• "I will assume that you are familiar with covering spaces" – Anixx Jan 4 at 18:19
• @Anixx: OK, I will then give an alternative description which does not use the covering theory. – Moishe Kohan Jan 4 at 18:51
• I think I understood your description, but it seems that the only point where two rays can intersect at angle greater than $\pi$ is the apex o. In all other points the angle between two rays cannot exceed $\pi$. I already had such "spiraling" model in mind, but can there be a model where all points have the property of o? – Anixx Jan 4 at 19:23