So what this means is that any causal network whose behavior obeys the Einstein equations must at the level of counting nodes in a cone have the same uniform structure as it would if it were going to correspond to ordinary flat space.

In ordinary flat space, regular polygons with more than 6 sides can never form a tessellation.

Pure gravity [theory] In the absence of matter, the Einstein equations always admit ordinary flat Minkowski space as a solution. … If one assumes that space is both homogeneous and isotropic then it turns out that only ordinary flat Minkowski space is allowed. … One form of solution to the vacuum Einstein equations is a gravitational wave consisting of a small perturbation propagating through flat space.

In the former category, it was recognized in the 1940s that a single atom is very unlikely to stick to a completely flat surface, so growth will always tend to occur at steps on a crystal surface, often associated with screw dislocations in the crystal structure.

But for ordinary flat Euclidean space it is always just IdentityMatrix[d] (at least with Cartesian coordinates). … In ordinary flat space there is no difference, but in general the difference is a vector that is defined to be Riemann . u . v. w . … In flat Euclidean space any two geodesics that start parallel always remain so.

The first idea was to represent gravity as a field that exists in flat spacetime, and by analogy with photons in quantum electrodynamics to introduce gravitons (at one point identified with neutrinos). … If one computes the product of Exp[  (j 1 + j 2 - j 3 )] for all triangles, then it turns out for example that this quantity is extremized exactly when the whole surface is flat. … And from this it turns out that limits of products of 6j symbols correspond essentially to Exp[  s] , where s is the discrete form of the Einstein–Hilbert action—extremized by flat 3D space.

But in updating networks a particularly straightforward implementation of one scheme can be obtained if one uses instead a more explicit symbolic representation such as u[1  v[2, 3, 4], 2  v[1, 3, 4], 3  v[1, 2, 4], 4  v[1, 2, 3]] This allows one to capture the basic character of networks by Attributes[u] = {Flat, Orderless}; Attributes[v] = Orderless Updating rules can then be written in terms of ordinary Mathematica patterns.

It is convenient to think of expressions in a language as having forms such as s["(", "(", ")", ")"] with Attributes[s] = Flat .

If f is associative (flat), so that f[f[i, j], k]  f[i, f[j, k]] , then the algebraic system is known as a semigroup.

Below 4D the vanishing of the Ricci tensor immediately implies the vanishing of all components of the Riemann tensor—so that the vacuum Einstein equations force space at least locally to have its ordinary flat form. (Even in 2D there can nevertheless still be non-trivial global topology—for example with flat space having its edges identified as on a torus.