A Klein bottle is a surface you cannot build in our world without it passing through itself. It is closed like a sphere and has no edge, but it also has no inside and no outside. Go once around it and you come back mirror-reversed.
I wanted to know whether the engine could handle one. Until this week it could curve spacetime: black holes, spinning black holes, wormhole throats, warp bubbles. But only in four dimensions, one coordinate patch at a time, with no idea of a shape's global structure. A Klein bottle is two-dimensional, and what makes it strange is not how it bends at any point but how its edges are glued.
So the curvature code now works in any dimension, and there is a new module for surfaces with glued edges: a sphere, a torus, a Klein bottle, a Moebius strip, and the pseudosphere, which is a piece of the hyperbolic plane you can hold in your hand.
The theorems are the tests
The way this engine earns trust is by computing things two ways that share no algebra and requiring them to agree. Geometry has some of the most beautiful two-way checks in mathematics, and each of them is now a test the code has to pass.
- Theorema Egregium. Gauss proved that a surface's curvature can be measured from inside it, from distances alone, and that it has to match the curvature you would measure from outside, by how it bends in space. The engine does both: once from the embedding, once from the metric through the same Riemann tensor code that curves spacetime. They agree to about two parts in a hundred thousand on all five shapes, the Klein bottle included.
- Gauss-Bonnet. Add up all the curvature on a closed surface and you get 2π times a whole number, the Euler characteristic, however the surface is bent. The engine gets 2.00003 for the sphere, and 0 for the torus and the Klein bottle, where 25 and 44 units of curvature respectively cancel exactly.
- Euler's count. Cut the surface into triangles and count: vertices minus edges plus faces. No calculus at all, and it gives the same 2, 0 and 0 at two different mesh sizes.
- The classification of surfaces. Those two numbers cannot tell a torus from a Klein bottle; both are 0. What tells them apart is orientability, whether you can give the surface a consistent inside and outside. The engine checks that twice as well, from the normals across each glued seam and by trying to orient every triangle of the mesh, and then names each shape correctly.
It has no inside
You can compute the volume inside a closed surface from the surface alone, with the divergence theorem. The engine does that for the sphere and gets 4/3 πR³ to four digits, and for the torus to six.
For the Klein bottle it refuses. The formula needs to know which way is out at every point, and a Klein bottle has no consistent answer. The refusal is not taken on trust: the integral it would have computed is still reported, and it comes out as 0, 3.0, 5.3 or 6.3 depending only on where the bottle was cut to lay it flat. Same object, four answers. That is a well formed number to a question with no answer, the recurring subject of this archive, and the right response is to say so.
Glass Klein bottles do hold water, because the glass has thickness and the place where it passes through itself is a hole. That makes them a different object from the mathematical one.
Experiment one: glue it with a flip
Here is the freakiest thing the engine found. Take a flat rectangle and glue its edges. Glue the left edge to the right edge straight across and you have a torus. Glue them with a flip, top of one side to the bottom of the other, and you have a Klein bottle. Both are perfectly flat everywhere. The only difference is one flip in the glue.
Now put heat on them and let it spread. Try "hot side, cold side" first, then the others.
A pattern that is hot on one side and cold on the other lasts twice as long on the torus: the engine measures it dying at 0.999π² on the torus and 2.001π² on the Klein bottle. Every trip across the glue turns that pattern inside out, so it cannot sit still. It is forced to twist along the glue as well, you can watch it do so in the right-hand panel, and twisting costs it.
A random start dies at the same rate on both, because the very slowest pattern is allowed on both shapes. The difference is how many ways each shape has to die slowly. The torus has two slowest patterns and the Klein bottle has one. The Klein bottle then has a level at twice the rate that the torus lacks entirely, and at four times the rate the torus has four patterns to the Klein bottle's three. In 1966 Mark Kac asked whether you can hear the shape of a drum, whether its frequencies pin down its shape. Here you can hear the twist.
Experiment two: walk a circle
On a curved surface you can measure the curvature without ever leaving it. Walk the same distance in every direction from one spot and measure the loop your walkers make. On flat ground it is 2π times the distance. On a sphere it is shorter, and on a saddle it is longer, and the shortfall tells you the curvature. That is the Bertrand-Diguet-Puiseux theorem, and the engine's walkers reproduce the exact circumferences on all three.
Keep walking on the sphere and something happens. The loop grows until you pass the equator, then shrinks, and the walkers converge on the single point opposite where they started. That is a conjugate point, and it comes from the Jacobi equation, which describes how neighbouring straight paths spread apart or refocus. The engine now integrates it directly: on a sphere of radius 2 the paths refocus at 6.283185, which is 2π to the digits shown. Along the outer equator of a torus they refocus at 5.4415, where theory says π√3 = 5.4414. On the saddle, and on the torus's inner equator, they never refocus at all.
Experiment three: flood a landscape
This one is about a different kind of shape. Take random, bumpy ground and let water rise over it. At first there are separate puddles. At some point one body of water connects the left side to the right. When?
Percolation theory has an answer. For random ground that looks statistically the same upside down, hills and valleys alike, the crossing happens at exactly half flooded. It is a symmetry argument: a wet crossing one way and a dry crossing the other way cannot both happen, and at the half-way mark each is as likely as the other. A proof announced this August shows how sharp the switch is for a broad class of networks. Try it: find the moment it connects, then press "new landscape" and do it again.
Over forty landscapes the engine finds the crossing at 50.9 percent. The switch sharpens as the ground gets bigger relative to its bumps: the spread between landscapes falls from 4.9 points to 2.7 when the ground is twice as wide. The largest puddle holds 8 percent of the water at 40 percent flooded and 97 percent at 60. Make the ground lopsided and the symmetry goes, and so does the half: it connects at 43 percent instead.
One more honest thing. The first results sat a little above half, every time. The cause was the grid: water here can only step to four neighbours, which makes crossing slightly harder than on real ground. The bias is 2.6 points when each bump spans four grid cells, 0.6 at eight and 0.1 at sixteen. It is the grid, and it goes away.
Are we modelling quantum gravity?
No, and nobody can yet, because there is no tested theory of quantum gravity to model. Everything above is classical geometry, and the spacetime side of the engine is classical general relativity.
What the engine can do is know where that stops being trustworthy. It now measures how sharply spacetime is curved, as a length, and compares it with the Planck length, about 1.6 × 10-35 metres, where quantum effects on gravity are expected to matter. At the horizon of a black hole the mass of the Sun, the curvature length is 1,587 metres, about 1038 Planck lengths, and classical geometry is fine. The engine keeps answering until 6 × 10-22 metres from the centre, and inside that it refuses. It refuses a black hole of one Planck mass outright. The line it draws, ten Planck lengths, is a stated choice rather than a result, because nobody knows exactly where it belongs.
That is the pattern in all of this. The theorems check the machinery, the refusals mark where the machinery ends, and in between you can now run the experiments yourself.