You can play with some examples of lambda terms (SKK, Y combinator, Omega combinator, Ackermann(2,2), some duplications of terms, lists, Church numbers multiplications). It is important to try several times, because the reduction algorithm uses randomness in an essential way! This justifies the “reload” button, the “start” which does the reduction for you (randomly), the “step” which choses a random reduction step and shows it to you. Or you may even use the mouse to reduce the graphs.

It may look kind of alike the other chemlambda reductions, but a bit different too, because the nodes are only apparently the usual ones (lambdas, applications, fanins and fanouts), in reality they are dilations, or homotheties, if you like, in a linear space.

I mean literary, that’s what they are.

That is why the name: anharmonic lambda calculus. I show you lambda terms because you are interested into those, but as well I could show you emergent (actually em-convex) reductions which have apparently nothing to do with lambda calculus.

But they are the same.

Here is my usual example Ackermann(2,2), you’ll notice that there are more colors than precedently:

The reason is that what you look at is called “kali24”, which for the moment uses 7 trivalent nodes, out of 24 possible from projective space considerations.

I will fiddle with it, probably I’ll make a full 24 nodes versions (of which lambda calculus alone would use only a part), there is still work to do, but I write all the time about the connections with geometry and what you look at does something very weird, with geometry.

Details will come. Relevant links:

- kali24, the last version
- kali, the initial version with 6 nodes, which almost works
- em-convex, the dilations enhanced lambda calculus which can be also done with kali
- and perhaps you would enjoy the pages to play and learn.

One more thing: when all fiddling will end, the next goal would be to go to the first interesting noncommutative example, the Heisenberg group. Its geometry, as a noncommutative linear space (in the sense of emergent algebras, because in the usual sense it is not a linear space), is different but worthy of investigation. The same treatment can be applied to it and it would be interesting to see what kind of lambda calculus is implied, in particular. As this approach is a machine of producing calculi, I have no preference towards the outcome, what can it be? Probably not quite a known variant of lambda, quantum or noncommutative, because the generalization does not come from a traditional treatment [**update:** which generalizes from a too particular example].

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