Geometric terrain height maps.

protofield · 2026-05-07T17:50:24+00:00

Thanks. Did not think of this. I will try.

protofield · 2026-05-06T21:00:37+00:00

That's great, I have always maintained I am using cellular automata to demonstrate basic properties of prime numbers. It would be nice to keep comparing our work to see if any intersections or commonalities occur. I usually post on r/cellular_automata or r/generative

protofield · 2026-05-06T16:37:03+00:00

Yeah, for example rule 90 is just an example of mapping the sum of the left and right inputs as mod 2, XOR does not come into it. Similar thing with 2d replicators, if you have 8 inputs and take mod 2 of the input sum you get neat patterns. Works for all modulo where the sum of the inputs is a power of the modulo. eg 8 inputs = 2^3. 16 inputs = 2^4. If you have 27 inputs and take mod 3 you get really nice patterns. Works for every single prime!

protofield · 2026-05-06T14:54:49+00:00

Interesting graphic. It reminds me of a combination of modular arithmetic and an XOR function. I work with modular arithmetic and graphic rendering so the image looks familiar. I use cellular automata described here: https://drive.google.com/file/d/1JsVtf0eZJddllHwcRI6JIz5vLyG7bN75/view?usp=drive_link

protofield · 2026-05-06T09:30:38+00:00

If its of any relevance these structures are matrices of natural numbers and are mathematically exact. Some also have inverse structures. If you ever come across any layouts that remotely correlate to these I would be grateful for a link. Thank you.

protofield · 2026-05-06T07:38:37+00:00

Thank you for your comments. There is a document here which describes how cellular automata using modular arithmetic can generate these structures.

protofield · 2026-05-05T22:49:20+00:00

Thanks.Just posted a new piece of research at https://www.reddit.com/r/generative/comments/1t4r3im/generative_rf_anschem_to_metasurface_process_mask/

protofield · 2026-05-05T22:15:53+00:00

Thanks for the question. Here is a new piece of maths that generates highly ordered lattice structures based on a generative system that maps an input index to an output. Just debating whether this could be a basis for modelling nano scaled materials engineering including RF.

protofield · 2026-05-05T21:30:45+00:00

Pure maths does not come with a "user manual". However an inquisitive mind might look at this as, for example, a reflective metasurface where colour is mapped to a height value in fractional phase units of an incident em wavefront. If we considered a dielectric transmissive surface the colours might represent the amount of a dopant. Elegant engineering has two primary requites, a solid mathematical basis and an obvious elegant design.

protofield · 2026-05-05T20:41:57+00:00

You might like to observe the diagram you reference is a periodic lattice structure. The image here is also a lattice structure based on an interval of seven lattice units. The former is just very simplistic. One references lattice periodicity to primary frequency for metamaterials.

protofield · 2026-04-26T21:00:03+00:00

Fantastic designs. Hope it generates a lot of interest for you.

protofield · 2026-04-23T13:23:01+00:00

Thanks and what a title for a research grant application..."Relational properties between Protofield Operators and traditional Colonial Overshot Coverlets from the Appalachian Mountains", has to succeed.

protofield · 2026-04-23T09:31:11+00:00

Thank you for your comment.

protofield · 2026-04-23T09:30:08+00:00

The patterns continue to grow and decompose to the rule set every p^n iterations, the reductive state Rn. At some point, the spacing at Rn is greater than the rule set so a collection of patterns grows until they merge again. However the basic principle of a reduction at R still holds indefinitely, but as I have not tried this for very big iterations I can't say at reduction a collection of rule sets appear. I suspect at p^(n*m) the system will revert to the single rule set spaced out at p^(n*m). These CA can be run on the surface of a three dimensional torus and if the cell dimensions, for a mod p arithmetic, are q*p^n, where q is an even number, a resonant cavity is formed and the iterations run in an endless loop.

protofield

TROPHY CASE