Breaking the infinite symmetry group of a rotating galactic disk produces the simplest consistent subgroup — a two-armed spiral — as the dominant emergent form in M51 and computer simulations.

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• A thin cylinder buckled by uniform pressure from above develops a symmetric pattern of dimples corresponding to a finite subgroup of its original infinite rotation group, preserving most structure.finding0.752
  Structural mechanics finding demonstrating structure-preserving transformation through symmetry-group reduction in buckling
• The deep symmetry between problem-solving in anatomical, physiological, transcriptional, and 3D spaces drives specific hypotheses.claim0.745
  Evolution pivoted the same problem-solving strategies across different domains.
• Even mathematical catastrophes always begin as features consistent with the symmetries of the earlier state and develop smoothly from within the existing wholeness.claim0.729
  Alexander's response to the apparent counter-examples from catastrophe theory: discontinuities are themselves structure-preserving at a deeper level
• Good shape arises in many natural systems such as leaves, bones, and Chladni patterns, where a strong center is intensified by minor centers.claim0.726
• In buildings, especially, the success of the process will be judged by the extent to which these middle-range entities appear with their own distinct symmetries, with their own definite and distinct force as strong centers.claim0.726
  Proposes middle-range entity quality as the criterion for judging the success of a building process
• There are two types of phenomenal time: inter-frame discrete (~40 Hz) and intra-frame continuous drift; transformers have analogous dual temporality: within-token and inter-token.claim0.725
  Cube Flipper and Imago found convergent phenomenology between human meditation and transformer structure.
• We hypothesize that a similar 'brutal' and purely geometric process always occurs somewhere in other kinds of unfolding that generate living order — in poetry, dance, social structure, planning, and family relationships.hypothesis0.722
  Extends the brutal geometry thesis beyond architecture into all creative and social domains; acknowledged as not yet confirmed with certainty
• Wind-blown sand ridges form with apparent wavelength equal to the distance an average grain is carried by wind; identical irregularity is then duplicated downwind, creating alternating repetition by a purely local mechanical process.finding0.722
  Classic mechanical explanation for alternating repetition in sand, used as a case where local mechanics suffices but cannot generalize