Art / materials formulas

0. Why this exists

Sheela Rahman’s premise is that PUM ratios are not cosmological symbols but resonant addresses in physical media — a frequency that “fits” a geometry produces coherent matter (clean nodes in cymatics, ordered crystallisation, lower-entropy assemblies); a frequency that doesn’t fit produces noise. This document collects the operational formulas to apply PUM directly to:

1. Cymatic plates (sized + driven so PUM ratios produce Chladni nodes, not blur)
2. Sacred-geometry generators (Seed/Flower/Metatron at PUM-rational angles)
3. Architectural ratios (rooms / walls / instruments designed in the PUM lattice)
4. Material lattices (the 32 × 64 torus as a tiling for woven, printed, or crystallised materials)
5. Generative visual identity (color, typography, layout entirely within the projection grid)
6. Crystal-growth + piezo experiments (driving solute solutions at elemental V frequencies)

Every formula here returns an exact rational — there is no place where you have to choose a number.

1. Cymatic plate sizing

A Chladni plate vibrating at frequency f shows nodal patterns at characteristic dimensions. For a circular plate of radius r made of material with longitudinal wave speed c, the m-th radial mode obeys

f_m  ≈  ( (α_m / r)² )  ·  ( h / 2  ·  √(E / (12 ρ (1-ν²))) )

(where α_m is the m-th root of J₀'; h plate thickness; E,ρ,ν Young’s modulus, density, Poisson ratio.)

Inverting for a plate that resonates at a specific PUM cardinal frequency:

r  =  α_m · sqrt( h · √(E / (12 ρ (1-ν²))) / (2 · f_PUM) )

PUM-suggested driving frequencies (audible band):

For a steel plate (E = 200 GPa, ρ = 7850 kg/m³, ν = 0.3) at h = 1 mm and the first axisymmetric mode (α₁ ≈ 3.196):

Buildable suggestion: a 365 mm steel plate driven at C = 800/3 Hz. The Chladni pattern will be the visual fingerprint of MUSIC_C. Mount a contact piezo at the centre, feed from any audio source.

For multi-frequency / chord plates: cut three concentric rings of these radii from one larger plate. Each ring vibrates at its own PUM cardinal — you get the projection grid as a visible, audible object.

2. Sacred-geometry generators (formulas)

2.1 Seed of Life at PUM angles

Standard Seed of Life uses 6-fold symmetry (60°). The PUM equivalent uses 16-fold symmetry (22.5° — the Vajar Torus Octave Sequence).

n_petals  = 16
angle     = 22.5°
center    = (0, 0)
radius    = R
petals_i  = circle(center = (R · cos(i · 22.5°), R · sin(i · 22.5°)), radius = R)
                for i in [0..15]

The intersection lattice has Σ k=1..15 (16-k) = 120 distinct double-circle vesicas. Each carries a PUM rational (sector_i × dim_fraction) for labelling. Scale R to any PUM cardinal divided by 2.

2.2 Flower of Life with PUM grid

Standard Flower has a 7-circle core; expand via n × 10/9 ratio for each successive ring of circles:

ring_k_radius   = R · (k × 10/9)         for k = 1..6  (6 dimension levels)
circles_in_ring = (1 + 6 × ring_k_radius / R)

This produces a Flower whose growth rate matches the dimensional ladder — visually similar to standard Flower of Life but with each ring carrying a labelled dimension.

2.3 Vesica Piscis from PUM right triangle

PART 14 of the patent series invokes Vesica Piscis explicitly. Bendall maps the 7.2 hours = 432 minutes identity to “VESICA PISCIS.” We can reverse-engineer:

vesica_height / vesica_width  =  √3 / 1
PUM equivalent:               =  432 / 259.2  =  5/3  =  1.6

So Bendall’s “PUM Vesica” has aspect ratio 5:3 (POS, the Sun:Time ratio), not the classical √3 : 1 (≈ 1.732). Drawing it produces a flatter almond, visually recognisable as a Bendall-Vesica.

2.4 Metatron’s Cube → PUM Cube

Standard Metatron contains all 5 Platonic solids as 2D projections. PUM offers a custom set: the cube whose internal angles = 2,160 (legend item: “internal angles of a cube = 2,160” is in PART 17 P 106). Projecting from 2,160 = 16 × 135 = 24 × 90 = 6 × 360 = MOON_DIAMETER lets the cube be rendered at any of those scaled sizes and remain “in-system.”

3. Architectural ratios

Apply the projection grid as room dimensions:

A circular pavilion with floor diameter 9 m, ceiling 1.5 m, 16 perimeter columns at 22.5°, central oculus diameter 540 mm, would have the PUM grid as its full design specification. Acoustic resonance at MUSIC_C if interior surface treatment is right.

3.1 Instrument body ratios

For built instruments — guitar bodies, drum shells, flute bores:

hollow_body_volume_L : hollow_body_volume_R  =  5 : 1  (Vajra outer:inner)
plate_count                                  =  15 + 15  (basket-weave)
node_intersect_angle                         =  90° (charge-source quadrature)
twist_handedness                             =  outer CW, inner CCW (Vajra convention)

A “PUM guitar” would have its sound-hole diameter = (body length) × 5/3 / something (POS), bridge position at 22.5° from the soundboard centre, etc. Each ratio from a single rational.

4. Material lattices — the 32 × 64 torus as a tiling

The plasmoid lattice (pum_lattice/lattice_full.scad) is a tile. Read out as material:

4.1 Woven textile

• Warp count: 32 (planes)
• Weft repeat: 64 (radial points)
• Weft pattern: spiral with 20 revolutions (PLASMOID_DEGREES = 7,200°) over the cloth length
• Twist direction: alternate clockwise / counter-clockwise per pass (Vajra convention)
• Thread tensioning: 5:1 outer:inner if you separate two plies

Result: a fabric whose visible pattern is the projection of the toroidal lattice.

4.2 3D-printed lattice

pum_lattice/lattice_full.scad already produces an STL. Print parameters to make it materially expressive:

• Nozzle: 0.3 mm (lower bound of Vajra plate thickness range)
• Layer height: 0.144 mm (= LIGHT × 10⁻³)
• Infill: 11.111 % (= AETHER_TO_MATTER × 10⁻¹)
• Print speed: 25.92 mm/s (= GREAT_YEAR × 10⁻³)
• Travel speed: 51.84 mm/s (= 2 × that, also = Kheops slope number)

Every machine setting is a PUM number. The print becomes a self-similar object across geometry × motion × material.

4.3 Crystal growth

For supersaturated solutions (sugar, salt, Rochelle salt, KDP crystals), apply audio at PUM frequencies during nucleation:

Argon at 2,592 Hz is the most-tested combination and the most likely candidate to produce visibly cleaner growth structures. Run a control + experiment in identical solutions, photograph at 24h / 48h / 72h, score morphology by hand. No physics claim — just an open experimental hypothesis from the model.

5. Generative visual identity (CSS / design tokens)

A complete brand system from PUM:

5.1 Color palette (HSL hues from compass angles)

(Rotation 22.5° per sector → 16 distinct hues; 4-cardinal scheme uses every 4th sector.)

5.2 Typographic scale (PUM modular)

Base font size = 16 px (= TORUS_SECTORS_BASE).

sizes = [16 × (n × 10 / 9)  for  n in 1..8]
      = [17.78, 35.56, 53.33, 71.11, 88.89, 106.67, 124.44, 142.22] px

Round to integer for legibility; the underlying ratio is exact.

5.3 Spacing scale

Spacing units in 22.5° divisions, scaled to a base of 9 px:

gap[k] = 9 × (k × 22.5°/360°)  =  9k/16  px       for k = 1..32

Result: 32 spacing tokens in [0.5625, 1.125, 1.6875, …, 18] px. Every margin/padding/gap in a PUM-designed layout snaps to this scale.

5.4 Layout grid

Page columns = 16 (TORUS_SECTORS_BASE). Page rows = 8 (TORUS_PLANES_BASE). Grid gutter = 22.5 px. The four major content zones map to the four cardinal quadrants of the projection grid.

pum_svg/projection_grid.svg and pum_svg/print_poster.svg are example pages built on this grid.

6. Reference build: the “PUM Cymatic Pavilion”

A combined art / acoustics / materials installation putting all 5 sections together:

• Floor: 9 m diameter circle, divided into 16 sectors of 22.5°, each tiled with a Chladni-pattern stone matched to a different PUM frequency.
• Walls: 16 vertical panels arranged at 22.5° intervals; each panel’s height = (cardinal scalar / 1000) mm; alternate panels are conductive (Cu) / insulating (ceramic) per Vajra convention.
• Ceiling: dome with 8 toroidal “ribs” projecting the plasmoid lattice; central oculus = 540 mm = 144,000 / 266.666.
• Sound system: 4-channel quadrature panner driven by the scope3d synthesizer; each channel tuned to one cardinal scalar (AETHER 144, SUN 864, TIME 518.4, MATTER 3456 — all in their kilo-units, audible).
• Cymatic centre: 365 mm steel plate at the exact dome centre, driven at C = 800/3 Hz from the sub-mix bus. Visitors see the Chladni pattern through a glass floor portal.
• Crystals: four KDP crystals seeded in the four quadrants, driven by their respective cardinal’s audio channel during the entire installation period; documentation photography at 12h intervals.

Total build is a PUM hardware patch: every dimension, frequency, count, ratio, and material property is a pum_core symbol. No magic numbers anywhere. The pavilion is the model in physical form.

7. Cross-link to scope3d + pum_core

Every formula above pulls from pum_core. Copy-paste-ready Python:

from pum_core import (AETHER, SUN, TIME, MATTER, MUSIC_C, RFEU, GREAT_YEAR,
                     ELEMENT_MELTING_POINTS, elemental_value, cascade_for_element,
                     TORUS_SECTORS_BASE, TORUS_PLANES_BASE)
from fractions import Fraction

# Cymatic plate radius for steel @ 1 mm thickness, mode α₁ = 3.196:
import math
def chladni_radius_mm(f_hz, h_mm=1.0, alpha=3.196, E=200e9, rho=7850, nu=0.3):
    return alpha * math.sqrt((h_mm/1000) * math.sqrt(E / (12 * rho * (1 - nu**2))) / (2 * f_hz)) * 1000

# 16-sector palette (HSL hues):
hues = [i * 360 / TORUS_SECTORS_BASE for i in range(TORUS_SECTORS_BASE)]

# Typographic scale (px), base 16:
typo = [16 * Fraction(n * 10, 9) for n in range(1, 9)]

# 22.5°-stepped spacing tokens:
spacing = [Fraction(9 * k, 16) for k in range(1, 33)]

scope3d already drives the cymatic plate. pum_lattice already prints the textile / 3D tile. pum_tuning already provides the .scl. This proposal binds them into one physical artefact discipline.