title: PUM formulas for art and material science audience: artists, architects, instrument builders, materials scientists working in the Sheela Rahman / Xosar / Hans Jenny / Dan Davidson lineage status: proposal v1 sources: pum_core seeds · Bendall PART 13 (Solomon’s Molten Sea Lense), PART 17 (Sacred Right Triangle, §C and §D) · Rahman lectures (32-step interpretation) · scope3d Chladni renderer
PUM formulas for art and material science
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:
- Cymatic plates (sized + driven so PUM ratios produce Chladni nodes, not blur)
- Sacred-geometry generators (Seed/Flower/Metatron at PUM-rational angles)
- Architectural ratios (rooms / walls / instruments designed in the PUM lattice)
- Material lattices (the 32 × 64 torus as a tiling for woven, printed, or crystallised materials)
- Generative visual identity (color, typography, layout entirely within the projection grid)
- 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):
| label | f (Hz) | nominal pitch |
|---|---|---|
| MUSIC_C | 266.6̄ | ≈ C₄ + 33 cents |
| RFEU / 100 | 1296 | ≈ E₆ |
| GREAT_YEAR / 10 | 2592 | ≈ E₇ |
| TIME / 100 | 5184 | ≈ E₈ - 28 cents |
| HYDROGEN cascade /60 sec | 0.5184 | sub-bass; use as the modulation envelope, not direct drive |
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):
| target f | required r |
|---|---|
| 266.6̄ Hz | ≈ 365 mm |
| 1,296 Hz | ≈ 165 mm |
| 2,592 Hz | ≈ 117 mm |
| 5,184 Hz | ≈ 83 mm |
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 = 1.6̄
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:
| element | ratio | example dimension (mm) |
|---|---|---|
| ceiling height : floor diameter | AETHER:SUN = 1:6 | 1500 : 9000 |
| wall thickness : room radius | LIGHT:AETHER = 1:1000 | 144 : 144000 |
| door height : window height | SUN:MOON = 864:216 = 4:1 | 2160 mm door : 540 mm window |
| beam spacing | 22.5° divisions of floor circle | radial 16 from centre |
| column count | 8 (TORUS_PLANES_BASE) or 16 (TORUS_SECTORS_BASE) | per plan |
| central acoustic resonance | C = 800/3 Hz at room volume | size for f₁ = 266.6̄ Hz |
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:
| element | drive Hz | claimed effect (Bendall PART 15 / Rahman) |
|---|---|---|
| H | 32,400 | nominally too high for liquid resonance; use V/2 = 16,200 |
| He | 3,136 | bass range; use directly |
| Ne | 2,304 (= 147,456 / 64) | string-like sustain |
| Ar | 2,592 (= 82,944 / 32 = GREAT_YEAR / 10) | “noble gas” cleanliness — the strongest Bendall claim |
| Kr | 1,225 | bell-like |
| Xe | 64 | sub-bass; pair with 4 kHz harmonic |
| Rn | 1,225 (same as Kr — coincidence in his digit-product² scheme) | as above |
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)
| name | hue (deg) | usage |
|---|---|---|
| AETHER | 0 | brand primary |
| SUN | 90 | accent |
| MATTER | 180 | neutral background |
| TIME | 270 | text body |
(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.