================================================================================ COSMIC STRUCTURE — OBSERVATIONAL DATA FOR CROSS-SCALE ANALYSIS ================================================================================ Compiled: 2026-03-18 Purpose: Quantitative geometric data across all cosmic scales, from stellar systems to the cosmic web. Numbers sourced from SDSS, Planck, Galaxy Zoo, and published surveys. For cross-referencing with TLT lattice predictions and {2,3} cipher geometric patterns. ================================================================================ 1. COSMIC WEB STRUCTURE ================================================================================ OVERVIEW: The cosmic web is the largest-scale structure in the observable universe, consisting of four morphological components: nodes, filaments, walls/sheets, and voids, arranged in a foam-like topology. VOLUME FRACTIONS (from NEXUS+ and SDSS analyses): Voids: ~77-80% of total volume (~15% of total mass) Walls/sheets: ~10-15% of total volume (~moderate mass fraction) Filaments: ~5-8% of total volume (~40-50% of total mass) Nodes: ~1-2% of total volume (~significant mass fraction) KEY: Most MASS is in filaments/nodes. Most VOLUME is void. This is a profound asymmetry — structure occupies ~20% of space but contains ~85% of matter. FILAMENTS: Geometry: NOT simple 1D lines. They are elongated overdense cylinders with internal substructure (galaxies, gas, dark matter). Typical length: 30-100 Mpc (most common range) Maximum length: up to ~130 Mpc (SDSS DR17 analysis) Typical width: 2-5 Mpc diameter Detection radius: ~0.5 h^-1 Mpc for galaxy-traced filaments Overdensity profile: drops to cosmic mean within ~3 Mpc from spine Galaxy content: 35-40% of total galaxy luminosity Cross-section: Roughly cylindrical, some are flattened ("walls") Source: Tempel+ (2014) SDSS filament catalog; MNRAS 438, 3465 Einasto+ (2022) SDSS DR17 analysis; MNRAS 519, 3227 WALLS / SHEETS: Geometry: Flattened 2D overdense structures forming boundaries between voids. Typical thickness: 5-10 Mpc Typical extent: up to ~100 Mpc across Density: 1-5x cosmic average Note: When a 3D sheet is sliced by a 2D plane, it appears as a "filament" — so 2D projections undercount sheets and overcount filaments. Source: NASA Imagine the Universe; SDSS large-scale structure analyses NODES / CLUSTERS: Located at intersections of filaments. Typically contain rich galaxy clusters (see Section 2). Cluster-cluster separation: 4-30 Mpc depending on mass threshold - Groups (~10^13 M_sun): separation 4-15 h^-1 Mpc (~6-22 Mpc) - Rich Abell clusters: characteristic distance ~21 h^-1 Mpc (~30 Mpc) CHARACTERISTIC SCALE: The "cell size" of the cosmic web: ~50-100 Mpc (This is the typical void diameter / filament-node-filament spacing) Superclusters (largest coherent structures): ~100 h^-1 Mpc ================================================================================ 2. GALAXY CLUSTERS ================================================================================ NUMBER OF GALAXIES: Galaxy groups: 10-50 galaxies (most common gravitationally bound systems) Poor clusters: 50-100 galaxies Rich clusters: 100-1,000 galaxies (typical "galaxy cluster") Massive clusters: 1,000-10,000+ galaxies (e.g., Coma Cluster ~10,000) CLUSTER GEOMETRY: Intrinsic shape: PREDOMINANTLY PROLATE (elongated), not spherical. - Prolate preferred by 60-76% of observed clusters - Triaxial models fit better than pure oblate or prolate - Inner regions: more spherical - Outer regions: more triaxial, trending oblate - Major axis tends to ALIGN WITH the filament connecting it to neighbors - This alignment is a key prediction of hierarchical structure formation Source: Sereno+ (2006) ApJ 645, 170; Cooray (1999) MNRAS 313, 783 CLUSTER SIZE: Virial radius: 1-3 Mpc Typical diameter: ~20 million light-years (~6 Mpc) for rich clusters Mass: 10^14 - 10^15 M_sun (including dark matter) COORDINATION NUMBER (nearest-neighbor clusters): From the cosmic web topology, a typical node connects to 3-6 filaments. This implies a COORDINATION NUMBER of approximately 3-6 for cluster nodes. (This is analogous to graph-theory degree in the cosmic web skeleton.) More specifically, from connectivity studies (Codis+ 2018, MNRAS 479, 973): - Mean connectivity (filaments per node) depends on peak height - Low-mass halos: ~3 filaments - High-mass halos (massive clusters): ~5-6 filaments - The connectivity scales with mass as kappa ~ 3 + 2*nu (where nu = peak height) SUPERCLUSTERS: Definition: The largest non-percolating structures; basins of gravitational attraction containing multiple clusters. Typical composition: 3-10 galaxy clusters + dozens of groups Larger superclusters: tens of clusters (e.g., Shapley: ~25 clusters) Size: 50-200 Mpc diameter Examples: - Virgo Supercluster: ~100 groups/clusters, 45 Mpc diameter - Laniakea Supercluster: ~300-500 groups/clusters, ~120 Mpc diameter, ~100,000 galaxies - Shapley Supercluster: ~25 rich clusters, ~200 Mpc away ================================================================================ 3. GALAXY MORPHOLOGY (Hubble Sequence) ================================================================================ SPIRAL ARM NUMBER (Galaxy Zoo 2 / Hart+ 2016, 2017): Survey: ~18,000 SDSS spiral galaxies classified by citizen scientists Classification options: 1, 2, 3, 4, 5+, or "can't count arms" Approximate distribution of classifiable spirals: Grand design (2-armed): ~30% of disc galaxies Multi-armed (3-4 arms): ~32% of disc galaxies Flocculent (fragmented): ~38% of disc galaxies (many short arm fragments) Key findings: - 2-armed spirals dominate in HIGHER-DENSITY environments - Multi-armed fraction INCREASES with stellar mass - 2-armed galaxies convert gas to stars ~10% more efficiently - 3-armed spirals are genuinely RARE (small fraction of multi-armed class) - 4-armed structure exists (the Milky Way has 4 major arms: Scutum-Centaurus, Perseus, Norma, Sagittarius) - 5+ arms: typically classified as flocculent rather than true grand design Source: Hart+ (2016) MNRAS 461, 3663; Hart+ (2017) MNRAS 468, 1850 LOGARITHMIC SPIRALS AND PITCH ANGLE: Spiral arms are well-modeled as LOGARITHMIC SPIRALS (r = a*e^(b*theta)) Pitch angle = arctan(b) = constant for a true logarithmic spiral Observed pitch angles: - Full range: 5-40 degrees - Majority: 10-25 degrees - Typical Sa (tightly wound): 5-15 degrees - Typical Sc (loosely wound): 15-30 degrees - Milky Way: ~12 degrees THE GOLDEN RATIO CONNECTION: Golden spiral pitch angle = 17.03239 degrees (The golden spiral grows by factor phi per 90 degrees of rotation) This falls squarely WITHIN the observed range (10-25 degrees) for typical spiral galaxies. However: - Galaxies are NOT precisely golden spirals - Pitch angle VARIES with radius (unlike ideal logarithmic spirals) - The 17-degree value is a coincidence within the range, not a tight fit - No published evidence for a preferred pitch angle at exactly phi Source: Davis+ (2012) ApJS 199, 33; Savchenko+ (2013) MNRAS 436, 1074 ELLIPTICAL GALAXIES (E0-E7): Classification: E_n where n = 10*(1 - b/a), a and b are projected axes E0: circular (spherical projection) E3: moderate ellipticity E7: maximum observed ellipticity (most flattened) IMPORTANT: E4 through E7 galaxies are now known to be largely MISCLASSIFIED LENTICULAR GALAXIES (S0) seen at various inclinations. True ellipticals top out around E4-E5. Intrinsic shapes: - TRIAXIAL models fit best (neither purely oblate nor prolate) - Bright/giant ellipticals: rounder, more triaxial - Faint/dwarf ellipticals: more flattened, more oblate - Distribution of intrinsic flattenings is broad and possibly BIMODAL Source: Binney (1978); Padilla & Strauss (2008); Krajnovic+ (2018) BARRED VS UNBARRED SPIRALS: Fraction barred: ~65-70% in the local universe (z~0) Fraction barred at z~0.8: ~20% (bars are a LATE development) What determines bar formation: - Disc stability (Toomre Q parameter): lower Q -> bars form more easily - Disc-to-halo mass ratio: disc-dominated galaxies bar more readily - Halo shape: prolate halos -> earlier bar formation; oblate halos -> delayed - Gas fraction: gas can arrest bar slowdown ("bar locking") - Angular momentum exchange with dark matter halo reinforces bars Source: Sheth+ (2008) ApJ 675, 1141; Nair & Abraham (2010) ================================================================================ 4. SOLAR SYSTEM ORGANIZATION ================================================================================ TITIUS-BODE LAW: Formula: a_n = 0.4 + 0.3 * 2^n (in AU), where n = -inf, 0, 1, 2, ... Planet | Predicted (AU) | Actual (AU) | Ratio ------------|----------------|-------------|------ Mercury | 0.4 | 0.39 | 1.03 Venus | 0.7 | 0.72 | 0.97 Earth | 1.0 | 1.00 | 1.00 Mars | 1.6 | 1.52 | 1.05 (Asteroids) | 2.8 | 2.77* | 1.01 Jupiter | 5.2 | 5.20 | 1.00 Saturn | 10.0 | 9.54 | 1.05 Uranus | 19.6 | 19.19 | 1.02 Neptune | 38.8 | 30.07 | 1.29 **FAILS** (* Ceres at 2.77 AU) Status: Now considered a mathematical coincidence / selection effect. Works for inner solar system, fails for Neptune. Has been tested on exoplanet systems — shows some statistical validity but is NOT a physical law. May reflect resonance-clearing during planet formation. Source: Exponential distance relation in MNRAS 538, 2730 (2025) ORBITAL RESONANCES (most common in solar system): Dominant resonance ratios: (3:2), (5:3), (2:1), (5:2), (3:1) Examples in our solar system: - Neptune:Pluto = 3:2 (plutinos trapped in this resonance) - Jupiter:Saturn = 5:2 (the "Great Inequality") - Io:Europa:Ganymede = 4:2:1 (Laplace resonance — remarkable!) - Titan:Hyperion = 4:3 - Mimas:Tethys = 2:1 - Enceladus:Dione = 2:1 The Io:Europa:Ganymede 4:2:1 Laplace resonance is one of the most remarkable geometric patterns in the solar system — three bodies locked in mutual resonance creating a repeating geometric pattern. KIRKWOOD GAPS (asteroid belt depletions at Jupiter resonances): Gap | Resonance | Semi-major axis (AU) | Period ratio --------|-----------|---------------------|------------- Inner | 4:1 | ~2.06 | 4 Jupiter orbits per 1 asteroid Major | 3:1 | ~2.50 | 3:1 Strong | 5:2 | ~2.82 | 5:2 Narrow | 7:3 | ~2.95 | 7:3 Outer | 2:1 | ~3.28 | 2:1 The 3:1 gap at 2.5 AU divides the belt into inner and outer zones. The 5:2 gap at 2.82 AU further divides the outer zone. Note: ONLY INTEGER RATIOS CREATE GAPS. This is because orbital resonance requires commensurable periods — a geometric/number-theoretic constraint, not just a dynamical one. Source: Kirkwood (1866); Wisdom (1982); JPL Small Body Dynamics LAGRANGE POINTS: Number: exactly FIVE (L1 through L5) for any two-body gravitational system. This "5" is a topological necessity, not tunable. Geometry: L1, L2, L3: COLLINEAR points along the line connecting the two masses - L1: between the two bodies - L2: beyond the smaller body (away from the larger) - L3: beyond the larger body (opposite side from smaller) L4, L5: EQUILATERAL TRIANGLE vertices - Each forms an equilateral triangle with the two massive bodies - 60 degrees ahead of (L4) and behind (L5) the smaller body Stability: L1, L2, L3: UNSTABLE (saddle points in the effective potential) - Objects drift away on ~23-day timescale for Sun-Earth system L4, L5: STABLE (for mass ratio M1/M2 > 24.96) - Sun-Earth: stable (Jupiter's L4/L5 hold ~7000 Trojan asteroids) - Earth-Moon: marginally stable The equilateral triangle geometry of L4/L5 is exact and derives from the three-body problem — it is NOT an approximation. ================================================================================ 5. STELLAR STRUCTURE ================================================================================ BINARY STAR SYSTEMS (Raghavan+ 2010, definitive survey): Sample: 454 solar-type stars (F6-K3 dwarfs) within 25 pc System multiplicity: Single stars: 56% +/- 2% Binary (double): 33% +/- 2% Triple: 8% +/- 1% Quadruple+: 3% +/- 1% Companion fraction (companions per primary): 0.46 +/- 0.02 So roughly HALF of solar-type stars have at least one companion. Mass dependence: - O/B stars (massive): multiplicity fraction ~70-80%, triple fraction ~50-70% - Solar-type (FGK): multiplicity fraction ~44% - M dwarfs (low mass): multiplicity fraction ~26% KEY: Multiplicity INCREASES with stellar mass. Massive stars are almost never single. Source: Raghavan+ (2010) ApJS 190, 1 (454-star survey) Duchene & Kraus (2013) ARA&A 51, 269 (review) TRIPLE STAR SYSTEMS: ~8% of solar-type systems are triples (Raghavan+ 2010) For massive stars: triple fraction reaches 50-70% Architecture: almost always HIERARCHICAL — a close binary orbited by a distant third star. Non-hierarchical triples are unstable. Even low triple fractions are dynamically important: triple interactions occur as often as binary-single encounters in clusters. STAR CLUSTERS: GLOBULAR CLUSTERS: Shape: SPHERICAL (strongly self-gravitating, pressure-supported) Age: ~10-13 billion years (among the oldest objects in the universe) Number of stars: 10,000 to several MILLION (typically 100,000-1,000,000) Size: 10-30 light-years across (core radius ~1-5 pc) Location: in the HALO of galaxies, orbiting the galactic center Milky Way: ~150 known globular clusters Binary fraction in cores: only ~5-10% (dynamical destruction) Stellar population: Population II (metal-poor, old) OPEN CLUSTERS: Shape: IRREGULAR (loosely bound, eventually dispersed by tidal forces) Age: 10 million to ~1 billion years (young) Number of stars: ~100-1,000 (typically a few hundred) Size: up to ~30 light-years across Location: in the DISC of the galaxy (near star-forming regions) Milky Way: ~1,000 known open clusters (actual number much higher) Stellar population: Population I (metal-rich, young) KEY CONTRAST: Globular: spherical, old, 10^5-10^6 stars, halo, metal-poor Open: irregular, young, 10^2-10^3 stars, disc, metal-rich ================================================================================ 6. VOID FRACTION AND DISTRIBUTION ================================================================================ VOLUME FRACTION: Voids occupy 60-80% of the universe's volume. Most commonly cited: ~77-80%. Some analyses push this to ~90% depending on density threshold used. The void fraction depends critically on the definition: - "Below mean density": ~80% - "Below 20% of mean density": ~50-60% - "Below 10% of mean density": ~30-40% VOID SIZES: Small voids: 10-30 Mpc effective radius (most common) Typical voids: 30-80 Mpc diameter Large voids: up to ~100 Mpc diameter Supervoids: 300-500+ Mpc (e.g., Eridanus supervoid, Bootes void) Examples: - Bootes Void: ~100 Mpc diameter (one of the largest known) - Local Void: ~45 Mpc diameter - Eridanus Supervoid: ~500 Mpc (the Cold Spot) VOID SHAPES: Initial: highly ELLIPSOIDAL (reflecting the Gaussian random field peaks) Evolution: voids become MORE SPHERICAL over time as they expand Late time: merge to form complex, connected, non-spherical supervoid structures The "bubble theorem": isolated expanding voids approach spherical shape regardless of initial conditions. This is an attractor geometry. VOID INTERNAL DENSITY: Even the "emptiest" voids contain >15% of the average cosmic matter density. Voids are never truly empty — they contain tenuous galaxy populations, diffuse gas, and dark matter. VOID HIERARCHY (void-in-void): Voids exhibit hierarchical structure — smaller voids nested within larger underdense regions, connected by a watershed-like topology. Analysis tools construct hierarchical trees of void mergers. Source: Pan+ (2012) MNRAS 421, 926 (SDSS DR7 void catalog) Sutter+ (2012) ApJ 761, 44 (VIDE toolkit) ================================================================================ 7. THE COSMOLOGICAL PRINCIPLE ================================================================================ HOMOGENEITY SCALE: Below ~100-115 Mpc: the universe is INHOMOGENEOUS - Filamentary, anisotropic, clustered - Galaxy clusters, superclusters, voids, walls all visible - Fractal-like structure with dimension D ~ 2.0-2.5 Above ~100-115 Mpc: the universe is HOMOGENEOUS AND ISOTROPIC - Averaged over larger volumes, density fluctuations become negligible - This is confirmed by SDSS galaxy surveys and Planck CMB data - The CMB is uniform to 1 part in 100,000 (delta_T/T ~ 10^-5) The transition scale: ~100 Mpc (sometimes quoted as 70-120 Mpc) Source: Hogg+ (2005) ApJ 624, 54 (SDSS homogeneity test) Scrimgeour+ (2012) MNRAS 425, 116 (WiggleZ survey) EVIDENCE: - SDSS galaxy distribution: homogeneous above ~70 h^-1 Mpc - Planck CMB: isotropic to 10^-5 across the full sky - Baryon Acoustic Oscillations: characteristic scale 150 Mpc visible as a statistical excess in the galaxy correlation function - BAO scale serves as a "standard ruler" — only works if the universe is statistically homogeneous on those scales CHALLENGES: Some observed structures push the homogeneity scale: - Sloan Great Wall: ~420 Mpc long - Hercules-Corona Borealis Great Wall: ~3,000 Mpc (if confirmed) - These are "statistical structures" — may not violate homogeneity if they represent low-amplitude density fluctuations on large scales BARYON ACOUSTIC OSCILLATION (BAO) SCALE: Characteristic scale: ~150 Mpc (comoving) This is the sound horizon at recombination — the maximum distance a sound wave could travel in the primordial plasma before it froze out. Visible as a bump in the galaxy correlation function at ~150 Mpc separation. ================================================================================ 8. CROSS-SCALE GEOMETRIC SUMMARY TABLE ================================================================================ Scale | Size | Dominant Geometry | Key Number -----------------|-------------|--------------------------|------------ Stellar binary | ~AU | 2-body orbit | 2 (pair) Lagrange points | ~AU | equilateral triangles | 5 points Kirkwood gaps | ~AU | integer resonance ratios | 3:1, 5:2, 2:1 Planetary system | ~50 AU | coplanar ellipses | ~8 (planets) Open cluster | ~10 pc | irregular | 100-1000 stars Globular cluster | ~10 pc | SPHERICAL | 10^5-10^6 stars Spiral galaxy | ~30 kpc | logarithmic spirals | 2 arms (grand design) Galaxy group | ~1 Mpc | irregular | 10-50 galaxies Galaxy cluster | ~3-6 Mpc | PROLATE (60-76%) | 100-1000 galaxies Supercluster | ~50-200 Mpc| flattened / filamentary | 3-10 clusters Filament | 30-100 Mpc | cylindrical (1D+) | ~2-5 Mpc wide Wall/sheet | ~100 Mpc | planar (2D) | ~5-10 Mpc thick Void | 30-100 Mpc | spherical (attractor) | 77-80% of volume Cosmic web cell | ~100 Mpc | foam-like topology | ~100 Mpc scale BAO scale | 150 Mpc | spherical shell | 150 Mpc (fixed) Homogeneity | >100 Mpc | isotropic, featureless | (no structure) ================================================================================ 9. NUMBERS POTENTIALLY RELEVANT TO {2,3} CIPHER ================================================================================ The following observed numbers may connect to the {2,3} packing framework: TWOS (binary/pair geometry): - Binary stars: the dominant multi-star configuration (33% of systems) - Grand design spirals: 2 arms (the dominant arm count) - Kirkwood gaps: 2:1 resonance is a major gap - L4/L5: equilateral triangle = 2 triangular stable points THREES: - Triple stars: 8% of solar-type systems (the next after binaries) - 3:1 resonance: the MAJOR Kirkwood gap (strongest depletion) - 3:2 resonance: the most common orbital resonance (Neptune:Pluto) - Laplace resonance: 4:2:1 = chain of 2:1 resonances (3 bodies) - Cluster connectivity: ~3 filaments per low-mass node - Superclusters: 3-10 clusters (lower bound is 3) FIVES: - Lagrange points: exactly 5 (topological necessity) - Cluster connectivity: ~5-6 filaments per massive node - 5:2 resonance: strong Kirkwood gap - 5:3 resonance: one of the dominant planetary resonances - Galaxy Zoo: 5+ arms is a classification category GEOMETRY PATTERNS: - Voids evolve TOWARD spheres (the "bubble theorem" attractor) - Globular clusters are SPHERICAL - Galaxy clusters are PROLATE (elongated along filaments) - Filaments are CYLINDRICAL - Walls are PLANAR - This is a progression: 0D (nodes) -> 1D (filaments) -> 2D (walls) -> 3D (voids) The cosmic web naturally organizes into structures of EVERY dimensionality. VOID FRACTION (~80%): - If the universe is ~80% void by volume, then ~20% is structure - 20% = 1/5 of the volume contains essentially all the mass - The structure is concentrated in a 2D+1D network (walls + filaments) that encloses 3D voids ================================================================================ SOURCES ================================================================================ COSMIC WEB: - Tempel+ (2014) "Detecting filamentary pattern in the cosmic web" MNRAS 438, 3465 - Einasto+ (2022) "Maximum extent of filaments and sheets" MNRAS 519, 3227 - Cautun+ (2014) "Evolution of the cosmic web" MNRAS 441, 2923 - Codis+ (2018) "On the connectivity of the cosmic web" MNRAS 479, 973 VOIDS: - Pan+ (2012) "Cosmic voids in SDSS DR7" MNRAS 421, 926 - Sutter+ (2012) "A public void catalog" ApJ 761, 44 GALAXY CLUSTERS: - Sereno+ (2006) "Are clusters oblate or prolate?" ApJ 645, 170 - Cooray (1999) "Galaxy clusters: oblate or prolate?" MNRAS 313, 783 GALAXY MORPHOLOGY: - Hart+ (2016) "Galaxy Zoo: spiral arm demographics" MNRAS 461, 3663 - Hart+ (2017) "Galaxy Zoo: star formation vs spiral arm number" MNRAS 468, 1850 - Davis+ (2012) "Galactic spiral arm pitch angle" ApJS 199, 33 - Sheth+ (2008) "Evolution of the bar fraction" ApJ 675, 1141 STELLAR MULTIPLICITY: - Raghavan+ (2010) "Multiplicity of solar-type stars" ApJS 190, 1 - Duchene & Kraus (2013) "Stellar multiplicity" ARA&A 51, 269 COSMOLOGICAL PRINCIPLE: - Hogg+ (2005) "Cosmic homogeneity from SDSS" ApJ 624, 54 - Scrimgeour+ (2012) "The WiggleZ survey: homogeneity" MNRAS 425, 116 PLANETARY DYNAMICS: - Kirkwood gap: Wikipedia + JPL Small Body Database - Titius-Bode: Lara+ (2025) MNRAS 538, 2730 ================================================================================ END OF FILE ================================================================================