Umair Fiaz — notebook https://umairfiaz.com/ Notes from a kept notebook. Fri, 10 Jul 2026 07:24:42 +0000 The graticule https://umairfiaz.com/notes/the-graticule/ https://umairfiaz.com/notes/the-graticule/ Sun, 05 Jul 2026 00:00:00 +0000

Demo note — delete me. I exist to prove the machinery: math, code, wikilinks, backlinks, footnotes, tags.

Every measurement needs a reference frame. On a photographic plate it is the réseau; in an expanding universe it is the comoving grid. The scale factor a(t)a(t) carries all the dynamics, and the Friedmann equation ties it to what the universe contains:

(a˙a)2=8πG3ρ−ka2+Λ3\left(\frac{\dot a}{a}\right)^{2} = \frac{8\pi G}{3}\rho - \frac{k}{a^{2}} + \frac{\Lambda}{3}

Distances follow from integrating along the grid1:

from scipy.integrate import quad
from math import sqrt

C_KM_S = 299_792.458

def comoving_distance(z, h0=67.4, om=0.315):
    """Flat ΛCDM, matter + Λ only."""
    integrand = lambda zp: 1 / sqrt(om * (1 + zp) ** 3 + (1 - om))
    return (C_KM_S / h0) * quad(integrand, 0, z)[0]

This note links to The plate archive — open it and you’ll find this note listed under its backlinks. A link to a note that doesn’t exist yet looks like belief dynamics: planted, not grown.

  1. Footnotes render below a hairline. This one exists to prove it. ↩

]]> The plate archive https://umairfiaz.com/notes/the-plate-archive/ https://umairfiaz.com/notes/the-plate-archive/ Sun, 05 Jul 2026 00:00:00 +0000

Demo note — delete me. I exist so The graticule has something to link to — check my margin for the backlink count.

Harvard’s plate stacks hold half a million glass photographs of the sky, annotated in ink by the computers who measured them. A notebook is the same instrument at personal scale: expose, annotate, file, revisit.

Inline math also works: the Hubble rate is H(z)=H0Ωm(1+z)3+ΩΛH(z) = H_0 \sqrt{\Omega_m (1+z)^3 + \Omega_\Lambda}, set in KaTeX with variables italic and everything else upright — as it should be.

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