# Perfect Cuboid Float Geometry Point — G2 Seed

## Exact geometry

$$
(a,b,c)=(448,495,840)
$$

with space diagonal

$$
g=1073.
$$

The face diagonals are

$$
d_{ab}=\sqrt{445729}\approx 667.629388208757632,
$$

$$
d_{ac}=952,
$$

$$
d_{bc}=975.
$$

All four geometric identities hold exactly:

$$
448^2+495^2=d_{ab}^2,
$$

$$
448^2+840^2=952^2,
$$

$$
495^2+840^2=975^2,
$$

$$
448^2+495^2+840^2=1073^2.
$$

Thus this is an exact real-valued cuboid. Its only failure relative to the classical Perfect Cuboid condition is arithmetic:

$$
d_{ab}\notin\mathbb Z.
$$

## X-face coordinates

### Face $ab$

$$
\theta_{ab}
=
2\arctan\frac{448}{495}
\approx 84.293371949754^\circ.
$$

### Face $ac$

$$
\theta_{ac}
=
2\arctan\frac{8}{15}
\approx 56.144973871706^\circ.
$$

with coupling

$$
(\kappa_a,\kappa_c)
=
\left(\frac8{17},\frac{15}{17}\right).
$$

### Face $bc$

$$
\theta_{bc}
=
2\arctan\frac{33}{56}
\approx 61.020474812231^\circ.
$$

with coupling

$$
(\kappa_b,\kappa_c)
=
\left(\frac{33}{65},\frac{56}{65}\right).
$$

## Normalized location

With

$$
g=1,
$$

the shape point is

$$
\left(
\frac{448}{1073},
\frac{495}{1073},
\frac{840}{1073}
\right)
\approx
(0.417520969245,0.461323392358,0.782851817335).
$$

This point lies exactly on

$$
x^2+y^2+z^2=1.
$$

It is therefore a useful Geometry Branch seed: two face-X channels are already rational/integer Pythagorean closures, and only one face-diagonal arithmetic channel remains open.