Unit V · Executive Geometric Summary

Core Symbolic Constructions & Laws (Epistola 1597 & Harmonices Mundi Book V, 1619)

  1. The Kepler Right Triangle over $\mathbb{Q}(\sqrt{\phi})$: Constructed purely by straightedge and compass from $O(0,0)$ and $A(1,0)$ in geometor.kepler.harmonices_geometry.construct_kepler_right_triangle (@proposition("Epistola.1597.KeplerTriangle"), auto_intersect=False) over geometor.model.get_kepler_field() (quartic minimal polynomial $x^4 - x^2 - 1 = 0$):

Exact Invariants of the Kepler Right Triangle $\triangle OAB$

Geometric Element

Symbolic Length

Exact Quadrance $Q$

Defining Property

Base leg $OA$

$1$

$1$

Given unit segment

Vertical leg $OB$

$\sqrt{\phi}$

$\phi = \frac{1 + \sqrt{5}}{2}$

Geometric mean $\sqrt{1 \cdot \phi}$ (Eucl. VI.13)

Hypotenuse $AB$

$\phi$

$\phi^2 = \phi + 1 = \frac{3 + \sqrt{5}}{2}$

$Q(OA) + Q(OB) = 1 + \phi = \phi^2$ (Eucl. I.47 & II.11)

Altitude $OH \perp AB$

$\frac{1}{\sqrt{\phi}}$

$\phi^{-1} = \phi - 1 = \frac{\sqrt{5} - 1}{2}$

Divides $AB$ into $BH = 1$ and $HA = \phi^{-1}$

  1. Planetary Extremal Consonances & Kepler’s Third Law: Verified symbolically and against Kepler’s Book V tables in analyze_planetary_harmonies and verify_kepler_third_law (every naked-eye Tychonic orbit satisfies $(T/T_\oplus)^2 = (a/a_\oplus)^3$ to better than $0.89\%$):

Six-Planet Consonances & Sesquialterate Ratio (Harmonices Mundi Book V)

Planet

$\omega_{\text{aph}}$ ($''/\text{day}$)

$\omega_{\text{per}}$ ($''/\text{day}$)

Harmonic Consonance

$a$ (AU, Kepler)

$T$ (days, Kepler)

Saturn

$106''$ ($1'46''$)

$135''$ ($2'15''$)

$4 : 5$ (Major Third)

$9.510$

$10759.2$

Jupiter

$270''$ ($4'30''$)

$330''$ ($5'30''$)

$5 : 6$ (Minor Third)

$5.200$

$4332.6$

Mars

$1574''$ ($26'14''$)

$2281''$ ($38'01''$)

$2 : 3$ (Perfect Fifth)

$1.524$

$686.98$

Earth

$3433''$ ($57'03''$)

$3673''$ ($61'13''$)

$15 : 16$ (Semitone mi–fa)

$1.000$

$365.25$

Venus

$5690''$ ($94'50''$)

$5867''$ ($97'47''$)

$24 : 25$ (Chromatic Diesis)

$0.724$

$224.70$

Mercury

$9840''$ ($164'00''$)

$23040''$ ($384'00''$)

$5 : 12$ (Octave + Minor Third)

$0.388$

$87.97$