Universal Golden Gnomon & Dioptra Mountain Tunnel

Self-Similar Accretion in Q(φ) (Definitiones §58) & Subterranean Geodesy (Dioptra Ch. 15) — Codex Reference: Definitiones, Catoptrica & Dioptra: Genetic Axiomatics, Universal Gnomon, Shortest-Path Optics & Mountain Tunneling

1. Universal Golden Gnomon (heronian-golden-gnomon.svg)

Realizing Hero's Definition 58 (“a gnomon is anything which, added to any number or figure, makes the whole similar to that to which it was added”), adjoining the unit square gnomon ABCD to the golden seed BEFC via compass swing (MC) creates the whole rectangle AEFD with identical aspect quadrance \(\phi^2 = \frac{3 + \sqrt{5}}{2}\):

Hero Definitiones §58 — Universal Gnomon Realized on the Golden Rectangle in Q(phi) ( M C ): y**2 + (x - 1/2)**2 - 5/4 = 0 ( E B ): y**2 + (x - sqrt(5)/2 - 1/2)**2 - (-1/2 + sqrt(5)/2)**2 = 0 [ A B ]: y = 0 [ D E ]: x = 0 [ G H ]: x - 1 = 0 [ D C ]: y - 1 = 0 [ P Q ]: x - sqrt(5)/2 - 1/2 = 0 [ U V ]: y - sqrt(5)/2 + 1/2 = 0 / A E /: |seg| = 1/2 + sqrt(5)/2 / B E /: |seg| = -1/2 + sqrt(5)/2 A = (0, 0) B = (1, 0) D = (0, 1) C = (1, 1) M = (1/2, 0) E = (1/2 + sqrt(5)/2, 0) F = (1/2 + sqrt(5)/2, 1) G = (1/2 + sqrt(5)/2, -1/2 + sqrt(5)/2) H = (1, -1/2 + sqrt(5)/2) E1 = (1/2 - sqrt(5)/2, 0) A B D C M E F G H E1

2. Two-Portal Mountain Tunneling (dioptra-mountain-tunnel.svg)

Realizing Dioptra Chapter 15 (Schöne 1903, pp. 238–240), orthogonal dioptra stations E, Z, H, G, K, M determine the subterranean displacement \((\Delta x, \Delta y) = (12, 5)\) of BND, aligning exterior sighting triangles BOX and DPQ collinearly along XQ:

Hero Dioptra Ch. 15 — Two-Portal Mountain Tunneling by Orthogonal Traverse [ E B ]: y = 0 [ E Z ]: x + 2 = 0 [ Z H ]: y + 3 = 0 [ H G ]: x - 16 = 0 [ G K ]: y - 8 = 0 [ C F ]: x - 14 = 0 [ D J ]: y - 5 = 0 [ D N ]: x - 12 = 0 [ X Q ]: -10*x + 24*y = 0 / B D /: |seg| = 13 / B E /: |seg| = 2 / E Z /: |seg| = 3 / Z H /: |seg| = 18 / H G /: |seg| = 11 / G K /: |seg| = 2 / K M /: |seg| = 3 / M D /: |seg| = 2 B = (0, 0) D = (12, 5) E = (-2, 0) Z = (-2, -3) H = (16, -3) G = (16, 8) K = (14, 8) M = (14, 5) N = (12, 0) O = (-6, 0) X = (-6, -5/2) P = (18, 5) Q = (18, 15/2) B D E Z H G K M N O X P Q

Executable Python API

import sympy as sp
        from geometor.heron import PHI, construct_dioptra_tunnel, construct_heronian_gnomon

        _, gnom = construct_heronian_gnomon()
        assert sp.simplify(gnom["whole_aspect_quadrance"] - PHI**2) == 0
        assert sp.simplify(gnom["seed_aspect_quadrance"] - PHI**2) == 0

        _, diop = construct_dioptra_tunnel()
        assert diop["east_displacement"] == 12 and diop["north_displacement"] == 5
        assert diop["tunnel_length"] == 13