GC-03 · Squares, Lattice Grids & Golden Crosses

Inscribed squares, 2×2 and 3×3 lattice grids, Suppa’s incircles, the Greek Cross, and rotated squares

Orthogonal squares and \(2 \times 2\) or \(3 \times 3\) lattice grids naturally furnish \(1 : 2\) and \(1 : 3\) right triangles whose hypotenuses have quadrances \(5\) and \(10\). Through circumcircles, triangle incenters, and \(30^\circ\) rotations, geometers including Tran Quang Hung, Ercole Suppa, Bùi Quang Tuân, Alexandre Borovik, and John Arioni uncovered rich golden chains across elementary square configurations.

Cataloged Constructions (8)

GC-03.01 · Tran Quang Hung’s Square Inscribed in a Right Isosceles Semicircle

Let ABC be a right isosceles triangle inscribed in a semicircle of cO on diameter AB, and let square DEGF sit on AB with vertices D and E on legs AC and BC. Extending top side DE to meet circle cO at Q and P divides DP in the golden ratio: \(|DE|/|EP| = |DP|/|DE| = \phi\).

GC-03.02 · Tran Quang Hung’s Square Midpoint Circle Division

Let ABCD be a square with center F and side midpoints M, N, P, Q. The circle cKA on diameter AP intersects midline QN at X between F and N such that \(|FX|/|XN| = \phi\), while chord AX cuts MP at Z with \(|MZ|/|ZF| = \phi\).

GC-03.03 · Nguyen Thanh Dung’s 2×2 Grid Compass Arc

On a \(2 \times 2\) square grid P1P2P3P4 with horizontal midline AB, center O, and top midpoint D, extend AB by one grid unit to C. Circle cCD centered at C through D cuts AB at M in the golden ratio: \(|MB|/|AM| = |AB|/|MB| = \phi\).

GC-03.04 · Tran Quang Hung & Elliot McGucken’s 3×3 Grid Circle

In a \(3 \times 3\) square grid S1AS3S4 centered at O, let C be the upper-right vertex of the central grid cell, A the outer upper-right corner, and T the adjacent trisection vertex on the top edge. Circle cOT cuts diagonal segment CA at B such that \(|CB|/|BA| = \phi\).

GC-03.05 · Tran Quang Hung’s 3×3 Grid Incenter Division

In a \(3 \times 3\) grid AQ1Q2Q3 on \([0,3]^2\), let S \(= (1,1)\) and T \(= (2,2)\) be opposite corners of the central unit cell, and let U \(= (3,1)\) and V \(= (1,3)\). The incenter P of isosceles triangle AUV divides diagonal ST of the central cell in the golden ratio: \(|PT|/|SP| = \phi\).

GC-03.06 · Ercole Suppa’s Square Midpoint Inradii Ratio (2015)

In square ABCD, let M be the midpoint of side AD. The inradii \(R\) and \(r\) of triangles \(\triangle BCM\) (incircle cEH) and \(\triangle CDM\) (incircle cFP) satisfy \(R / r = \phi\), and segment EF joining their incenters is divided by CM at K with \(|EK|/|KF| = \phi\).

GC-03.07 · Bùi Quang Tuân & Alexandre Borovik’s Greek Cross & Square

Inscribe a symmetrical five-square Greek cross CROSS and a concentric axis-aligned square S1S2S3S4 in the same circle cO. Along any inner arm line AC of the cross, the square edge S1S2 cuts AC at B such that \(|BC|/|AB| = \phi\), while \(|BD|/|CD| = \phi\).

GC-03.08 · John Arioni’s 30°-Rotated Unit Square & Diagonal Circle

Rotate unit square DGHJ clockwise by \(30^\circ\) around vertex D to obtain congruent square DCBA. Circle cAC centered at A through opposite vertex C intersects line DG at E and F such that \(|DG|/|GE| = |DE|/|DG| = \phi\).