Magic Squaresproof atlas
← Back to the partial-configurations series

Partial square configurations · 3.7

7/9 Patterns: A Complete Trigonometric Atlas

Up to rotations and reflections, seven square entries occur in eight positional patterns. For every orbit, four quadrics collapse to one equation in three rational angles; all eight models are derived below, together with the exact scope of their completeness.

1. Setup and the eight orbits

The general coordinate form of a magic square of order 3 is

(ABCDEFGHJ)=(E+xE−x+yE−yE−x−yEE+x+yE+yE+x−yE−x). \begin{pmatrix} A&B&C\\D&E&F\\G&H&J \end{pmatrix} = \begin{pmatrix} E+x&E-x+y&E-y\\ E-x-y&E&E+x+y\\ E+y&E+x-y&E-x \end{pmatrix}.

For a selected seven-entry pattern S, write P=p² for every P∈S. The seven linear forms in E,x,y have rank 3, so their left kernel has dimension 7−3=4. Thus every 7/9 pattern is defined by exactly four independent homogeneous quadrics among its seven roots.

dim⁡ker⁡LS T=7−rank⁡LS=4. \dim\ker L_S^{\,T}=7-\operatorname{rank}L_S=4.

Orbit-count theorem

There are exactly eight D4 orbits of seven-entry patterns.

The complement has two entries, so it is enough to classify unordered pairs of positions. By Burnside's lemma, the identity fixes C(9,2)=36 pairs, the ±90° rotations fix none, and the 180° rotation fixes the four opposite pairs. Each of the four reflections has three fixed entries and three exchanged pairs, hence fixes C(3,2)+3=6 two-element sets.

N7/9=N2/9=36+0+4+0+4⋅68=8. N_{7/9}=N_{2/9} =\frac{36+0+4+0+4\cdot6}{8}=8.

The eight complement types are an adjacent corner-edge pair, two corners on one side, a corner with the center, a knight move, opposite corners, two edge cells adjacent to a corner, an edge cell with the center, and opposite edge cells. Their orbit sizes are 8,4,4,8,2,4,4,2, summing to all 36 pairs.

2. Universal one-angle coordinates

Let t=n/m=tan θ∈P¹(ℚ). Introduce four rational functions:

Ct=1−t21+t2=cos⁡2θ,St=2t1+t2=sin⁡2θ,Lt=St−Ct=t2+2t−11+t2,Rt=St+Ct=1+2t−t21+t2. \begin{aligned} C_t&=\frac{1-t^2}{1+t^2}=\cos 2\theta,& S_t&=\frac{2t}{1+t^2}=\sin 2\theta,\\ L_t&=S_t-C_t=\frac{t^2+2t-1}{1+t^2},& R_t&=S_t+C_t=\frac{1+2t-t^2}{1+t^2}. \end{aligned}

They simultaneously parametrize a red progression of squares and a yellow equality of two norms:

Lt2+Rt2=2,Ct2+St2=1, L_t^2+R_t^2=2,\qquad C_t^2+S_t^2=1,X2+Z2=2Y2⟸(X,Z)=Y(Lt,Rt),P2+Q2=U2+V2⟸(UV)=(Ct−StStCt)(PQ). \begin{aligned} X^2+Z^2=2Y^2 &\Longleftarrow (X,Z)=Y(L_t,R_t),\\ P^2+Q^2=U^2+V^2 &\Longleftarrow \binom UV= \begin{pmatrix}C_t&-S_t\\S_t&C_t\end{pmatrix}\binom PQ. \end{aligned}

The first line is the rational parametrization of the conic of arithmetic progressions of squares. Its normalized oriented difference is

δθ:=Rt2−1=1−Lt2=sin⁡4θ=4dir⁡(m,n)=4t(1−t2)(1+t2)2. \delta_\theta:=R_t^2-1=1-L_t^2 =\sin4\theta =4\operatorname{dir}(m,n) =\frac{4t(1-t^2)}{(1+t^2)^2}.

Thus, in the convention used on this site, the angular coordinate satisfies δθ=4dir(m,n). The factor 4 matters: for a progression of squares, Δ/V=4dir(m,n)=δθ.

The second line is a rational rotation of the circle. Both parametrizations are complete over Q in the projective sense: t=∞ supplies the point missing from the ordinary affine chart, while root signs allow the determinant +1 component of the orthogonal transformation to be chosen.

The inverse parameter is explicit. For a normalized red progression l²+r²=2, put S=(l+r)/2 and C=(r−l)/2; for two nonzero vectors of the same norm N, put C=(PU+QV)/N and S=(PV−QU)/N. In both cases

t=S1+C(C≠−1),C=−1⟺t=∞. t=\frac{S}{1+C}\quad(C\ne-1),\qquad C=-1\Longleftrightarrow t=\infty.

3. From four quadrics to one equation

For every orbit, a basis of the left kernel is selected with lexicographic priority red > yellow. The resulting profiles are

RRRY, RRRR, RRRY, RRRR, RRRR, RRRY, RRYY, RRRY. RRRY,\ RRRR,\ RRRY,\ RRRR,\ RRRR,\ RRRY,\ RRYY,\ RRRY.

The first three relations in every atlas row are used constructively. Each introduces one rational angle and new roots while preserving the already constructed shared entries. After one common normalization, all seven roots become rational functions of u,v,w, where u=tan α, v=tan β, and w=tan γ. The fourth independent quadric gives one equation Fᵢ(α,β,γ)=0.

δα=sin⁡4α,δβ=sin⁡4β,δγ=sin⁡4γ. \delta_\alpha=\sin4\alpha,\qquad \delta_\beta=\sin4\beta,\qquad \delta_\gamma=\sin4\gamma.

The identities L²=1−δ and R²=1+δ reduce six of the eight final models to relations among the three sin 4 functions that are linear in δγ. In types CDEFGHJ and ACDFGHJ, a yellow norm is joined to the red progressions by a rational rotation, so sin 2 and cos 2 also remain in their final formulas. These are two mixed trigonometric models, not unsimplified versions of the other six.

Completeness theorem for the three-angle charts

On the nondegenerate rational locus of each of the eight patterns, every solution of the four original quadrics can, after choosing a common scale and root signs, be represented by a rational triple (u,v,w), equivalently by three rationally parametrized angles (α,β,γ), satisfying the highlighted trigonometric equation in the corresponding card. Conversely, every such triple reconstructs a solution through the coordinate formulas.

Proof

Necessity follows by successively applying the complete parametrization of the red conic and the rational rotation for a yellow norm. In every chain, a shared nonzero entry determines the next scale. After three steps, the fourth basis quadric remains. Substituting the coordinates and using L²=1−sin 4θ and R²=1+sin 4θ converts it to the stated trigonometric form. Conversely, the coordinate formulas identically satisfy the first three relations, and the trigonometric equation is equivalent to the fourth. Since the four coefficient vectors have rank 4=7−3, the vector of seven squares lies in the image of Lₛ and recovers unique E,x,y.

The denominators Lᵥ, L𝑤, Rᵥ, R𝑤, and S𝑤 occur only when a scale is recovered from a known root. On the positive pairwise-distinct locus, that shared root is nonzero; at the sole occurrence of S𝑤 in a denominator, S𝑤=0 would force A=C and leave this locus as well. Adjacent charts cover the projective boundary by swapping progression endpoints, changing a root sign, or using the parameter value ∞.

4. The atlas as the main object

The cards are ordered by the geometric type of the two nonspecified positions. The colored strip shows the four basis relations, while the large framed formula is the simplified trigonometric equation of the resulting three-angle surface. Coordinate substitutions remain available in an expandable technical block. The number 7 remains a lower guarantee: a point on a surface may produce an additional eighth or ninth square entry.

Complete trigonometric atlas

Eight orbits and eight equations for 7/9

The main object in every card is a simplified equation in three rational angles. Six orbits use only sin 4α, sin 4β, and sin 4γ; two mixed orbits also retain double-angle functions. Coordinate reconstruction of the roots is placed in an expandable technical block.

red progression of squaresyellow norm equalityThe R/Y profile follows the priority red > yellow.
01
CDEFGHJ

complement AB · orbit size 8

an adjacent corner and edge cell
simplified trigonometric equation(1−sin⁡4α)(1−sin⁡4β)+[cos⁡2γ−sin⁡2γ(sin⁡2α+cos⁡2α)]2=2 \boxed{\begin{gathered} (1-\sin4\alpha)(1-\sin4\beta)\\ +\bigl[\cos2\gamma-\sin2\gamma (\sin2\alpha+\cos2\alpha)\bigr]^2=2 \end{gathered}}
RRRYbuilds CDHbuilds CEGbuilds EFGJcloses DEF
CDHd2+h2=2c2d^2+h^2=2c^2
CEGc2+g2=2e2c^2+g^2=2e^2
EFGJe2+g2=f2+j2e^2+g^2=f^2+j^2
DEFd2+f2=2e2d^2+f^2=2e^2
Coordinate reconstruction of the roots

Normalization: e=1e=1

c=Lu,g=Ru,d=LuLv,h=LuRv,f=Cw−SwRu,j=Sw+CwRu, \begin{aligned} c&=L_u,&g&=R_u,\\ d&=L_uL_v,&h&=L_uR_v,\\ f&=C_w-S_wR_u,&j&=S_w+C_wR_u, \end{aligned}

Coordinate residual

 (LuLv)2+(Cw−SwRu)2=2  \boxed{\ (L_uL_v)^2+(C_w-S_wR_u)^2=2\ }

CEG gives the first progression, and CDH the second one centered at c². The yellow norm EFGJ rotates the rational vector (1,Rᵤ) to (f,j). The only unused condition is then the DEF progression.

02
BDEFGHJ

complement AC · orbit size 4

the two corners of one side
simplified trigonometric equation(1−2sin⁡4α+sin⁡4αsin⁡4β)sin⁡4γ=sin⁡4β−sin⁡4α \boxed{\begin{gathered} \bigl(1-2\sin4\alpha+\sin4\alpha\sin4\beta\bigr)\sin4\gamma\\ =\sin4\beta-\sin4\alpha \end{gathered}}
RRRRbuilds BDJbuilds BEHbuilds BFGcloses DEF
BDJb2+d2=2j2b^2+d^2=2j^2
BEHb2+h2=2e2b^2+h^2=2e^2
BFGb2+f2=2g2b^2+f^2=2g^2
DEFd2+f2=2e2d^2+f^2=2e^2
Coordinate reconstruction of the roots

Normalization: j=1j=1

b=Lu,d=Ru,e=LuLv,h=LuRvLv,g=LuLw,f=LuRwLw, \begin{aligned} b&=L_u,&d&=R_u,\\ e&=\frac{L_u}{L_v},&h&=\frac{L_uR_v}{L_v},\\ g&=\frac{L_u}{L_w},&f&=\frac{L_uR_w}{L_w}, \end{aligned}

Coordinate residual

 Ru2+(LuRwLw)2=2(LuLv)2  \boxed{\ R_u^2+\left(\frac{L_uR_w}{L_w}\right)^2 =2\left(\frac{L_u}{L_v}\right)^2\ }

BDJ fixes b and d after j=1. In BEH and BFG the known root b is used as the left endpoint of two new progressions, giving scales Lᵤ/Lᵥ and Lᵤ/L𝑤. The DEF progression closes the compatibility cycle.

03
BCDFGHJ

complement AE · orbit size 4

a corner and the center
simplified trigonometric equation(1+sin⁡4α−2sin⁡4αsin⁡4β)sin⁡4γ=sin⁡4β(1−sin⁡4α) \boxed{\begin{gathered} \bigl(1+\sin4\alpha-2\sin4\alpha\sin4\beta\bigr)\sin4\gamma\\ =\sin4\beta(1-\sin4\alpha) \end{gathered}}
RRRYbuilds BDJbuilds BFGbuilds CDHcloses BCGH
BDJb2+d2=2j2b^2+d^2=2j^2
BFGb2+f2=2g2b^2+f^2=2g^2
CDHd2+h2=2c2d^2+h^2=2c^2
BCGHc2+g2=b2+h2c^2+g^2=b^2+h^2
Coordinate reconstruction of the roots

Normalization: j=1j=1

b=Lu,d=Ru,g=LuLv,f=LuRvLv,c=RuLw,h=RuRwLw, \begin{aligned} b&=L_u,&d&=R_u,\\ g&=\frac{L_u}{L_v},&f&=\frac{L_uR_v}{L_v},\\ c&=\frac{R_u}{L_w},&h&=\frac{R_uR_w}{L_w}, \end{aligned}

Coordinate residual

 (RuLw)2+(LuLv)2=Lu2+(RuRwLw)2  \boxed{\ \left(\frac{R_u}{L_w}\right)^2+ \left(\frac{L_u}{L_v}\right)^2 =L_u^2+\left(\frac{R_uR_w}{L_w}\right)^2\ }

BDJ supplies the two shared endpoints for the independent BFG and CDH progressions. The three red conics therefore express all seven roots through u,v,w. The remaining Gaussian norm BCGH becomes the single equation.

04
BCDEGHJ

complement AF · orbit size 8

a knight-move pair
simplified trigonometric equation (1−sin⁡4α)sin⁡4γ=−sin⁡4α(1−sin⁡4β)  \boxed{\ (1-\sin4\alpha)\sin4\gamma =-\sin4\alpha(1-\sin4\beta)\ }
RRRRbuilds BDJbuilds BEHbuilds CEGcloses CDH
BDJb2+d2=2j2b^2+d^2=2j^2
BEHb2+h2=2e2b^2+h^2=2e^2
CEGc2+g2=2e2c^2+g^2=2e^2
CDHd2+h2=2c2d^2+h^2=2c^2
Coordinate reconstruction of the roots

Normalization: j=1j=1

b=Lu,d=Ru,e=LuLv,h=LuRvLv,c=LuLwLv,g=LuRwLv, \begin{aligned} b&=L_u,&d&=R_u,\\ e&=\frac{L_u}{L_v},&h&=\frac{L_uR_v}{L_v},\\ c&=\frac{L_uL_w}{L_v},&g&=\frac{L_uR_w}{L_v}, \end{aligned}

Coordinate residual

 Ru2+(LuRvLv)2=2(LuLwLv)2  \boxed{\ R_u^2+\left(\frac{L_uR_v}{L_v}\right)^2 =2\left(\frac{L_uL_w}{L_v}\right)^2\ }

BDJ constructs the first dir line. Through the shared root b, BEH determines the scale e, while CEG with the same center e² introduces c and g. The fourth dir line CDH gives the final equation. This is the D4 orbit containing the known Bremner–Sallows square with pattern ABCDEHJ.

Orbit of the known integral 7/9 class
05
BCDEFGH

complement AJ · orbit size 2

opposite corners
simplified trigonometric equation (1+sin⁡4α−2sin⁡4β)sin⁡4γ=sin⁡4β(1−sin⁡4α)  \boxed{\ (1+\sin4\alpha-2\sin4\beta)\sin4\gamma =\sin4\beta(1-\sin4\alpha)\ }
RRRRbuilds BEHbuilds BFGbuilds CDHcloses CEG
BEHb2+h2=2e2b^2+h^2=2e^2
BFGb2+f2=2g2b^2+f^2=2g^2
CDHd2+h2=2c2d^2+h^2=2c^2
CEGc2+g2=2e2c^2+g^2=2e^2
Coordinate reconstruction of the roots

Normalization: e=1e=1

b=Lu,h=Ru,g=LuLv,f=LuRvLv,c=RuRw,d=RuLwRw, \begin{aligned} b&=L_u,&h&=R_u,\\ g&=\frac{L_u}{L_v},&f&=\frac{L_uR_v}{L_v},\\ c&=\frac{R_u}{R_w},&d&=\frac{R_uL_w}{R_w}, \end{aligned}

Coordinate residual

 (RuRw)2+(LuLv)2=2  \boxed{\ \left(\frac{R_u}{R_w}\right)^2+ \left(\frac{L_u}{L_v}\right)^2=2\ }

BEH gives b and h around the normalized center e=1. Those two endpoints extend independently through BFG and CDH. The final central progression CEG requires the resulting c² and g² to have midpoint e²=1.

06
ACEFGHJ

complement BD · orbit size 4

two edge cells adjacent to one corner
simplified trigonometric equation sin⁡4γ=−(1−sin⁡4α)sin⁡4β  \boxed{\ \sin4\gamma=-(1-\sin4\alpha)\sin4\beta\ }
RRRYbuilds AEJbuilds AFHbuilds CEGcloses ACEH
AEJa2+j2=2e2a^2+j^2=2e^2
AFHf2+h2=2a2f^2+h^2=2a^2
CEGc2+g2=2e2c^2+g^2=2e^2
ACEHa2+c2=e2+h2a^2+c^2=e^2+h^2
Coordinate reconstruction of the roots

Normalization: e=1e=1

a=Lu,j=Ru,f=LuLv,h=LuRv,c=Lw,g=Rw, \begin{aligned} a&=L_u,&j&=R_u,\\ f&=L_uL_v,&h&=L_uR_v,\\ c&=L_w,&g&=R_w, \end{aligned}

Coordinate residual

 Lu2+Lw2=1+(LuRv)2  \boxed{\ L_u^2+L_w^2=1+(L_uR_v)^2\ }

AEJ and CEG are two progressions with the common center e². The AFH progression uses a² as its new center. After parametrizing all three conics, only the norm equality ACEH remains.

07
ACDFGHJ

complement BE · orbit size 4

an edge cell and the center
simplified trigonometric equation(cos⁡2γ−sin⁡2α−cos⁡2αsin⁡2β+cos⁡2βsin⁡2γ)2=1−sin⁡4β+2sin⁡4αsin⁡4β1+sin⁡4β \boxed{\begin{gathered} \left( \frac{\displaystyle\cos2\gamma- \frac{\sin2\alpha-\cos2\alpha}{\sin2\beta+\cos2\beta}} {\displaystyle\sin2\gamma} \right)^2\\ =\frac{1-\sin4\beta+2\sin4\alpha\sin4\beta} {1+\sin4\beta} \end{gathered}}
RRYYbuilds AFHbuilds CDHbuilds ACGJcloses ADHJ
AFHf2+h2=2a2f^2+h^2=2a^2
CDHd2+h2=2c2d^2+h^2=2c^2
ACGJa2+j2=c2+g2a^2+j^2=c^2+g^2
ADHJa2+d2=h2+j2a^2+d^2=h^2+j^2
Coordinate reconstruction of the roots

Normalization: a=1a=1

h=Lu,f=Ru,c=LuRv,d=LuLvRv,j=Cw−Lu/RvSw,g=Sw+Cwj, \begin{aligned} h&=L_u,&f&=R_u,\\ c&=\frac{L_u}{R_v},&d&=\frac{L_uL_v}{R_v},\\ j&=\frac{C_w-L_u/R_v}{S_w},& g&=S_w+C_wj, \end{aligned}

Coordinate residual

 1+(LuLvRv)2=Lu2+(Cw−Lu/RvSw)2  \boxed{\ 1+\left(\frac{L_uL_v}{R_v}\right)^2 =L_u^2+\left(\frac{C_w-L_u/R_v}{S_w}\right)^2\ }

This is the only orbit whose optimal profile is RRYY. AFH and CDH produce f,h,c,d. The norm ACGJ is represented by the rational rotation (1,j)↦(c,g); the known first coordinate c determines j. The second norm ADHJ remains as the compatibility equation.

08
ACDEFGJ

complement BH · orbit size 2

opposite edge cells
simplified trigonometric equation sin⁡4γ=sin⁡4β−sin⁡4α  \boxed{\ \sin4\gamma=\sin4\beta-\sin4\alpha\ }
RRRYbuilds AEJbuilds CEGbuilds DEFcloses ACDE
AEJa2+j2=2e2a^2+j^2=2e^2
CEGc2+g2=2e2c^2+g^2=2e^2
DEFd2+f2=2e2d^2+f^2=2e^2
ACDEa2+d2=c2+e2a^2+d^2=c^2+e^2
Coordinate reconstruction of the roots

Normalization: e=1e=1

a=Lu,j=Ru,c=Lv,g=Rv,d=Lw,f=Rw, \begin{aligned} a&=L_u,&j&=R_u,\\ c&=L_v,&g&=R_v,\\ d&=L_w,&f&=R_w, \end{aligned}

Coordinate residual

 Lu2+Lw2=Lv2+1  \boxed{\ L_u^2+L_w^2=L_v^2+1\ }

The three progressions AEJ, CEG, and DEF share the center e² and are independently parametrized by three angles. The yellow quadric ACDE is their only compatibility relation, so this orbit has the shortest final equation.

5. Step-by-step derivation of all eight equations

1. CDEFGHJ · RRRY

Complement AB: an adjacent corner and edge cell. CEG gives the first progression, and CDH the second one centered at c². The yellow norm EFGJ rotates the rational vector (1,Rᵤ) to (f,j). The only unused condition is then the DEF progression.

Step 1: CDHd2+h2=2c2d^2+h^2=2c^2
Step 2: CEGc2+g2=2e2c^2+g^2=2e^2
Step 3: EFGJe2+g2=f2+j2e^2+g^2=f^2+j^2
Residual: DEFd2+f2=2e2d^2+f^2=2e^2

After normalization e=1e=1, the DEF residual simplifies to

(1−sin⁡4α)(1−sin⁡4β)+[cos⁡2γ−sin⁡2γ(sin⁡2α+cos⁡2α)]2=2 \boxed{\begin{gathered} (1-\sin4\alpha)(1-\sin4\beta)\\ +\bigl[\cos2\gamma-\sin2\gamma (\sin2\alpha+\cos2\alpha)\bigr]^2=2 \end{gathered}}
Coordinate substitution and inverse reconstruction

The first three relations give the roots

c=Lu,g=Ru,d=LuLv,h=LuRv,f=Cw−SwRu,j=Sw+CwRu, \begin{aligned} c&=L_u,&g&=R_u,\\ d&=L_uL_v,&h&=L_uR_v,\\ f&=C_w-S_wR_u,&j&=S_w+C_wR_u, \end{aligned}

Before trigonometric simplification, the DEF residual is

 (LuLv)2+(Cw−SwRu)2=2  \boxed{\ (L_uL_v)^2+(C_w-S_wR_u)^2=2\ }

2. BDEFGHJ · RRRR

Complement AC: the two corners of one side. BDJ fixes b and d after j=1. In BEH and BFG the known root b is used as the left endpoint of two new progressions, giving scales Lᵤ/Lᵥ and Lᵤ/L𝑤. The DEF progression closes the compatibility cycle.

Step 1: BDJb2+d2=2j2b^2+d^2=2j^2
Step 2: BEHb2+h2=2e2b^2+h^2=2e^2
Step 3: BFGb2+f2=2g2b^2+f^2=2g^2
Residual: DEFd2+f2=2e2d^2+f^2=2e^2

After normalization j=1j=1, the DEF residual simplifies to

(1−2sin⁡4α+sin⁡4αsin⁡4β)sin⁡4γ=sin⁡4β−sin⁡4α \boxed{\begin{gathered} \bigl(1-2\sin4\alpha+\sin4\alpha\sin4\beta\bigr)\sin4\gamma\\ =\sin4\beta-\sin4\alpha \end{gathered}}
Coordinate substitution and inverse reconstruction

The first three relations give the roots

b=Lu,d=Ru,e=LuLv,h=LuRvLv,g=LuLw,f=LuRwLw, \begin{aligned} b&=L_u,&d&=R_u,\\ e&=\frac{L_u}{L_v},&h&=\frac{L_uR_v}{L_v},\\ g&=\frac{L_u}{L_w},&f&=\frac{L_uR_w}{L_w}, \end{aligned}

Before trigonometric simplification, the DEF residual is

 Ru2+(LuRwLw)2=2(LuLv)2  \boxed{\ R_u^2+\left(\frac{L_uR_w}{L_w}\right)^2 =2\left(\frac{L_u}{L_v}\right)^2\ }

3. BCDFGHJ · RRRY

Complement AE: a corner and the center. BDJ supplies the two shared endpoints for the independent BFG and CDH progressions. The three red conics therefore express all seven roots through u,v,w. The remaining Gaussian norm BCGH becomes the single equation.

Step 1: BDJb2+d2=2j2b^2+d^2=2j^2
Step 2: BFGb2+f2=2g2b^2+f^2=2g^2
Step 3: CDHd2+h2=2c2d^2+h^2=2c^2
Residual: BCGHc2+g2=b2+h2c^2+g^2=b^2+h^2

After normalization j=1j=1, the BCGH residual simplifies to

(1+sin⁡4α−2sin⁡4αsin⁡4β)sin⁡4γ=sin⁡4β(1−sin⁡4α) \boxed{\begin{gathered} \bigl(1+\sin4\alpha-2\sin4\alpha\sin4\beta\bigr)\sin4\gamma\\ =\sin4\beta(1-\sin4\alpha) \end{gathered}}
Coordinate substitution and inverse reconstruction

The first three relations give the roots

b=Lu,d=Ru,g=LuLv,f=LuRvLv,c=RuLw,h=RuRwLw, \begin{aligned} b&=L_u,&d&=R_u,\\ g&=\frac{L_u}{L_v},&f&=\frac{L_uR_v}{L_v},\\ c&=\frac{R_u}{L_w},&h&=\frac{R_uR_w}{L_w}, \end{aligned}

Before trigonometric simplification, the BCGH residual is

 (RuLw)2+(LuLv)2=Lu2+(RuRwLw)2  \boxed{\ \left(\frac{R_u}{L_w}\right)^2+ \left(\frac{L_u}{L_v}\right)^2 =L_u^2+\left(\frac{R_uR_w}{L_w}\right)^2\ }

4. BCDEGHJ · RRRR

Complement AF: a knight-move pair. BDJ constructs the first dir line. Through the shared root b, BEH determines the scale e, while CEG with the same center e² introduces c and g. The fourth dir line CDH gives the final equation. This is the D4 orbit containing the known Bremner–Sallows square with pattern ABCDEHJ.

Step 1: BDJb2+d2=2j2b^2+d^2=2j^2
Step 2: BEHb2+h2=2e2b^2+h^2=2e^2
Step 3: CEGc2+g2=2e2c^2+g^2=2e^2
Residual: CDHd2+h2=2c2d^2+h^2=2c^2

After normalization j=1j=1, the CDH residual simplifies to

 (1−sin⁡4α)sin⁡4γ=−sin⁡4α(1−sin⁡4β)  \boxed{\ (1-\sin4\alpha)\sin4\gamma =-\sin4\alpha(1-\sin4\beta)\ }
Coordinate substitution and inverse reconstruction

The first three relations give the roots

b=Lu,d=Ru,e=LuLv,h=LuRvLv,c=LuLwLv,g=LuRwLv, \begin{aligned} b&=L_u,&d&=R_u,\\ e&=\frac{L_u}{L_v},&h&=\frac{L_uR_v}{L_v},\\ c&=\frac{L_uL_w}{L_v},&g&=\frac{L_uR_w}{L_v}, \end{aligned}

Before trigonometric simplification, the CDH residual is

 Ru2+(LuRvLv)2=2(LuLwLv)2  \boxed{\ R_u^2+\left(\frac{L_uR_v}{L_v}\right)^2 =2\left(\frac{L_uL_w}{L_v}\right)^2\ }

5. BCDEFGH · RRRR

Complement AJ: opposite corners. BEH gives b and h around the normalized center e=1. Those two endpoints extend independently through BFG and CDH. The final central progression CEG requires the resulting c² and g² to have midpoint e²=1.

Step 1: BEHb2+h2=2e2b^2+h^2=2e^2
Step 2: BFGb2+f2=2g2b^2+f^2=2g^2
Step 3: CDHd2+h2=2c2d^2+h^2=2c^2
Residual: CEGc2+g2=2e2c^2+g^2=2e^2

After normalization e=1e=1, the CEG residual simplifies to

 (1+sin⁡4α−2sin⁡4β)sin⁡4γ=sin⁡4β(1−sin⁡4α)  \boxed{\ (1+\sin4\alpha-2\sin4\beta)\sin4\gamma =\sin4\beta(1-\sin4\alpha)\ }
Coordinate substitution and inverse reconstruction

The first three relations give the roots

b=Lu,h=Ru,g=LuLv,f=LuRvLv,c=RuRw,d=RuLwRw, \begin{aligned} b&=L_u,&h&=R_u,\\ g&=\frac{L_u}{L_v},&f&=\frac{L_uR_v}{L_v},\\ c&=\frac{R_u}{R_w},&d&=\frac{R_uL_w}{R_w}, \end{aligned}

Before trigonometric simplification, the CEG residual is

 (RuRw)2+(LuLv)2=2  \boxed{\ \left(\frac{R_u}{R_w}\right)^2+ \left(\frac{L_u}{L_v}\right)^2=2\ }

6. ACEFGHJ · RRRY

Complement BD: two edge cells adjacent to one corner. AEJ and CEG are two progressions with the common center e². The AFH progression uses a² as its new center. After parametrizing all three conics, only the norm equality ACEH remains.

Step 1: AEJa2+j2=2e2a^2+j^2=2e^2
Step 2: AFHf2+h2=2a2f^2+h^2=2a^2
Step 3: CEGc2+g2=2e2c^2+g^2=2e^2
Residual: ACEHa2+c2=e2+h2a^2+c^2=e^2+h^2

After normalization e=1e=1, the ACEH residual simplifies to

 sin⁡4γ=−(1−sin⁡4α)sin⁡4β  \boxed{\ \sin4\gamma=-(1-\sin4\alpha)\sin4\beta\ }
Coordinate substitution and inverse reconstruction

The first three relations give the roots

a=Lu,j=Ru,f=LuLv,h=LuRv,c=Lw,g=Rw, \begin{aligned} a&=L_u,&j&=R_u,\\ f&=L_uL_v,&h&=L_uR_v,\\ c&=L_w,&g&=R_w, \end{aligned}

Before trigonometric simplification, the ACEH residual is

 Lu2+Lw2=1+(LuRv)2  \boxed{\ L_u^2+L_w^2=1+(L_uR_v)^2\ }

7. ACDFGHJ · RRYY

Complement BE: an edge cell and the center. This is the only orbit whose optimal profile is RRYY. AFH and CDH produce f,h,c,d. The norm ACGJ is represented by the rational rotation (1,j)↦(c,g); the known first coordinate c determines j. The second norm ADHJ remains as the compatibility equation.

Step 1: AFHf2+h2=2a2f^2+h^2=2a^2
Step 2: CDHd2+h2=2c2d^2+h^2=2c^2
Step 3: ACGJa2+j2=c2+g2a^2+j^2=c^2+g^2
Residual: ADHJa2+d2=h2+j2a^2+d^2=h^2+j^2

After normalization a=1a=1, the ADHJ residual simplifies to

(cos⁡2γ−sin⁡2α−cos⁡2αsin⁡2β+cos⁡2βsin⁡2γ)2=1−sin⁡4β+2sin⁡4αsin⁡4β1+sin⁡4β \boxed{\begin{gathered} \left( \frac{\displaystyle\cos2\gamma- \frac{\sin2\alpha-\cos2\alpha}{\sin2\beta+\cos2\beta}} {\displaystyle\sin2\gamma} \right)^2\\ =\frac{1-\sin4\beta+2\sin4\alpha\sin4\beta} {1+\sin4\beta} \end{gathered}}
Coordinate substitution and inverse reconstruction

The first three relations give the roots

h=Lu,f=Ru,c=LuRv,d=LuLvRv,j=Cw−Lu/RvSw,g=Sw+Cwj, \begin{aligned} h&=L_u,&f&=R_u,\\ c&=\frac{L_u}{R_v},&d&=\frac{L_uL_v}{R_v},\\ j&=\frac{C_w-L_u/R_v}{S_w},& g&=S_w+C_wj, \end{aligned}

Before trigonometric simplification, the ADHJ residual is

 1+(LuLvRv)2=Lu2+(Cw−Lu/RvSw)2  \boxed{\ 1+\left(\frac{L_uL_v}{R_v}\right)^2 =L_u^2+\left(\frac{C_w-L_u/R_v}{S_w}\right)^2\ }

8. ACDEFGJ · RRRY

Complement BH: opposite edge cells. The three progressions AEJ, CEG, and DEF share the center e² and are independently parametrized by three angles. The yellow quadric ACDE is their only compatibility relation, so this orbit has the shortest final equation.

Step 1: AEJa2+j2=2e2a^2+j^2=2e^2
Step 2: CEGc2+g2=2e2c^2+g^2=2e^2
Step 3: DEFd2+f2=2e2d^2+f^2=2e^2
Residual: ACDEa2+d2=c2+e2a^2+d^2=c^2+e^2

After normalization e=1e=1, the ACDE residual simplifies to

 sin⁡4γ=sin⁡4β−sin⁡4α  \boxed{\ \sin4\gamma=\sin4\beta-\sin4\alpha\ }
Coordinate substitution and inverse reconstruction

The first three relations give the roots

a=Lu,j=Ru,c=Lv,g=Rv,d=Lw,f=Rw, \begin{aligned} a&=L_u,&j&=R_u,\\ c&=L_v,&g&=R_v,\\ d&=L_w,&f&=R_w, \end{aligned}

Before trigonometric simplification, the ACDE residual is

 Lu2+Lw2=Lv2+1  \boxed{\ L_u^2+L_w^2=L_v^2+1\ }

6. What the atlas solves—and what remains open

The atlas completes the positional and geometric reduction of 7/9 on the nondegenerate locus: no pattern or inverse reconstruction is left unspecified. For any specified seven-entry pattern in a rational solution, one can determine its D4 orbit, recover three angles, and obtain a point on the corresponding surface; the roots and the magic square are then reconstructed from that point. If a square has eight or nine square entries, its different seven-entry subpatterns naturally occur in several cards.

This is not yet a classification of every rational point on Fᵢ=0, still less of all positive integral points. The known Bremner–Sallows square belongs to the BCDEGHJ orbit, equivalent to its pattern ABCDEHJ, and uses four red dir lines. Proving that no other integral class exists would require solving the arithmetic problem on all eight surfaces subject to positivity, pairwise distinctness, and integrality.

Open the standalone atlas →Theory of dir linesThe 6/9 atlas