Magic Squaresproof atlas
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Complete D4 classification

Seven Square Entries: 8 Three-Angle Surfaces

Every card represents one positional orbit, a complete basis of four red and yellow quadrics, and one simplified trigonometric equation in three rational angles. Reversible coordinate formulas for the seven roots remain available in an expandable technical block.

Completeness concerns the orbit census and the reduction of the original system: no branch of the four quadrics is lost on the nondegenerate rational locus. Classifying rational or integral points on the eight resulting surfaces remains a separate arithmetic problem.

Proof of the orbit count, chart completeness, and step-by-step derivation →

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.