red(DEF) yellow(ACDE) blue(BCDE)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E = e 2 E + x + y = f 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E E + x + y = a 2 = b 2 = c 2 = d 2 = e 2 = f 2 d 2 + f 2 = 2 e 2 d^2+f^2=2e^2 d 2 + f 2 = 2 e 2 a 2 + d 2 = c 2 + e 2 a^2+d^2=c^2+e^2 a 2 + d 2 = c 2 + e 2 b 2 + 2 c 2 = d 2 + 2 e 2 b^2+2c^2=d^2+2e^2 b 2 + 2 c 2 = d 2 + 2 e 2 One progression and two quadrics on the shared C,D,E block.
K3: 12I₂; 1≤rank≤6
red(CEG) yellow(ACDE) yellow(ABEG)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E = e 2 E + y = g 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+y&=g^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E E + y = a 2 = b 2 = c 2 = d 2 = e 2 = g 2 c 2 + g 2 = 2 e 2 c^2+g^2=2e^2 c 2 + g 2 = 2 e 2 a 2 + d 2 = c 2 + e 2 a^2+d^2=c^2+e^2 a 2 + d 2 = c 2 + e 2 a 2 + b 2 = e 2 + g 2 a^2+b^2=e^2+g^2 a 2 + b 2 = e 2 + g 2 The CEG progression and two independent Gaussian norms.
K3: 12I₂; 1≤rank≤6
red(CDH) red(BEH) yellow(ACEH)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E = e 2 E + x − y = h 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E+x-y&=h^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E E + x − y = a 2 = b 2 = c 2 = d 2 = e 2 = h 2 d 2 + h 2 = 2 c 2 d^2+h^2=2c^2 d 2 + h 2 = 2 c 2 b 2 + h 2 = 2 e 2 b^2+h^2=2e^2 b 2 + h 2 = 2 e 2 a 2 + c 2 = e 2 + h 2 a^2+c^2=e^2+h^2 a 2 + c 2 = e 2 + h 2 An intersecting red-red-yellow type with shared entry H.
Rectangular K3: 2I₄+8I₂; 2≤rank≤4
red(BDJ) red(AEJ) yellow(ACDE)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E = e 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E&=e^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E E − x = a 2 = b 2 = c 2 = d 2 = e 2 = j 2 b 2 + d 2 = 2 j 2 b^2+d^2=2j^2 b 2 + d 2 = 2 j 2 a 2 + j 2 = 2 e 2 a^2+j^2=2e^2 a 2 + j 2 = 2 e 2 a 2 + d 2 = c 2 + e 2 a^2+d^2=c^2+e^2 a 2 + d 2 = c 2 + e 2 Two intersecting progressions and a yellow compatibility relation.
Rectangular K3: 2I₄+8I₂; 1≤rank≤4
red(BFG) yellow(BCDG) blue(ACFG)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E + x + y = f 2 E + y = g 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+y&=g^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E + x + y E + y = a 2 = b 2 = c 2 = d 2 = f 2 = g 2 b 2 + f 2 = 2 g 2 b^2+f^2=2g^2 b 2 + f 2 = 2 g 2 b 2 + c 2 = d 2 + g 2 b^2+c^2=d^2+g^2 b 2 + c 2 = d 2 + g 2 2 a 2 + g 2 = c 2 + 2 f 2 2a^2+g^2=c^2+2f^2 2 a 2 + g 2 = c 2 + 2 f 2 A progression, a Gaussian norm, and an x²+2y² norm without the center.
K3: 12I₂; 1≤rank≤6
red(AFH) red(CDH) yellow(BDFH)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E + x + y = f 2 E + x − y = h 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+x-y&=h^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E + x + y E + x − y = a 2 = b 2 = c 2 = d 2 = f 2 = h 2 f 2 + h 2 = 2 a 2 f^2+h^2=2a^2 f 2 + h 2 = 2 a 2 d 2 + h 2 = 2 c 2 d^2+h^2=2c^2 d 2 + h 2 = 2 c 2 b 2 + h 2 = d 2 + f 2 b^2+h^2=d^2+f^2 b 2 + h 2 = d 2 + f 2 An intersecting red-red-yellow type with shared entry H.
Rectangular K3: 2I₄+8I₂; 1≤rank≤4
red(BDJ) yellow(ACGJ) yellow(BCDG)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E + y = g 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E + y E − x = a 2 = b 2 = c 2 = d 2 = g 2 = j 2 b 2 + d 2 = 2 j 2 b^2+d^2=2j^2 b 2 + d 2 = 2 j 2 a 2 + j 2 = c 2 + g 2 a^2+j^2=c^2+g^2 a 2 + j 2 = c 2 + g 2 b 2 + c 2 = d 2 + g 2 b^2+c^2=d^2+g^2 b 2 + c 2 = d 2 + g 2 One progression and two independent Gaussian norms.
K3: 12I₂; 1≤rank≤6
red(BDJ) red(CDH) yellow(ABHJ)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E − x − y = d 2 E + x − y = h 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E-x-y&=d^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E − x − y E + x − y E − x = a 2 = b 2 = c 2 = d 2 = h 2 = j 2 b 2 + d 2 = 2 j 2 b^2+d^2=2j^2 b 2 + d 2 = 2 j 2 d 2 + h 2 = 2 c 2 d^2+h^2=2c^2 d 2 + h 2 = 2 c 2 a 2 + j 2 = b 2 + h 2 a^2+j^2=b^2+h^2 a 2 + j 2 = b 2 + h 2 An intersecting pair of progressions with shared entry D.
Rectangular K3: 2I₄+8I₂; 1≤rank≤4
red(CEG) red(BEH) yellow(ACEH)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E = e 2 E + y = g 2 E + x − y = h 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E&=e^2\\E+y&=g^2\\E+x-y&=h^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E E + y E + x − y = a 2 = b 2 = c 2 = e 2 = g 2 = h 2 c 2 + g 2 = 2 e 2 c^2+g^2=2e^2 c 2 + g 2 = 2 e 2 b 2 + h 2 = 2 e 2 b^2+h^2=2e^2 b 2 + h 2 = 2 e 2 a 2 + c 2 = e 2 + h 2 a^2+c^2=e^2+h^2 a 2 + c 2 = e 2 + h 2 The same K3 quartic as ABCEGJ, with a different cell interpretation.
Rectangular K3: 2I₄+8I₂; 2≤rank≤4
red(CEG) red(AEJ) yellow(BEGJ)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E = e 2 E + y = g 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E&=e^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E E + y E − x = a 2 = b 2 = c 2 = e 2 = g 2 = j 2 c 2 + g 2 = 2 e 2 c^2+g^2=2e^2 c 2 + g 2 = 2 e 2 a 2 + j 2 = 2 e 2 a^2+j^2=2e^2 a 2 + j 2 = 2 e 2 b 2 + e 2 = g 2 + j 2 b^2+e^2=g^2+j^2 b 2 + e 2 = g 2 + j 2 The same K3 quartic as ABCEGH, with a different cell interpretation.
Rectangular K3: 2I₄+8I₂; 2≤rank≤4
yellow(ACGJ) yellow(ABHJ) blue(ACHJ)
{ E + x = a 2 E − x + y = b 2 E − y = c 2 E + y = g 2 E + x − y = h 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-y&=c^2\\E+y&=g^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − y E + y E + x − y E − x = a 2 = b 2 = c 2 = g 2 = h 2 = j 2 a 2 + j 2 = c 2 + g 2 a^2+j^2=c^2+g^2 a 2 + j 2 = c 2 + g 2 a 2 + j 2 = b 2 + h 2 a^2+j^2=b^2+h^2 a 2 + j 2 = b 2 + h 2 a 2 + 2 c 2 = 2 h 2 + j 2 a^2+2c^2=2h^2+j^2 a 2 + 2 c 2 = 2 h 2 + j 2 The unique pattern without a red progression: two Gaussian and one blue norm.
K3: 4I₄+4I₂; 1≤rank≤2
red(AFH) red(DEF) red(BEH)
{ E + x = a 2 E − x + y = b 2 E − x − y = d 2 E = e 2 E + x + y = f 2 E + x − y = h 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\\E+x-y&=h^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − x − y E E + x + y E + x − y = a 2 = b 2 = d 2 = e 2 = f 2 = h 2 f 2 + h 2 = 2 a 2 f^2+h^2=2a^2 f 2 + h 2 = 2 a 2 d 2 + f 2 = 2 e 2 d^2+f^2=2e^2 d 2 + f 2 = 2 e 2 b 2 + h 2 = 2 e 2 b^2+h^2=2e^2 b 2 + h 2 = 2 e 2 Three red conditions with two shared centers.
Triangular K3: 4I₄+4I₂; 1≤rank≤2
red(BDJ) red(DEF) red(AEJ)
{ E + x = a 2 E − x + y = b 2 E − x − y = d 2 E = e 2 E + x + y = f 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E&=e^2\\E+x+y&=f^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − x − y E E + x + y E − x = a 2 = b 2 = d 2 = e 2 = f 2 = j 2 b 2 + d 2 = 2 j 2 b^2+d^2=2j^2 b 2 + d 2 = 2 j 2 d 2 + f 2 = 2 e 2 d^2+f^2=2e^2 d 2 + f 2 = 2 e 2 a 2 + j 2 = 2 e 2 a^2+j^2=2e^2 a 2 + j 2 = 2 e 2 A different topology of three red conditions.
Triangular K3: 4I₄+4I₂; 1≤rank≤2
red(AFH) red(BDJ) yellow(BDFH)
{ E + x = a 2 E − x + y = b 2 E − x − y = d 2 E + x + y = f 2 E + x − y = h 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E-x-y&=d^2\\E+x+y&=f^2\\E+x-y&=h^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E − x − y E + x + y E + x − y E − x = a 2 = b 2 = d 2 = f 2 = h 2 = j 2 f 2 + h 2 = 2 a 2 f^2+h^2=2a^2 f 2 + h 2 = 2 a 2 b 2 + d 2 = 2 j 2 b^2+d^2=2j^2 b 2 + d 2 = 2 j 2 b 2 + h 2 = d 2 + f 2 b^2+h^2=d^2+f^2 b 2 + h 2 = d 2 + f 2 A parallel red-red-yellow type without the central entry.
Rectangular Kummer K3; complete tfmn
red(AFH) red(BFG) red(BEH)
{ E + x = a 2 E − x + y = b 2 E = e 2 E + x + y = f 2 E + y = g 2 E + x − y = h 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E+x+y&=f^2\\E+y&=g^2\\E+x-y&=h^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E E + x + y E + y E + x − y = a 2 = b 2 = e 2 = f 2 = g 2 = h 2 f 2 + h 2 = 2 a 2 f^2+h^2=2a^2 f 2 + h 2 = 2 a 2 b 2 + f 2 = 2 g 2 b^2+f^2=2g^2 b 2 + f 2 = 2 g 2 b 2 + h 2 = 2 e 2 b^2+h^2=2e^2 b 2 + h 2 = 2 e 2 A triangle of three pairwise square means.
Triangular K3: 4I₄+4I₂; 1≤rank≤2
red(BFG) red(AEJ) yellow(ABFJ)
{ E + x = a 2 E − x + y = b 2 E = e 2 E + x + y = f 2 E + y = g 2 E − x = j 2 \left\{\begin{aligned}E+x&=a^2\\E-x+y&=b^2\\E&=e^2\\E+x+y&=f^2\\E+y&=g^2\\E-x&=j^2\end{aligned}\right. ⎩ ⎨ ⎧ E + x E − x + y E E + x + y E + y E − x = a 2 = b 2 = e 2 = f 2 = g 2 = j 2 b 2 + f 2 = 2 g 2 b^2+f^2=2g^2 b 2 + f 2 = 2 g 2 a 2 + j 2 = 2 e 2 a^2+j^2=2e^2 a 2 + j 2 = 2 e 2 a 2 + b 2 = f 2 + j 2 a^2+b^2=f^2+j^2 a 2 + b 2 = f 2 + j 2 A parallel red-red-yellow type containing the center.
Rectangular Kummer K3; complete tfmn