Home.

The following tables describe quintright-like multiplication for a right angle divided into various numbers of parts. Squares are shaded in red, and antisquares in green. The larger tables will help the reader surmise the underlying pattern.

Omitted are any tables where the number of parts is a multiple of three, as these will contain the element 2 cos 60° which, being rational, gives multiple ways to write zero. An open question is whether anything similar can occur when the number of parts is not a multiple of three.

 5-partMultiplication(quintrights) Q2 cos (90° / 5)2 sin (360° / 5)1.902113 R2 cos (180° / 5)2 sin (270° / 5)1.618034 S2 cos (270° / 5)2 sin (180° / 5)1.175571 T2 cos (360° / 5)2 sin (90° / 5)0.618034 Q1.902113 2 + R3.618034 Q + S3.077684 R + T2.236068 S1.175571 R1.618034 Q + S3.077684 2 + T2.618034 Q1.902113 R − T1.000000 S1.175571 R + T2.236068 Q1.902113 2 − T1.381966 Q − S0.726543 T0.618034 S1.175571 R − T1.000000 Q − S0.726543 2 − R0.381966 R − T = RT = 1 QS = √5

 7-partMultiplication(septrights) Q2 cos (90° / 7)2 sin (540° / 7)1.949856 R2 cos (180° / 7)2 sin (450° / 7)1.801938 S2 cos (270° / 7)2 sin (360° / 7)1.563663 T2 cos (360° / 7)2 sin (270° / 7)1.246980 U2 cos (450° / 7)2 sin (180° / 7)0.867767 V2 cos (540° / 7)2 sin (90° / 7)0.445042 Q1.949856 2 + R3.801938 Q + S3.513519 R + T3.048917 S + U2.431430 T + V1.692021 U0.867767 R1.801938 Q + S3.513519 2 + T3.246980 Q + U2.817623 R + V2.246980 S1.563663 T − V0.801938 S1.563663 R + T3.048917 Q + U2.817623 2 + V2.445042 Q1.949856 R − V1.356896 S − U0.695895 T1.246980 S + U2.431430 R + V2.246980 Q1.949856 2 − V1.554958 Q − U1.082088 R − T0.554958 U0.867767 T + V1.692021 S1.563663 R − V1.356896 Q − U1.082088 2 − T0.753020 Q − S0.386193 V0.445042 U0.867767 T − V0.801938 S − U0.695895 R − T0.554958 Q − S0.386193 2 − R0.198062 R − T + V = RTV = 1 QSU = √7 Many identities can be found on a page devoted to septrights.

 11-partMultiplication Q2 cos (90° / 11)2 sin (900° / 11)1.979643 R2 cos (180° / 11)2 sin (810° / 11)1.918986 S2 cos (270° / 11)2 sin (720° / 11)1.819264 T2 cos (360° / 11)2 sin (630° / 11)1.682507 U2 cos (450° / 11)2 sin (540° / 11)1.511499 V2 cos (540° / 11)2 sin (450° / 11)1.309721 W2 cos (630° / 11)2 sin (360° / 11)1.081282 X2 cos (720° / 11)2 sin (270° / 11)0.830830 Y2 cos (810° / 11)2 sin (180° / 11)0.563465 Z2 cos (900° / 11)2 sin (90° / 11)0.284630 Q1.979643 2 + R3.918986 Q + S3.798907 R + T3.601493 S + U3.330763 T + V2.992229 U + W2.592781 V + X2.140551 W + Y1.644747 X + Z1.115460 Y0.563465 R1.918986 Q + S3.798907 2 + T3.682507 Q + U3.491142 R + V3.228707 S + W2.900546 T + X2.513337 U + Y2.074964 V + Z1.594351 W1.081282 X − Z0.546200 S1.819264 R + T3.601493 Q + U3.491142 2 + V3.309721 Q + W3.060925 R + X2.749816 S + Y2.382729 T + Z1.967137 U1.511499 V − Z1.025092 W − Y0.517817 T1.682507 S + U3.330763 R + V3.228707 Q + W3.060925 2 + X2.830830 Q + Y2.543108 R + Z2.203616 S1.819264 T − Z1.397877 U − Y0.948034 V − X0.478891 U1.511499 T + V2.992229 S + W2.900546 R + X2.749816 Q + Y2.543108 2 + Z2.284630 Q1.979643 R − Z1.634356 S − Y1.255799 T − X0.851677 U − W0.430218 V1.309721 U + W2.592781 T + X2.513337 S + Y2.382729 R + Z2.203616 Q1.979643 2 − Z1.715370 Q − Y1.416178 R − X1.088156 S − W0.737982 T − V0.372786 W1.081282 V + X2.140551 U + Y2.074964 T + Z1.967137 S1.819264 R − Z1.634356 Q − Y1.416178 2 − X1.169170 Q − W0.898361 R − V0.609264 S − U0.307765 X0.830830 W + Y1.644747 V + Z1.594351 U1.511499 T − Z1.397877 S − Y1.255799 R − X1.088156 Q − W0.898361 2 − V0.690279 Q − U0.468144 R − T0.236479 Y0.563465 X + Z1.115460 W1.081282 V − Z1.025092 U − Y0.948034 T − X0.851677 S − W0.737982 R − V0.609264 Q − U0.468144 2 − T0.317493 Q − S0.160379 Z0.284630 Y0.563465 X − Z0.546200 W − Y0.517817 V − X0.478891 U − W0.430218 T − V0.372786 S − U0.307765 R − T0.236479 Q − S0.160379 2 − R0.081014 R − T + V − X + Z = RTVXZ = 1 QSUWY = √11

 13-partMultiplication Q2 cos (90° / 13)2 sin (1080° / 13)1.985418 R2 cos (180° / 13)2 sin (990° / 13)1.941884 S2 cos (270° / 13)2 sin (900° / 13)1.870032 T2 cos (360° / 13)2 sin (810° / 13)1.770912 U2 cos (450° / 13)2 sin (720° / 13)1.645968 V2 cos (540° / 13)2 sin (630° / 13)1.497021 W2 cos (630° / 13)2 sin (540° / 13)1.326245 X2 cos (720° / 13)2 sin (450° / 13)1.136129 Y2 cos (810° / 13)2 sin (360° / 13)0.929446 Z2 cos (900° / 13)2 sin (270° / 13)0.709210 A2 cos (990° / 13)2 sin (180° / 13)0.478631 B2 cos (1080° / 13)2 sin (90° / 13)0.241073 Q1.985418 2 + R3.941884 Q + S3.855450 R + T3.712796 S + U3.516000 T + V3.267934 U + W2.972213 V + X2.633151 W + Y2.255692 X + Z1.845339 Y + A1.408078 Z + B0.950283 A0.478631 R1.941884 Q + S3.855450 2 + T3.770912 Q + U3.631385 R + V3.438905 S + W3.196278 T + X2.907042 U + Y2.575414 V + Z2.206231 W + A1.804877 X + B1.377203 Y0.929446 Z − B0.468136 S1.870032 R + T3.712796 Q + U3.631385 2 + V3.497021 Q + W3.311663 R + X3.078013 S + Y2.799479 T + Z2.480122 U + A2.124599 V + B1.738095 W1.326245 X − B0.895056 Y − A0.450815 T1.770912 S + U3.516000 R + V3.438905 Q + W3.311663 2 + X3.136129 Q + Y2.914864 R + Z2.651093 S + A2.348664 T + B2.011985 U1.645968 V − B1.255948 W − A0.847614 X − Z0.426920 U1.645968 T + V3.267934 S + W3.196278 R + X3.078013 Q + Y2.914864 2 + Z2.709210 Q + A2.464049 R + B2.182957 S1.870032 T − B1.529839 U − A1.167336 V − Z0.787812 W − Y0.396799 V1.497021 U + W2.972213 T + X2.907042 S + Y2.799479 R + Z2.651093 Q + A2.464049 2 + B2.241073 Q1.985418 R − B1.700810 S − A1.391401 T − Z1.061702 U − Y0.716521 V − X0.360892 W1.326245 V + X2.633151 U + Y2.575414 T + Z2.480122 S + A2.348664 R + B2.182957 Q1.985418 2 − B1.758927 Q − A1.506786 R − Z1.232674 S − Y0.940586 T − X0.634783 U − W0.319722 X1.136129 W + Y2.255692 V + Z2.206231 U + A2.124599 T + B2.011985 S1.870032 R − B1.700810 Q − A1.506786 2 − Z 1.290790 Q − Y1.055971 R − X0.805754 S − W0.543787 T − V0.273891 Y0.929446 X + Z1.845339 W + A1.804877 V + B1.738095 U1.645968 T − B1.529839 S − A1.391401 R − Z1.232674 Q − Y1.055971 2 − X0.863871 Q − W0.659172 R − V0.444862 S − U0.224065 Z0.709210 Y + A1.408078 X + B1.377203 W1.326245 V − B1.255948 U − A1.167336 T − Z1.061702 S − Y0.940586 R − X0.805754 Q − W0.659172 2 − V0.502979 Q − U0.339450 R − T0.170972 A0.478631 Z + B0.950283 Y0.929446 X − B0.895056 W − A0.847614 V − Z0.787812 U − Y0.716521 T − X0.634783 S − W0.543787 R − V0.444862 Q − U0.339450 2 − T0.229088 Q − S0.115385 B0.241073 A0.478631 Z − B0.468136 Y − A0.450815 X − Z0.426920 W − Y0.396799 V − X0.360892 U − W0.319722 T − V0.273891 S − U0.224065 R − T0.170972 Q − S0.115385 2 − R0.058116 R − T + V − X + Z − B = RTVXZB = 1 QSUWYA = √13

Multiplication can be extended to systems having a number of parts that is twice an odd number.

Equivalences of some constants

In the
10- and 14-part
tables
In the
5- and 7-part
tables
RQ
TR
VS
XT
ZU
BV

 10-partMultiplication Q2 cos (90° / 10)2 sin (810° / 10)1.975377 R2 cos (180° / 10)2 sin (720° / 10)1.902113 S2 cos (270° / 10)2 sin (630° / 10)1.782013 T2 cos (360° / 10)2 sin (540° / 10)1.618034 U2 cos (450° / 10)2 sin (450° / 10)1.414214 V2 cos (540° / 10)2 sin (360° / 10)1.175571 W2 cos (630° / 10)2 sin (270° / 10)0.907981 X2 cos (720° / 10)2 sin (180° / 10)0.618034 Y2 cos (810° / 10)2 sin (90° / 10)0.312869 Q1.975377 2 + R3.902113 Q + S3.757390 R + T3.520147 S + U3.196227 T + V2.793604 U + W2.322195 V + X1.793604 W + Y1.220850 X0.618034 R1.902113 Q + S3.757390 2 + T3.618034 Q + U3.389590 R + V3.077684 S + W2.689994 T + X2.236068 U + Y1.727082 V1.175571 W − Y0.595112 S1.782013 R + T3.520147 Q + U3.389590 2 + V3.175571 Q + W2.883358 R + X2.520147 S + Y2.094882 T1.618034 U − Y1.101345 V − X0.557537 T1.618034 S + U3.196227 R + V3.077684 Q + W2.883358 2 + X2.618034 Q + Y2.288246 R1.902113 S − Y1.469144 T − X1.000000 U − W0.506233 U1.414214 T + V2.793604 S + W2.689994 R + X2.520147 Q + Y2.288246 22.000000 Q − Y1.662508 R − X1.284079 S − W0.874032 T − V0.442463 V1.175571 U + W2.322195 T + X2.236068 S + Y2.094882 R1.902113 Q − Y1.662508 2 − X1.381966 Q − W1.067396 R − V0.726543 S − U0.367799 W0.907981 V + X1.793604 U + Y1.727082 T1.618034 S − Y1.469144 R − X1.284079 Q − W1.067396 2 − V0.824429 Q − U0.561163 R − T0.284079 X0.618034 W + Y1.220850 V1.175571 U − Y1.101345 T − X1.000000 S − W0.874032 R − V0.726543 Q − U0.561163 2 − T0.381966 Q − S0.193364 Y0.312869 X0.618034 W − Y0.595112 V − X0.557537 U − W0.506233 T − V0.442463 S − U0.367799 R − T0.284079 Q − S0.193364 2 − R0.097887 T − X = TX = 1 RV = √5

 14-partMultiplication Q2 cos (90° / 14)2 sin (1170° / 14)1.987424 R2 cos (180° / 14)2 sin (1080° / 14)1.949856 S2 cos (270° / 14)2 sin (990° / 14)1.887767 T2 cos (360° / 14)2 sin (900° / 14)1.801938 U2 cos (450° / 14)2 sin (810° / 14)1.693448 V2 cos (540° / 14)2 sin (720° / 14)1.563663 W2 cos (630° / 14)2 sin (630° / 14)1.414214 X2 cos (720° / 14)2 sin (540° / 14)1.246980 Y2 cos (810° / 14)2 sin (450° / 14)1.064064 Z2 cos (900° / 14)2 sin (360° / 14)0.867767 A2 cos (990° / 14)2 sin (270° / 14)0.660558 B2 cos (1080° / 14)2 sin (180° / 14)0.445042 C2 cos (1170° / 14)2 sin (90° / 14)0.223929 Q1.987424 2 + R3.949856 Q + S3.875191 R + T3.751794 S + U3.581215 T + V3.365601 U + W3.107662 V + X2.810643 W + Y2.478278 X + Z2.114747 Y + A1.724622 Z + B1.312809 A + C0.884487 B0.445042 R1.949856 Q + S3.875191 2 + T3.801938 Q + U3.680873 R + V3.513519 S + W3.301980 T + X3.048917 U + Y2.757513 V + Z2.431430 W + A2.074772 X + B1.692021 Y + C1.287993 Z0.867767 A − C0.436629 S1.887767 R + T3.751794 Q + U3.680873 2 + V3.563663 Q + W3.401638 R + X3.196835 S + Y2.951831 T + Z2.669705 U + A2.354007 V + B2.008705 W + C1.638143 X1.246980 Y − C0.840135 Z − B0.422726 T1.801938 S + U3.581215 R + V3.513519 Q + W3.401638 2 + X3.246980 Q + Y3.051489 R + Z2.817623 S + A2.548325 T + B2.246980 U + C1.917377 V1.563663 W − C1.190285 X − B0.801938 Y − A0.403506 U1.693448 T + V3.365601 S + W3.301980 R + X3.196835 Q + Y3.051489 2 + Z2.867767 Q + A2.647983 R + B2.394898 S + C2.111696 T1.801938 U − C1.469519 V − B1.118621 W − A0.753655 X − Z0.379212 V1.563663 U + W3.107662 T + X3.048917 S + Y2.951831 R + Z2.817623 Q + A2.647983 2 + B2.445042 Q + C2.211353 R1.949856 S − C1.663838 T − B1.356896 U − A1.032890 V − Z0.695895 W − Y0.350149 W1.414214 V + X2.810643 U + Y2.757513 T + Z2.669705 S + A2.548325 R + B2.394898 Q + C2.211353 22.000000 Q − C1.763495 R − B1.504814 S − A1.227209 T − Z0.934170 U − Y0.629384 V − X0.316683 X1.246980 W + Y2.478278 V + Z2.431430 U + A2.354007 T + B2.246980 S + C2.111696 R1.949856 Q − C1.763495 2 − B1.554958 Q − A1.326866 R − Z1.082088 S − Y0.823703 T − X0.554958 U − W0.279235 Y1.064064 X + Z2.114747 W + A2.074772 V + B2.008705 U + C1.917377 T1.801938 S − C1.663838 R − B1.504814 Q − A1.326866 2 − Z1.132233 Q − Y0.923360 R − X0.702876 S − W0.473553 T − V0.238275 Z0.867767 Y + A1.724622 X + B1.692021 W + C1.638143 V1.563663 U − C1.469519 T − B1.356896 S − A1.227209 R − Z1.082088 Q − Y0.923360 2 − X0.753020 Q − W0.573211 R − V0.386193 S − U0.194318 A0.660558 Z + B1.312809 Y + C1.287993 X1.246980 W − C1.190285 V − B1.118621 U − A1.032890 T − Z0.934170 S − Y0.823703 R − X0.702876 Q − W0.573211 2 − V0.436337 Q − U0.293976 R − T0.147918 B0.445042 A + C0.884487 Z0.867767 Y − C0.840135 X − B0.801938 W − A0.753655 V − Z0.695895 U − Y0.629384 T − X0.554958 S − W0.473553 R − V0.386193 Q − U0.293976 2 − T0.198062 Q − S0.099658 C0.223929 B0.445042 A − C0.436629 Z − B0.422726 Y − A0.403506 X − Z0.379212 W − Y0.350149 V − X0.316683 U − W0.279235 T − V0.238275 S − U0.194318 R − T0.147918 Q − S0.099658 2 − R0.050144 T − X + B = TXB = 1 RVZ = √7