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-part Multiplication (quintrights) | Q 2 cos (90° / 5) 2 sin (360° / 5) 1.902113 | R 2 cos (180° / 5) 2 sin (270° / 5) 1.618034 | S 2 cos (270° / 5) 2 sin (180° / 5) 1.175571 | T 2 cos (360° / 5) 2 sin (90° / 5) 0.618034 |
Q 1.902113 | 2 + R 3.618034 | Q + S 3.077684 | R + T 2.236068 | S 1.175571 |
R 1.618034 | Q + S 3.077684 | 2 + T 2.618034 | Q 1.902113 | R − T 1.000000 |
S 1.175571 | R + T 2.236068 | Q 1.902113 | 2 − T 1.381966 | Q − S 0.726543 |
T 0.618034 | S 1.175571 | R − T 1.000000 | Q − S 0.726543 | 2 − R 0.381966 |
R − T = RT = 1
QS = √5 |
7-part Multiplication (septrights) | Q 2 cos (90° / 7) 2 sin (540° / 7) 1.949856 | R 2 cos (180° / 7) 2 sin (450° / 7) 1.801938 | S 2 cos (270° / 7) 2 sin (360° / 7) 1.563663 | T 2 cos (360° / 7) 2 sin (270° / 7) 1.246980 | U 2 cos (450° / 7) 2 sin (180° / 7) 0.867767 | V 2 cos (540° / 7) 2 sin (90° / 7) 0.445042 |
Q 1.949856 | 2 + R 3.801938 | Q + S 3.513519 | R + T 3.048917 | S + U 2.431430 | T + V 1.692021 | U 0.867767 |
R 1.801938 | Q + S 3.513519 | 2 + T 3.246980 | Q + U 2.817623 | R + V 2.246980 | S 1.563663 | T − V 0.801938 |
S 1.563663 | R + T 3.048917 | Q + U 2.817623 | 2 + V 2.445042 | Q 1.949856 | R − V 1.356896 | S − U 0.695895 |
T 1.246980 | S + U 2.431430 | R + V 2.246980 | Q 1.949856 | 2 − V 1.554958 | Q − U 1.082088 | R − T 0.554958 |
U 0.867767 | T + V 1.692021 | S 1.563663 | R − V 1.356896 | Q − U 1.082088 | 2 − T 0.753020 | Q − S 0.386193 |
V 0.445042 | U 0.867767 | T − V 0.801938 | S − U 0.695895 | R − T 0.554958 | Q − S 0.386193 | 2 − R 0.198062 |
R − T + V = RTV = 1
QSU = √7 | ||||||
Many identities can be found on a page devoted to septrights. |
11-part Multiplication | Q 2 cos (90° / 11) 2 sin (900° / 11) 1.979643 | R 2 cos (180° / 11) 2 sin (810° / 11) 1.918986 | S 2 cos (270° / 11) 2 sin (720° / 11) 1.819264 | T 2 cos (360° / 11) 2 sin (630° / 11) 1.682507 | U 2 cos (450° / 11) 2 sin (540° / 11) 1.511499 | V 2 cos (540° / 11) 2 sin (450° / 11) 1.309721 | W 2 cos (630° / 11) 2 sin (360° / 11) 1.081282 | X 2 cos (720° / 11) 2 sin (270° / 11) 0.830830 | Y 2 cos (810° / 11) 2 sin (180° / 11) 0.563465 | Z 2 cos (900° / 11) 2 sin (90° / 11) 0.284630 |
---|---|---|---|---|---|---|---|---|---|---|
Q 1.979643 | 2 + R 3.918986 | Q + S 3.798907 | R + T 3.601493 | S + U 3.330763 | T + V 2.992229 | U + W 2.592781 | V + X 2.140551 | W + Y 1.644747 | X + Z 1.115460 | Y 0.563465 |
R 1.918986 | Q + S 3.798907 | 2 + T 3.682507 | Q + U 3.491142 | R + V 3.228707 | S + W 2.900546 | T + X 2.513337 | U + Y 2.074964 | V + Z 1.594351 | W 1.081282 | X − Z 0.546200 |
S 1.819264 | R + T 3.601493 | Q + U 3.491142 | 2 + V 3.309721 | Q + W 3.060925 | R + X 2.749816 | S + Y 2.382729 | T + Z 1.967137 | U 1.511499 | V − Z 1.025092 | W − Y 0.517817 |
T 1.682507 | S + U 3.330763 | R + V 3.228707 | Q + W 3.060925 | 2 + X 2.830830 | Q + Y 2.543108 | R + Z 2.203616 | S 1.819264 | T − Z 1.397877 | U − Y 0.948034 | V − X 0.478891 |
U 1.511499 | T + V 2.992229 | S + W 2.900546 | R + X 2.749816 | Q + Y 2.543108 | 2 + Z 2.284630 | Q 1.979643 | R − Z 1.634356 | S − Y 1.255799 | T − X 0.851677 | U − W 0.430218 |
V 1.309721 | U + W 2.592781 | T + X 2.513337 | S + Y 2.382729 | R + Z 2.203616 | Q 1.979643 | 2 − Z 1.715370 | Q − Y 1.416178 | R − X 1.088156 | S − W 0.737982 | T − V 0.372786 |
W 1.081282 | V + X 2.140551 | U + Y 2.074964 | T + Z 1.967137 | S 1.819264 | R − Z 1.634356 | Q − Y 1.416178 | 2 − X 1.169170 | Q − W 0.898361 | R − V 0.609264 | S − U 0.307765 |
X 0.830830 | W + Y 1.644747 | V + Z 1.594351 | U 1.511499 | T − Z 1.397877 | S − Y 1.255799 | R − X 1.088156 | Q − W 0.898361 | 2 − V 0.690279 | Q − U 0.468144 | R − T 0.236479 |
Y 0.563465 | X + Z 1.115460 | W 1.081282 | V − Z 1.025092 | U − Y 0.948034 | T − X 0.851677 | S − W 0.737982 | R − V 0.609264 | Q − U 0.468144 | 2 − T 0.317493 | Q − S 0.160379 |
Z 0.284630 | Y 0.563465 | X − Z 0.546200 | W − Y 0.517817 | V − X 0.478891 | U − W 0.430218 | T − V 0.372786 | S − U 0.307765 | R − T 0.236479 | Q − S 0.160379 | 2 − R 0.081014 |
R − T + V − X + Z = RTVXZ = 1
QSUWY = √11 |
13-part Multiplication | Q 2 cos (90° / 13) 2 sin (1080° / 13) 1.985418 | R 2 cos (180° / 13) 2 sin (990° / 13) 1.941884 | S 2 cos (270° / 13) 2 sin (900° / 13) 1.870032 | T 2 cos (360° / 13) 2 sin (810° / 13) 1.770912 | U 2 cos (450° / 13) 2 sin (720° / 13) 1.645968 | V 2 cos (540° / 13) 2 sin (630° / 13) 1.497021 | W 2 cos (630° / 13) 2 sin (540° / 13) 1.326245 | X 2 cos (720° / 13) 2 sin (450° / 13) 1.136129 | Y 2 cos (810° / 13) 2 sin (360° / 13) 0.929446 | Z 2 cos (900° / 13) 2 sin (270° / 13) 0.709210 | A 2 cos (990° / 13) 2 sin (180° / 13) 0.478631 | B 2 cos (1080° / 13) 2 sin (90° / 13) 0.241073 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Q 1.985418 | 2 + R 3.941884 | Q + S 3.855450 | R + T 3.712796 | S + U 3.516000 | T + V 3.267934 | U + W 2.972213 | V + X 2.633151 | W + Y 2.255692 | X + Z 1.845339 | Y + A 1.408078 | Z + B 0.950283 | A 0.478631 |
R 1.941884 | Q + S 3.855450 | 2 + T 3.770912 | Q + U 3.631385 | R + V 3.438905 | S + W 3.196278 | T + X 2.907042 | U + Y 2.575414 | V + Z 2.206231 | W + A 1.804877 | X + B 1.377203 | Y 0.929446 | Z − B 0.468136 |
S 1.870032 | R + T 3.712796 | Q + U 3.631385 | 2 + V 3.497021 | Q + W 3.311663 | R + X 3.078013 | S + Y 2.799479 | T + Z 2.480122 | U + A 2.124599 | V + B 1.738095 | W 1.326245 | X − B 0.895056 | Y − A 0.450815 |
T 1.770912 | S + U 3.516000 | R + V 3.438905 | Q + W 3.311663 | 2 + X 3.136129 | Q + Y 2.914864 | R + Z 2.651093 | S + A 2.348664 | T + B 2.011985 | U 1.645968 | V − B 1.255948 | W − A 0.847614 | X − Z 0.426920 |
U 1.645968 | T + V 3.267934 | S + W 3.196278 | R + X 3.078013 | Q + Y 2.914864 | 2 + Z 2.709210 | Q + A 2.464049 | R + B 2.182957 | S 1.870032 | T − B 1.529839 | U − A 1.167336 | V − Z 0.787812 | W − Y 0.396799 |
V 1.497021 | U + W 2.972213 | T + X 2.907042 | S + Y 2.799479 | R + Z 2.651093 | Q + A 2.464049 | 2 + B 2.241073 | Q 1.985418 | R − B 1.700810 | S − A 1.391401 | T − Z 1.061702 | U − Y 0.716521 | V − X 0.360892 |
W 1.326245 | V + X 2.633151 | U + Y 2.575414 | T + Z 2.480122 | S + A 2.348664 | R + B 2.182957 | Q 1.985418 | 2 − B 1.758927 | Q − A 1.506786 | R − Z 1.232674 | S − Y 0.940586 | T − X 0.634783 | U − W 0.319722 |
X 1.136129 | W + Y 2.255692 | V + Z 2.206231 | U + A 2.124599 | T + B 2.011985 | S 1.870032 | R − B 1.700810 | Q − A 1.506786 | 2 − Z 1.290790 | Q − Y 1.055971 | R − X 0.805754 | S − W 0.543787 | T − V 0.273891 |
Y 0.929446 | X + Z 1.845339 | W + A 1.804877 | V + B 1.738095 | U 1.645968 | T − B 1.529839 | S − A 1.391401 | R − Z 1.232674 | Q − Y 1.055971 | 2 − X 0.863871 | Q − W 0.659172 | R − V 0.444862 | S − U 0.224065 |
Z 0.709210 | Y + A 1.408078 | X + B 1.377203 | W 1.326245 | V − B 1.255948 | U − A 1.167336 | T − Z 1.061702 | S − Y 0.940586 | R − X 0.805754 | Q − W 0.659172 | 2 − V 0.502979 | Q − U 0.339450 | R − T 0.170972 |
A 0.478631 | Z + B 0.950283 | Y 0.929446 | X − B 0.895056 | W − A 0.847614 | V − Z 0.787812 | U − Y 0.716521 | T − X 0.634783 | S − W 0.543787 | R − V 0.444862 | Q − U 0.339450 | 2 − T 0.229088 | Q − S 0.115385 |
B 0.241073 | A 0.478631 | Z − B 0.468136 | Y − A 0.450815 | X − Z 0.426920 | W − Y 0.396799 | V − X 0.360892 | U − W 0.319722 | T − V 0.273891 | S − U 0.224065 | R − T 0.170972 | Q − S 0.115385 | 2 − R 0.058116 |
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 |
R | Q |
T | R |
V | S |
X | T |
Z | U |
B | V |
10-part Multiplication | Q 2 cos (90° / 10) 2 sin (810° / 10) 1.975377 | R 2 cos (180° / 10) 2 sin (720° / 10) 1.902113 | S 2 cos (270° / 10) 2 sin (630° / 10) 1.782013 | T 2 cos (360° / 10) 2 sin (540° / 10) 1.618034 | U 2 cos (450° / 10) 2 sin (450° / 10) 1.414214 | V 2 cos (540° / 10) 2 sin (360° / 10) 1.175571 | W 2 cos (630° / 10) 2 sin (270° / 10) 0.907981 | X 2 cos (720° / 10) 2 sin (180° / 10) 0.618034 | Y 2 cos (810° / 10) 2 sin (90° / 10) 0.312869 |
---|---|---|---|---|---|---|---|---|---|
Q 1.975377 | 2 + R 3.902113 | Q + S 3.757390 | R + T 3.520147 | S + U 3.196227 | T + V 2.793604 | U + W 2.322195 | V + X 1.793604 | W + Y 1.220850 | X 0.618034 |
R 1.902113 | Q + S 3.757390 | 2 + T 3.618034 | Q + U 3.389590 | R + V 3.077684 | S + W 2.689994 | T + X 2.236068 | U + Y 1.727082 | V 1.175571 | W − Y 0.595112 |
S 1.782013 | R + T 3.520147 | Q + U 3.389590 | 2 + V 3.175571 | Q + W 2.883358 | R + X 2.520147 | S + Y 2.094882 | T 1.618034 | U − Y 1.101345 | V − X 0.557537 |
T 1.618034 | S + U 3.196227 | R + V 3.077684 | Q + W 2.883358 | 2 + X 2.618034 | Q + Y 2.288246 | R 1.902113 | S − Y 1.469144 | T − X 1.000000 | U − W 0.506233 |
U 1.414214 | T + V 2.793604 | S + W 2.689994 | R + X 2.520147 | Q + Y 2.288246 | 2 2.000000 | Q − Y 1.662508 | R − X 1.284079 | S − W 0.874032 | T − V 0.442463 |
V 1.175571 | U + W 2.322195 | T + X 2.236068 | S + Y 2.094882 | R 1.902113 | Q − Y 1.662508 | 2 − X 1.381966 | Q − W 1.067396 | R − V 0.726543 | S − U 0.367799 |
W 0.907981 | V + X 1.793604 | U + Y 1.727082 | T 1.618034 | S − Y 1.469144 | R − X 1.284079 | Q − W 1.067396 | 2 − V 0.824429 | Q − U 0.561163 | R − T 0.284079 |
X 0.618034 | W + Y 1.220850 | V 1.175571 | U − Y 1.101345 | T − X 1.000000 | S − W 0.874032 | R − V 0.726543 | Q − U 0.561163 | 2 − T 0.381966 | Q − S 0.193364 |
Y 0.312869 | X 0.618034 | W − Y 0.595112 | V − X 0.557537 | U − W 0.506233 | T − V 0.442463 | S − U 0.367799 | R − T 0.284079 | Q − S 0.193364 | 2 − R 0.097887 |
T − X = TX = 1
RV = √5 |
14-part Multiplication | Q 2 cos (90° / 14) 2 sin (1170° / 14) 1.987424 | R 2 cos (180° / 14) 2 sin (1080° / 14) 1.949856 | S 2 cos (270° / 14) 2 sin (990° / 14) 1.887767 | T 2 cos (360° / 14) 2 sin (900° / 14) 1.801938 | U 2 cos (450° / 14) 2 sin (810° / 14) 1.693448 | V 2 cos (540° / 14) 2 sin (720° / 14) 1.563663 | W 2 cos (630° / 14) 2 sin (630° / 14) 1.414214 | X 2 cos (720° / 14) 2 sin (540° / 14) 1.246980 | Y 2 cos (810° / 14) 2 sin (450° / 14) 1.064064 | Z 2 cos (900° / 14) 2 sin (360° / 14) 0.867767 | A 2 cos (990° / 14) 2 sin (270° / 14) 0.660558 | B 2 cos (1080° / 14) 2 sin (180° / 14) 0.445042 | C 2 cos (1170° / 14) 2 sin (90° / 14) 0.223929 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Q 1.987424 | 2 + R 3.949856 | Q + S 3.875191 | R + T 3.751794 | S + U 3.581215 | T + V 3.365601 | U + W 3.107662 | V + X 2.810643 | W + Y 2.478278 | X + Z 2.114747 | Y + A 1.724622 | Z + B 1.312809 | A + C 0.884487 | B 0.445042 |
R 1.949856 | Q + S 3.875191 | 2 + T 3.801938 | Q + U 3.680873 | R + V 3.513519 | S + W 3.301980 | T + X 3.048917 | U + Y 2.757513 | V + Z 2.431430 | W + A 2.074772 | X + B 1.692021 | Y + C 1.287993 | Z 0.867767 | A − C 0.436629 |
S 1.887767 | R + T 3.751794 | Q + U 3.680873 | 2 + V 3.563663 | Q + W 3.401638 | R + X 3.196835 | S + Y 2.951831 | T + Z 2.669705 | U + A 2.354007 | V + B 2.008705 | W + C 1.638143 | X 1.246980 | Y − C 0.840135 | Z − B 0.422726 |
T 1.801938 | S + U 3.581215 | R + V 3.513519 | Q + W 3.401638 | 2 + X 3.246980 | Q + Y 3.051489 | R + Z 2.817623 | S + A 2.548325 | T + B 2.246980 | U + C 1.917377 | V 1.563663 | W − C 1.190285 | X − B 0.801938 | Y − A 0.403506 |
U 1.693448 | T + V 3.365601 | S + W 3.301980 | R + X 3.196835 | Q + Y 3.051489 | 2 + Z 2.867767 | Q + A 2.647983 | R + B 2.394898 | S + C 2.111696 | T 1.801938 | U − C 1.469519 | V − B 1.118621 | W − A 0.753655 | X − Z 0.379212 |
V 1.563663 | U + W 3.107662 | T + X 3.048917 | S + Y 2.951831 | R + Z 2.817623 | Q + A 2.647983 | 2 + B 2.445042 | Q + C 2.211353 | R 1.949856 | S − C 1.663838 | T − B 1.356896 | U − A 1.032890 | V − Z 0.695895 | W − Y 0.350149 |
W 1.414214 | V + X 2.810643 | U + Y 2.757513 | T + Z 2.669705 | S + A 2.548325 | R + B 2.394898 | Q + C 2.211353 | 2 2.000000 | Q − C 1.763495 | R − B 1.504814 | S − A 1.227209 | T − Z 0.934170 | U − Y 0.629384 | V − X 0.316683 |
X 1.246980 | W + Y 2.478278 | V + Z 2.431430 | U + A 2.354007 | T + B 2.246980 | S + C 2.111696 | R 1.949856 | Q − C 1.763495 | 2 − B 1.554958 | Q − A 1.326866 | R − Z 1.082088 | S − Y 0.823703 | T − X 0.554958 | U − W 0.279235 |
Y 1.064064 | X + Z 2.114747 | W + A 2.074772 | V + B 2.008705 | U + C 1.917377 | T 1.801938 | S − C 1.663838 | R − B 1.504814 | Q − A 1.326866 | 2 − Z 1.132233 | Q − Y 0.923360 | R − X 0.702876 | S − W 0.473553 | T − V 0.238275 |
Z 0.867767 | Y + A 1.724622 | X + B 1.692021 | W + C 1.638143 | V 1.563663 | U − C 1.469519 | T − B 1.356896 | S − A 1.227209 | R − Z 1.082088 | Q − Y 0.923360 | 2 − X 0.753020 | Q − W 0.573211 | R − V 0.386193 | S − U 0.194318 |
A 0.660558 | Z + B 1.312809 | Y + C 1.287993 | X 1.246980 | W − C 1.190285 | V − B 1.118621 | U − A 1.032890 | T − Z 0.934170 | S − Y 0.823703 | R − X 0.702876 | Q − W 0.573211 | 2 − V 0.436337 | Q − U 0.293976 | R − T 0.147918 |
B 0.445042 | A + C 0.884487 | Z 0.867767 | Y − C 0.840135 | X − B 0.801938 | W − A 0.753655 | V − Z 0.695895 | U − Y 0.629384 | T − X 0.554958 | S − W 0.473553 | R − V 0.386193 | Q − U 0.293976 | 2 − T 0.198062 | Q − S 0.099658 |
C 0.223929 | B 0.445042 | A − C 0.436629 | Z − B 0.422726 | Y − A 0.403506 | X − Z 0.379212 | W − Y 0.350149 | V − X 0.316683 | U − W 0.279235 | T − V 0.238275 | S − U 0.194318 | R − T 0.147918 | Q − S 0.099658 | 2 − R 0.050144 |
T − X + B = TXB = 1
RVZ = √7 |