Generalization of multiplication.

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-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

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

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

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

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

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.

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