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Complex licht multiplication example with various gons.

With multiplication, the gon(s) of the output may or may not equal the gons of the inputs; what follows is an example using 5-lichts. Numerous preliminary definitions are necessary to present in a clear manner the final pattern, which turns out to be a Latin square.

To establish the divisors, define two ordered quintuples, which came from a random number generator:

A𝗗 = (	[−2.157831, +2.036379],	[−1.613613, −0.530356],	[+1.266265, +0.314028],
		[+1.609747, +0.060905],	[+2.574440, +2.426379]	)

B𝗗 = (	[−0.310587, +0.642595],	[+1.938439, +1.503547],	[−0.865090, −0.358493],
		[+0.062550, +1.217095],	[+0.185946, −1.706754]	)

whence a third:

C𝗗 = A𝗗 × B𝗗 = (	[−0.638373, −2.019083],	[−2.330475, −3.454207],	[−0.982857, −0.725609],
		[+0.026562, +1.963024]	[+4.619939, −3.942761]	)

To establish the lengths, define two complex numbers:

A_𝗟0 = [−0.447087, +2.276521]
B_𝗟0 = [+0.982247, −1.109308]

whence a third:

C_𝗟0 = A_𝗟0 × B_𝗟0 = [+2.086212, +2.732064]

Using those, define fifteen ordered quintuples:

A0𝗟 = (	A𝗟0,	[+1.415754, −0.775016],	[−1.501595, +1.261835],
		[−2.111657, +2.273926],	[−2.934627, +2.683005]	)

A1𝗟 = (	A𝗟0,	[+1.174576, +1.106969],	[+0.473128, −1.903461],
		[+3.044946, −0.598444],	[+1.644840, +3.620090]	)

A2𝗟 = (	A𝗟0,	[−0.689826, +1.459160],	[+0.736058, +1.818030],
		[−2.815170, −1.305623],	[+3.951194, −0.445667]	)

A3𝗟 = (	A𝗟0,	[−1.600912, −0.205158],	[−1.664095, −1.038173],
		[+1.510094, +2.710987],	[+0.797132, −3.895527]	)

A4𝗟 = (	A𝗟0,	[−0.299592, −1.585955],	[+1.956504, −0.138231],
		[+0.371786, −3.080845],	[−3.458539, −1.961902]	)

B0𝗟 = (	B𝗟0,	[−0.111076, −2.654123],	[+0.727699, −1.179138],
		[−1.870319, +0.294643],	[−1.814216, +0.812855]	)

B1𝗟 = (	B𝗟0,	[+2.489897, −0.925809],	[+0.104359, +1.381673],
		[+1.686307, +0.860975],	[+0.212447, +1.976608]	)

B2𝗟 = (	B𝗟0,	[+1.649917, +2.081942],	[−0.896556, −1.056456],
		[−0.858182, −1.687729],	[+1.945515, +0.408756]	)

B3𝗟 = (	B𝗟0,	[−1.470192, +2.212520],	[+1.346298, +0.327709],
		[−0.297738, +1.869829],	[+0.989947, −1.723983]	)

B4𝗟 = (	B𝗟0,	[−2.558546, −0.714530],	[−1.281801, +0.526212],
		[+1.339933, −1.337717],	[−1.333694, −1.474236]	)

C0𝗟 = (	C𝗟0,	[−4.176042, +0.971315],	[+1.260752, −2.407579],
		[+0.212390, +5.871708],	[+7.869277, +0.748023]	)

C1𝗟 = (	C𝗟0,	[−2.214243, −3.671499],	[+0.395169, +2.688824],
		[+3.279476, −4.875151],	[+3.143152, −7.252976]	)

C2𝗟 = (	C𝗟0,	[+2.807564, −3.240426],	[−1.900150, −1.943029],
		[−5.518694, +2.016452],	[−5.926702, −5.230608]	)

C3𝗟 = (	C𝗟0,	[+3.949414, +1.668805],	[+2.679337, +0.455064],
		[+5.649958, +1.612463],	[−6.806056, +4.020282]	)

C4𝗟 = (	C𝗟0,	[−0.366692, +4.271805],	[−2.435109, +1.206721],
		[−3.623130, −4.625472],	[+1.720328, +7.715279]	)

Now can be defined fifteen lichts:

A0 = ⟨ A𝗗; A0𝗟; 0 ⟩ A1 = ⟨ A𝗗; A1𝗟; 1 ⟩ A2 = ⟨ A𝗗; A2𝗟; 2 ⟩ A3 = ⟨ A𝗗; A3𝗟; 3 ⟩ A4 = ⟨ A𝗗; A4𝗟; 4 ⟩

B0 = ⟨ B𝗗; B0𝗟; 0 ⟩ B1 = ⟨ B𝗗; B1𝗟; 1 ⟩ B2 = ⟨ B𝗗; B2𝗟; 2 ⟩ B3 = ⟨ B𝗗; B3𝗟; 3 ⟩ B4 = ⟨ B𝗗; B4𝗟; 4 ⟩

C0 = ⟨ C𝗗; C0𝗟; 0 ⟩ C1 = ⟨ C𝗗; C1𝗟; 1 ⟩ C2 = ⟨ C𝗗; C2𝗟; 2 ⟩ C3 = ⟨ C𝗗; C3𝗟; 3 ⟩ C4 = ⟨ C𝗗; C4𝗟; 4 ⟩

which lead to twenty-five products:

A0 × B0 = C1	A0 × B1 = C2	A0 × B2 = C3	A0 × B3 = C4	A0 × B4 = C0
A1 × B0 = C2	A1 × B1 = C3	A1 × B2 = C4	A1 × B3 = C0	A1 × B4 = C1
A2 × B0 = C3	A2 × B1 = C4	A2 × B2 = C0	A2 × B3 = C1	A2 × B4 = C2
A3 × B0 = C4	A3 × B1 = C0	A3 × B2 = C1	A3 × B3 = C2	A3 × B4 = C3
A4 × B0 = C0	A4 × B1 = C1	A4 × B2 = C2	A4 × B3 = C3	A4 × B4 = C4

The subscripts of the Cs, which are the lichts' respective gon numbers, form a Latin square:

1	2	3	4	0
2	3	4	0	1
3	4	0	1	2
4	0	1	2	3
0	1	2	3	4

Yet unknown is whether Latin squares are always formed in the general nonzero case.