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A method for constructing some nontrivial integer b-lichts.

Arbitrarily select quantity b of positive real integers:

d₀, d₁, d₂ … d_b−1

Let A_𝗗 be this ordered b-tuple:

( (d₀)^b, (d₁)^b, (d₂)^b … (d_b−1)^b )

Let A_𝗟0 equal this:

(d₀)^b−1 × (d₁)^b−2 × (d₂)^b−3 × … × (d_b−2)¹

Note that d_b−1 is intentionally omitted from this product — or it could have been written with an exponent of zero.

These are the A_𝗗 and A_𝗟0 to supply to the usual method of creating a licht, as in §R2B. If the members of A_𝗗 are pairwise relatively prime, so will be the members of A_𝗟.

Example 1. Let d₀ = 5, d₁ = 3, d₂ = 7, d₃ = 2.

The resultant 4-licht is ⟨ 625, 81, 2401, 16; 7875, 2646, 6860, 600 ⟩.

The lengths have related factorizations:

7875	= 53 × 32 × 71 × 20
2646	= 50 × 33 × 72 × 21
6860	= 51 × 30 × 73 × 22
600	= 52 × 31 × 70 × 23

Example 2. Exchange d₁ and d₂ from example 1.

The resultant 4-licht is ⟨ 625, 2401, 81, 16; 18375, 6174, 540, 1400 ⟩. Although the divisors have been permuted, the lengths are different figures entirely. Still, they display factorizations resembling those of example 1:

18375	= 53 × 72 × 31 × 20
6174	= 50 × 73 × 32 × 21
540	= 51 × 70 × 33 × 22
1400	= 52 × 71 × 30 × 23

This method also works with Gaussian integers, which are those complex numbers whose real and imaginary parts are integers.

Example 3. Define these arbitrary Gaussian integers:

d0 = [ +5, −2 ]

d1 = [ −3, −7 ]

d2 = [ +2, +3 ]

d3 = [ +7, −5 ]

Now create 4-licht A with A_𝗚 = 0, and with the following divisors, which are the respective fourth powers of the d_n:

A𝗗0 =	[	+41,	−840 ]
A𝗗1 =	[	−164,	−3360 ]
A𝗗2 =	[	−119,	−120 ]
A𝗗3 =	[	−4324,	−3360 ]

Using the method from above, the following lengths are produced, themselves Gaussian integers:

A𝗚 = 0:	A𝗟0 =	[	−18502,	+26912 ]
	A𝗟1 =	[	+6436,	−48976 ]
	A𝗟2 =	[	+4502,	+18128 ]
	A𝗟3 =	[	+16124,	−139664 ]

Licht B uses the same divisors as A, but is calculated with B_𝗚 = 1:

B𝗚 = 1:	B𝗟0 =	[	−18502,	+26912 ]
	B𝗟1 =	[	+48976,	+6436 ]
	B𝗟2 =	[	−4502,	−18128 ]
	B𝗟3 =	[	−139664,	−16124 ]

Licht C with C_𝗚 = 2:

C𝗚 = 2:	C𝗟0 =	[	−18502,	+26912 ]
	C𝗟1 =	[	−6436,	+48976 ]
	C𝗟2 =	[	+4502,	+18128 ]
	C𝗟3 =	[	−16124,	+139664 ]

Licht D with D_𝗚 = 3:

D𝗚 = 3:	D𝗟0 =	[	−18502,	+26912 ]
	D𝗟1 =	[	−48976,	−6436 ]
	D𝗟2 =	[	−4502,	−18128 ]
	D𝗟3 =	[	+139664,	+16124 ]

When the respective lengths differ, they differ by a factor which is a fourth root of unity, four being the breadth.