Lichtenberg ratios for complex numbers.
§C1A Introduction. Complex numbers are suitable for lichs. All multiple roots of a complex number must be recognized, and they require surprisingly delicate management.
As with real lichts, divisors must never be zero. All lengths are zero, or none are.
It helps to establish a particular notation for the complex numbers. In this report, the real and imaginary parts of the rectangular representation are enclosed in square brackets; the radius and angle of the polar representation in round brackets. Hence:
[ real, imag ] or ( rad, ang )
[ 6.7 cos (1.2π), 6.7 sin (1.2π) ] = ( 6.7, 1.2π )
Here, the angle portion will always be written in the interval 0 ≤ ang < 2π. This means that it will be the minimum possible nonnegative representation. With this constraint, the multiple roots of a complex number, all of which have the same radius, can be unambiguously sorted in ascending order by angle. Each root is then addressed by a non-negative integer called a gon (from the Greek word for angle) starting with zero. Example:
| seventh roots of [−2.572799, +2.427990] = (3.537574, 2.385144) | gon |
|---|---|
| [+1.128943, +0.400283] = (1.197806, 0.340735) | 0 |
| [+0.390931, +1.132216] = (1.197806, 1.238333) | 1 |
| [−0.641460, +1.011567] = (1.197806, 2.135931) | 2 |
| [−1.190819, +0.129188] = (1.197806, 3.033529) | 3 |
| [−0.843467, −0.850472] = (1.197806, 3.931126) | 4 |
| [+0.139033, −1.189710] = (1.197806, 4.828724) | 5 |
| [+1.016838, −0.633071] = (1.197806, 5.726322) | 6 |
Every nonzero complex n-licht has precisely one gon value, at least zero but less than n (the monogon case); every zero complex n-licht has all n possible gon values (the panagon case).
As with real numbers, any zero licht is certainly flat; a nonzero licht may or may not be flat.
A gon of A is denoted A𝗚. Two phenomena:
Thus unsuccessful will be the attempt to form disjoint, but isomorphic, algebras by partitioning complex lichts according to their gons.
Some researchers might instead opt for modular behavior, allowing a gon g of an n-licht to be any integer, under the equivalence g ± n ≡ g. This corresponds exactly to the complex identity (rad, ang ± 2π) = (rad, ang).
§C1B Notation. The components of a generic complex n-licht A can be written between two shallow angle brackets, as the following:
A = ⟨ A𝗗; A𝗟; A𝗚 ⟩
which is often split between two lines for readability:
| A = ⟨ | A𝗗0, | A𝗗1, | A𝗗2 | … | A𝗗n−1 | ; | |
| A𝗟0, | A𝗟1, | A𝗟2 | … | A𝗟n−1 | ; | A𝗚 ⟩ |
A𝗗 and A𝗟 are entirely analogous to the real-licht case.
There is no generally useful default value for the gon. Exception: If a real licht happens to be implemented as a complex licht with all of the imaginary parts set to zero, its gon should be zero to ensure that its step will be a nonnegative real number.
Associated with every complex b-licht A is a complex number called the wedge, symbolized A𝗪. The wedge, which is entirely dependent on the breadth, is the root of unity whose gon is one. In polar coordinates it is:
A𝗪 = (1, 2π/A𝗕)
In the table of seventh roots in §C1A, the ratio of consecutive entries is the wedge associated with a breadth of seven.
§C1C Usual method to create a complex n-licht from scratch. This is adapted from §R2A and §R2B.
§C1C1. Needed to construct nonzero licht A are:
The step (A𝗦) will be one of the nth roots of the product of the divisors; A𝗚 specifies which root, as explained in §C1A.
Now the ratios can be calculated with the following familiar formula:
A𝗥i = A𝗦 ÷ A𝗗i
for each integer i: 0 ≤ i < n
The final formula is the same as in the real-licht case, except for adding the A𝗚 notation:
| A = ⟨ | A𝗗0, | A𝗗1, | A𝗗2, | A𝗗3 | … | A𝗗n−1 | ; | |
| A𝗟0, | A𝗟0 × A𝗥0, | A𝗟1 × A𝗥1, | A𝗟2 × A𝗥2, | … | A𝗟n−2 × A𝗥n−2 | ; | A𝗚 ⟩ |
If A is a complex n-licht, then A𝗩 is defined as the value of the product of the lengths, not the absolute value thereof. Also complex would be A𝗠, which is the nth root of A𝗩, the root selected according to the same gon used for the A𝗦. (Some researchers might choose to investigate what happens if a different gon is selected.)
§C1C2. Needed to construct zero licht Z is merely Z𝗗 containing any nonzero complex numbers. All lengths will be zero, and Z will be a panagon. Note that two zero lichts of the same breadth might have different divisors.
§C1D Gon search. If all divisors and lengths of a licht A have been found by whatever means, a linear search will reveal which gon(s) apply to it. Here is how:
Let i successively be each nonnegative integer less than n.
For each i, calculate a new licht named Bi according to the method of §C1C where:
If the lengths of (Bi)𝗟 equal the respective lengths of A𝗟, then i is a gon of A. For a zero licht, all possible gon values will work.
§C1E All divisors equal. In the case of a real licht, if all divisors are equal, all lengths will be equal. With complex lichts, this might not be true. For example, define these complex numbers in polar form:
| v = (2.692582, 2.594804) |
| a = (1.749286, 5.742766) |
| b = (1.749286, 3.229492) |
| c = (1.749286, 0.716218) |
| d = (1.749286, 4.486129) |
| e = (1.749286, 1.972855) |
Then:
| A0 = ⟨ | v, | v, | v, | v, | v; | a, | b, | c, | d, | e; | 0 ⟩ |
| A1 = ⟨ | v, | v, | v, | v, | v; | a, | d, | b, | e, | c; | 1 ⟩ |
| A2 = ⟨ | v, | v, | v, | v, | v; | a, | a, | a, | a, | a; | 2 ⟩ — flat |
| A3 = ⟨ | v, | v, | v, | v, | v; | a, | c, | e, | b, | d; | 3 ⟩ |
| A4 = ⟨ | v, | v, | v, | v, | v; | a, | e, | d, | c, | b; | 4 ⟩ |
§C2 Manipulators. Decrement, increment, and cycle work the same as with real numbers. There is no change in the gon(s).
§C2A Reverse of a zero licht is simple, consisting merely of reversing the sequence of divisors.
On the other hand, reverse of a nonzero licht is far more complicated because of the multiple-root situation. Specifically, rev (rev (A)) ≡ rev2 (A) will not equal A if the respective gons for the two reverse operations are incorrectly chosen.
Fortunately, there is a direct way to find the right gons to achieve rev2 (A) = A, which of course is how any proper reverse operation ought to behave. Unfortunately, the only method known to the present author is a brute-force, but not lengthy, search.
It helps to define an auxiliary operation called rev_cand, which takes an n-licht and a gon to produce a candidate for rev (A). Here are the steps to calculate it:
| How to find rev_cand (A, g) |
|---|
|
1. Create temporary licht T, setting its divisors to those of A, except in reverse sequence. T's lengths will come later. 2. Calculate the step of T as the nth root of the product of T's divisors, specifically that root corresponding to gon g. 3. From this root and the divisors, find the ratios normally. 4. Temporarily assign the complex value [1, 0] to T𝗟0, and figure the remaining lengths normally. 5. Find the product of T's lengths. Take the nth root of this product, specifically that root corresponding to gon g. This result is T's magnitude, T𝗠, which will be a complex number. 6. Multiply all of T's lengths by the quotient A𝗠 ÷ T𝗠, giving the final result for rev_cand (A, g). Essential is that the same gon be used for both of T's root extraction operations (steps 2 and 5). This might differ from A's gon. |
The rev_cand operation is now established. A less bulky notation for rev_cand (rev_cand (A, i), j) is rev_cand2 (A, i, j).
| How to find rev (A) |
|---|
| Let i and j be positive integers less than n.
For every combination of i and j, calculate rev_cand2 (A, i, j). If this value equals A, then rev_cand (A, i) is a value of rev (A). |
Comments:
§C3 Arithmetic can be performed on complex lichts, as follows.
§C3A Unary operations are simple. Unary plus is an identity operation, and unary minus negates the lengths:
| −A = ⟨ | A𝗗0, | A𝗗1, | A𝗗2 | … | A𝗗n−1 | ; | |
| −A𝗟0, | −A𝗟1, | −A𝗟2 | … | −A𝗟n−1 | ; | A𝗚 ⟩ |
In this report, the complex conjugate of a complex value such as A𝗗0 is symbolized with a tilde, for example ~A𝗗0, rather than with an overbar, because the presence of subscripts complicates the typography. Note that ~A𝗗0 should be interpreted as ~(A𝗗0) rather than (~A)𝗗0. Otherwise, the definition below becomes an infinite recursion.
With that, the complex conjugate of a licht is formed by conjugating all the divisors and lengths, and adjusting the gon:
| ~A = ⟨ | ~A𝗗0, | ~A𝗗1, | ~A𝗗2 | … | ~A𝗗n−1 | ; | |
| ~A𝗟0, | ~A𝗟1, | ~A𝗟2 | … | ~A𝗟n−1 | ; | A𝗕 − 1 − A𝗚⟩ |
§C3B Addition of two n-lichts requires caution become some lichts are monogons and some are not. Always required is that A𝗗 = B𝗗. Cases:
| if … | then … |
|---|---|
| A = 0 and B = 0 | A + B = 0, using A𝗗 |
| A = 0 and B ≠ 0 | A + B = B |
| A ≠ 0 and B = 0 | A + B = A |
| A ≠ 0 and B ≠ 0 and A𝗚 = B𝗚 | respective lengths are added, as below |
| A ≠ 0 and B ≠ 0 and A𝗚 ≠ B𝗚 | addition is not defined |
The formula to add two nonzero lichts is like the formula for real numbers, except to specify the gon:
| A + B = ⟨ | A𝗗0, | A𝗗1, | A𝗗2 | … | A𝗗n−1 | ; | |
| A𝗟0 + B𝗟0, | A𝗟1 + B𝗟1, | A𝗟2 + B𝗟2 | … | A𝗟n−1 + B𝗟n−1 | ; | A𝗚 ⟩ |
More concisely, A + B = ⟨ A𝗗; A𝗟 + B𝗟; A𝗚 ⟩.
Identities:
On the other hand, rev (A) + rev (B) might not equal rev (A + B). That is because even when A𝗚 = B𝗚, (rev (A))𝗚 might not equal (rev (B))𝗚. In that case, rev (A) + rev (B) is not even defined.
In the complex case, A𝗠 + B𝗠 bears no obvious relationship to (A + B)𝗠. This differs from the real case.
Subtraction is defined in the corresponding manner.
§C3C Multiplication of n-lichts A and B is possible even if A𝗗 ≠ B𝗗 or A𝗚 ≠ B𝗚. All respective components are multiplied:
| C = A × B = ⟨ | A𝗗0 × B𝗗0, | A𝗗1 × B𝗗1, | A𝗗2 × B𝗗2 | … | A𝗗n−1 × B𝗗n−1 | ; | |
| A𝗟0 × B𝗟0, | A𝗟1 × B𝗟1, | A𝗟2 × B𝗟2 | … | A𝗟n−1 × B𝗟n−1 | ; | C𝗚 ⟩ |
Concisely: C = A × B = ⟨ A𝗗 × B𝗗; A𝗟 × B𝗟; C𝗚 ⟩. Similarly, C𝗥 = A𝗥 × B𝗥.
Determining the value of C𝗚 requires the gon search of §C1D, as there is no obvious general relationship among the input gons and the output gon. However, patterns will sometimes develop.
In the multiplicative identity, all the divisors and lengths are one, and the gon is zero.
Division is defined in the corresponding manner.
§C3D Powers and roots. To raise a complex licht to an integer power p, raise every component to that power:
| A p = ⟨ | (A𝗗0) p, | (A𝗗1) p, | (A𝗗2) p | … | (A𝗗n−1) p | ; | |
| (A𝗟0) p, | (A𝗟1) p, | (A𝗟2) p | … | (A𝗟n−1) p | ; | C𝗚 ⟩ |
As with multiplication, determining the value of C𝗚 requires the gon search of §C1D.
The power p is limited to integers in this definition, because otherwise this would be a multiple-valued function. The researcher who needs a non-integral power can extract the divisors and lengths individually and take whatever steps might be desired.